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we can use a device designed to intentionally break that continuity (called a switch). This switch can be mounted at any convenient location that we can run wires to. It controls the ow of electrons in the hole circuit: 253 switch Battery - + It doesn’t matter how twisted or convoluted a route the wires take conducting current, so long as they form a complete, uninterrupted loop (circuit). This is how a switch mounted on the wall of a house can control a lamp that is mounted down a long hallway, or even in another room, far away from the switch. The switch itself is constructed of a pair of conductive contacts (usually made of some kind of metal) forced together by a mechanical lever actuator or pushbutton. When the contacts touch each other, electrons are able to ow from one to the other and the circuit’s continuity is established; when the contacts are separated, electron ow from one to the other is prevented by the insulation of the air between, and the circuit’s continuity is broken. In keeping with the "open" and "closed" terminology of circuits, a switch that is making contact from one connection terminal to the other provides continuity for electrons to ow through, and is called a closed switch. Conversely, a switch that is breaking continuity won’t allow electrons to pass through and is called an open switch. This terminology is often confusing to the new student of electronics, because the words "open" and "closed" are commonly understood in the context of a door, where "open" is equated with free passage and "closed" with blockage. With electrical switches, these terms have opposite meaning: "open" means no ow while "closed" means free passage of electrons. 14.5 Voltage and current in a practical circuit Because it takes energy to force electrons to ow against the opposition of a resistance, there will be voltage manifested (or "dropped") between any points in a circuit with resistance between them. It is important to note that although the amount of current is uniform in a simple circuit, the amount of voltage between diﬁerent sets of points in a single circuit may vary considerably: 254 same rate of current . . . Battery 1 - + 4 2 3 . . . at all points in this circuit (NOTE TO SELF: How do you actually measure currents and voltages? In the next two paragraphs there’s a lot of ado what we ’see’ at diﬁerent points of a circuit, but I don’t see anything!) Take this circuit as an example. We labelled four points with the numbers 1, 2, 3, and 4. The amount of current conducted through the wire between points 1 and 2 is exactly the same as the amount of current conducted through the lamp (between points 2 and 3). This same quantity of current passes through the wire between points 3 and 4, and through the battery (between points 1 and 4). However, we will ﬂnd the voltage appearing between any two of these points to be directly proportional to the resistance within the conductive path between those two points. In a normal lamp circuit, the resistance of a lamp will be much greater than the resistance of the connecting wires. So we should expect to see a substantial amount of voltage drop between points 2 and 3, and only a very small one between points 1 and 2, or between 3 and 4. The voltage drop between points 1 and 4, of course, will be the full voltage oﬁered by the battery. This will be only slightly higher than the voltage drop across the lamp (between points 2 and 3). This, again, is analogous to the water reservoir system: 255 2 Reservoir 1 Waterwheel (energy released) 3 (energy stored) Pump 4 Pond Between points 2 and 3, where the falling water is releasing energy at the water-wheel, there is a diﬁerence of pressure between the two points. This reects the opposition to the ow of water through the water-wheel. From point 1 to point 2, or from point 3 to point 4, where water is owing freely through reservoirs with little opposition, there is little or no diﬁerence of pressure (no potential energy). However, the rate of water ow in this continuous system is the same everywhere (assuming the water levels in both pond and reservoir are unchanging): through the pump, through the water-wheel, and through all the pipes. So it is with simple electric circuits: the rate of electron ow is the same at every point in the circuit, although voltages may diﬁer between diﬁerent sets of points. 14.6 Direction of current ow in a circuit We know now that the moving charges in an electrical ciruit are the negatively chargend electrons. These electrons naturally ow from the negative pole of a battery to the positive pole. This form of symbology became known as electron ow notation: Electron flow notation + - Electric charge moves from the negative (surplus) side of the battery to the positive (deficiency) side. 256 However, for historical reasons the current ow in a circuit is conventionally denoted in the opposite direction. That is, it ows from the positive pole to the negative one. This became known as conventional ow notation: Conventional flow notation + - Electric charge moves from the positive (surplus) side of the battery to the negative (deficiency) side. In conventional ow notation, we show the motion of charge according to the (technically . This way the labels make sense, but the direction of charge ow is incorrect) labels of + and incorrect. ¡ Does it matter, really, how we designate charge ow in a circuit? Not really, so long as we’re consistent in the use of our symbols. You may follow an imagined direction of current (conventional ow) or the actual (electron ow) with equal success insofar as circuit analysis is concerned. Concepts of voltage, current, resistance, continuity, and even mathematical treatments such as Ohm’s Law (section 2 (NOTE TO SELF: make this reference dynamic)) and Kirchhoﬁ’s Laws (section 6 (NOTE TO SELF: make this reference dynamic)) remain just as valid with either style of notation. Aside: Benjamin Franklin made a conjecture regarding the direction of charge ow when rubbing smooth wax with rough wool. By assuming that the observed charges ow from the wax to the wool, he set the precedent for electrical notation that exists to this day. Because Franklin assumed electric charge moved in the opposite direction that it actually does, electrons are said to have a negative charge, and so objects he called "negative" (representing a deﬂciency of charge) actually have a surplus of electrons. By the time the true direction of electron ow was discovered, the nomenclature of "positive" and "negative" had already been so well established in the scientiﬂc community that no eﬁort was made to change it. It would have made more sense to call electrons "positive" in referring to "excess" charge. You see, the terms "positive" and "negative" are human inventions, and as such have no absolute meaning beyond our own conventions of language and scientiﬂc description. Franklin could have just as easily referred to a surplus of charge as "black" and a deﬂciency as "white", in which case scientists would speak of electrons having a "white" charge. However, because we tend to associate the word "positive" with "surplus" and "negative" with "deﬂciency," the standard label for electron charge does seem backward. As discussed above, many engineers decided to retain the old concept of electricity with "positive" referring to a surplus of charge, and label charge ow (current) accordingly. 257 14.7 How voltage, current, and resistance relate First lets recap some of the ideas we have learnt so far. We will need these to understand how voltage, current and resistance relate. An electric circuit is formed when a conductive path is created to allow free electrons to continuously move. This continuous movement of free electrons through the conductors of a circuit is called a current. It is often referred to in terms of "ow," just like the ow of a liquid through a hollow pipe. The force motivating electrons to "ow" in a circuit is called voltage. Voltage is a speciﬂc measure of potential energy that is always relative between two points. When we speak of a certain amount of voltage being present in a circuit, we are referring to the measurement of how much potential energy exists to move electrons from one particular point in that circuit to another particular point. Without reference to two particular points, the term "voltage" has no meaning. Free electrons tend to move through conductors with some degree of friction, or opposition to motion. This opposition to motion is more properly called resistance. The amount of current in a circuit depends on the amount of voltage available to motivate the electrons (NOTE TO SELF: Motivated electrons?), and also the amount of resistance in the circuit to oppose electron ow. Just like voltage, resistance is a quantity relative between two points. For this reason, the quantities of voltage and resistance are often stated as being "between" or "across" two points in a circuit. To be able to make meaningful statements about these quantities in circuits, we need to be able to describe their quantities in the same way that we might quantify mass, temperature, volume, length, or any other kind of physical quantity. For mass we might use the units of "pound" or "gram". For temperature we might use degrees Fahrenheit or degrees Celsius. Here are the standard units of measurement for electrical current, voltage, and resistance: Quantity Current Voltage Resistance Charge Symbol I V (or E) R Q Unit of Measurement Abbreviation of Unit Ampere volt Ohm coulomb A V › C The "symbol" given for each quantity is the standard alphabetical letter used to represent that quantity in an algebraic equation. Standardized letters like these are common in the disciplines of physics and engineering, and are internationally recognized. The "unit abbreviation" for each quantity represents the alphabetical symbol used as a shorthand notation for its particular unit of measurement. And, yes, that strange-lo
oking "horseshoe" symbol is the capital Greek letter › (called omega), just a character in a foreign alphabet (apologies to any Greek readers here). Aside: Each unit of measurement is named after a famous experimenter in electricity: The amp after the Frenchman Andre M. Ampere, the volt after the Italian Alessandro Volta, and the ohm after the German Georg Simon Ohm. The mathematical symbol for each quantity is meaningful as well. The "R" for resistance and the "V " for voltage are both self-explanatory. The "I" is thought to have been meant to represent "Intensity" (of electron ow). The other symbol for voltage, "E", stands for "Electromotive force". The symbols "E" and "V " are interchangeable for the most part, although some texts reserve "E" to represent voltage across a source (such as a battery or generator) and "V " to represent voltage across anything else. 258 One foundational unit of electrical measurement is the unit of the coulomb. It is a measure of electric charge proportional to the number of electrons in an imbalanced state. One coulomb of charge is rougly equal to the charge of 6,250,000,000,000,000,000 electrons. The symbol for electric charge quantity is the capital letter "Q", with the unit of coulombs abbreviated by the capital letter "C". It so happens that the unit for electron ow, the ampere, is equal to 1 coulomb of electrons passing by a given point in a circuit in 1 second of time. Cast in these terms, current is the rate of electric charge motion through a conductor. As stated before, voltage is the measure of potential energy per unit charge available to motivate electrons from one point to another. Before we can precisely deﬂne what a "volt" is, we must understand how to measure this quantity we call "potential energy". The general metric unit for energy of any kind is the joule, equal to the amount of work performed by a force of 1 newton exerted through a motion of 1 meter (in the same direction). (NOTE TO SELF: Make a reference to the Mechanics chapter.) Deﬂned in these scientiﬂc terms, 1 volt is equal to 1 joule of electric potential energy per (divided by) 1 coulomb of charge. Thus, a 9 volt battery releases 9 joules of energy for every coulomb of electrons moved through a circuit. These units and symbols for electrical quantities will become very important to know as we begin to explore the relationships between them in circuits. The ﬂrst, and perhaps most important, relationship between current, voltage, and resistance is called Ohm’s Law. It states that the amount of electric current through a metal conductor in a circuit is directly proportional to the voltage impressed across it, for any given temperature. It can be expressed in the form of a simple equation, describing how voltage, current, and resistance interrelate: ¢ In this algebraic expression, voltage (V ) is equal to current (I) multiplied by resistance (R). V = I R : (14.1) Aside: Georg Simon Ohm published his law in his 1827 paper, The Galvanic Circuit Investigated Mathematically. Using algebra techniques, we can manipulate this equation into two variations, solving for I and for R, respectively: Let’s see how these equations might work to help us analyze simple circuits14.2) electron flow Battery + - Electric lamp (glowing) electron flow 259 (NOTE TO SELF: replace E by V) In the above circuit, there is only one source of voltage (the battery, on the left) and only one source of resistance to current (the lamp, on the right). This makes it very easy to apply Ohm’s Law. If we know the values of any two of the three quantities (voltage, current, and resistance) in this circuit, we can use Ohm’s Law to determine the third. In this ﬂrst example, we will calculate the amount of current (I) in a circuit, given values of voltage (V ) and resistance (R): Worked Example 81 Question: What is the amount of current (I) in this circuit? Battery E = 12 V + - I = ??? I = ??? Lamp R = 3 W (NOTE TO SELF: replace E by V) Answer: I = V R = 12 V 3 › = 4 A : (14.3) In the second example, we will calculate the amount of resistance (R) in a circuit, given values of voltage (V ) and current (I): Worked Example 82 Question: What is the amount of resistance (R) oﬁered by the lamp? 260 Battery E = 36 Lamp R = ??? (NOTE TO SELF: replace E by V) Answer: R = V I = 36 V 4 A = 9 › : (14.4) In the last example, we will calculate the amount of voltage supplied by a battery, given values of current (I) and resistance (R): Worked Example 83 Question: What is the amount of voltage provided by the battery? Battery E = ??? + - I = 2 A I = 2 A Lamp R = 7 W (NOTE TO SELF: replace E by V) Answer: V = I ¢ R = (2 A) ¢ (7 ›) = 14 V : (14.5) 261 Ohm’s Law is a very simple and useful tool for analyzing electric circuits. It is used so often in the study of electricity and electronics that it needs to be committed to memory by the serious student. All you need to do is commit V = I R to memory and derive the other two formulae from that when you need them! ¢ 14.8 Voltmeters, ammeters, and ohmmeters As we have seen in previous sections, an electric circuit is made up of a number of diﬁerent components such as batteries and resistors. In electronics, there are many types of meters used to measure the properties of the individual components of an electric circuit. For example, one may be interested in measuring the amount of current owing through a circuit, or measure the voltage provided by a battery. In this section we will discuss the practical usage of voltmeters, ammeters, and ohmmeters. A voltmeter is an instrument for measuring the voltage between two points in an electric circuit. In analogy with a water circuit, a voltmeter is like a meter designed to measure pressure diﬁerence. Since one is interested in measuring the voltage between two points in a circuit, a voltmeter must be connected in parallel with the portion of the circuit on which the measurement is made: V The above illustration shows a voltmeter connected in parallel with a battery. One lead of the voltmeter is connected to one end of the battery and the other lead is connected to the opposite end. The voltmeter may also be used to measure the voltage across a resistor or any other component of a circuit that has a voltage drop. An ammeter is an instrument used to measure the ow of electric current in a circuit. Since one is interested in measuring the current owing through a circuit component, the ammeter must be connected in series with the measured circuit component: A An ohmmeter is an instrument for measuring electrical resistance. The basic ohmmeter can function much like an ammeter. The ohmmeter works by suppling a constant voltage to the resistor and measuring the current owing through it. The measured current is then converted into a corresponding resistance reading through Ohm’s law. One cautionary detail needs to be 262 mentioned with regard to ohmmeters: they only function correctly when measuring resistance that is not being powered by a voltage or current source. In other words, you cannot measure the resistance of a component that is already connected to a circuit. The reason for this is simple: the ohmmeter’s accurate indication depends only on its own source of voltage. The presence of any other voltage across the measured circuit component interferes with the ohmmeter’s operation. The circuit diagram below shows an ohmmeter solely connected with a resistor: › The table below summarizes the use of each measuring instrument that we discussed and the way it should be connected to a circuit component. Instrument Measured Quantity Proper Connection Voltmeter Ammeter Ohmmeter In Parallel In Series Only with Resistor Voltage Current Resistance 14.9 An analogy for Ohm’s Law In our water-and-pipe analogy, Ohm’s Law also exists. Think of a water pump that exerts pressure (voltage) to push water around a "circuit" (current) through a restriction (resistance). If the resistance to water ow stays the same and the pump pressure increases, the ow rate must also increase. " V = " I R If the pressure stays the same and the resistance increases (making it more di–cult for the water to ow), then the ow rate must decrease: V = # I " R If the ow rate stays the same while the resistance to ow decreases, the required pressure from the pump decreases: # V = I # R As odd as it may seem, the actual mathematical relationship between pressure, ow, and resistance is actually more complex for uids like water than it is for electrons. If you pursue further studies in physics, you will discover this for yourself. Thankfully for the electronics student, the mathematics of Ohm’s Law is very simple. 263 14.10 Power in electric circuits In addition to voltage and current, there is another measure of free electron activity in a circuit: power. The concept of power was introduced in Chapter 8. Basically, it is a measure of how rapidly a standard amount of work is done. In electric circuits, power is a function of both voltage and current: P = IV: So power (P ) is exactly equal to current (I) multiplied by voltage (V ) and there is no extra constant of proportionality. The unit of measurement for power is the Watt (abbreviated W). Aside: You can verify for yourself that the eqution for power in an electric cicuit makes sense. Remember that voltage is the speciﬂc work (or potential energy) per unit charge, while current is the amount electric charge that ow though a conductor per time unit. So the product of those two qunatities is the oumount of work per time unit, which is exactly the power. It is important to realise that only the combination of a voltage drop and the ow of current corresponds to power. So, a circuit with high voltage and low current may be dissipating the same amount of power as a circuit with low voltage and high current. In an open circuit, where voltage is present between the terminals of the source and there is zero current, there is zero power dissipated, no matter how great that volta
ge may be. Since P = IV and I = 0, the power dissipated in any open circuit must be zero. 14.11 Calculating electric power We’ve seen the formula for determining the power in an electric circuit: by multiplying the voltage in volts by the current in Amperes we arrive at an answer in watts." Let’s apply this to a circuit example: Battery E = 18 V + - I = ??? I = ??? Lamp R = 3 W In the above circuit, we know we have a battery voltage of 18 Volts and a lamp resistance of 3 ›. Using Ohm’s Law to determine current, we get: I = V R = 18V 3› = 6A: 264 Now that we know the current, we can take that value and multiply it by the voltage to determine power: P = IV = (6A)(18V = 108W: Answer: the lamp is dissipating (releasing) 108 W of power, most likely in the form of both light and heat. Let’s try taking that same circuit and increasing the battery voltage to see what happens: Battery E = 36 V + - I = ??? I = ??? Lamp R = 3 W Since the resistance stays the same, the current will increase when we increase the voltage: 36V 3› Note that Ohm’s Law is linear, so the current exactly doubles when we double the voltage. = 12A: V R I = = Now, let’s calculate the power: P = IV = (12A)(36V) = 432W: Notice that the power has increased just as we might have suspected, but it increased quite a bit more than the current. Why is this? Because power is a function of voltage multiplied by current, and both voltage and current doubled from their previous values, the power will increase by a factor of 2 x 2, or 4: the ratio of the new power 432 W and the old power 108 W, is exactly 4. We could in fact have arrived at this result without the intermediate step of calculating the current. From we can expres power directly as a function of voltage: I = V R and P = IV The analogous relation between power and current is P = IIR = I 2R: 265 Interesting Fact: It was James Prescott Joule, not Georg Simon Ohm, who ﬂrst discovered the mathematical relationship between power dissipation and current through a resistance. This discovery, published in 1841, followed the form of the last equation (P = I2R), and is properly known as Joule’s Law. However, these power equations are so commonly associated with the Ohm’s Law equations relating voltage, current, and resistance (V = IR ; I = V =R ; and R = V =I) that they are frequently credited to Ohm. 14.12 Resistors Because the relationship between voltage, current, and resistance in any circuit is so regular, we can reliably control any variable in a circuit simply by controlling the other two. Perhaps the easiest variable in any circuit to control is its resistance. This can be done by changing the material, size, and shape of its conductive components (remember how the thin metal ﬂlament of a lamp created more electrical resistance than a thick wire?). Special components called resistors are made for the express purpose of creating a precise quantity of resistance for insertion into a circuit. They are typically constructed of metal wire or carbon, and engineered to maintain a stable resistance value over a wide range of environmental conditions. Unlike lamps, they do not produce light, but they do produce heat as electric power is dissipated by them in a working circuit. Typically, though, the purpose of a resistor is not to produce usable heat, but simply to provide a precise quantity of electrical resistance. The most common schematic symbol for a resistor is a zig-zag line: Resistor values in ohms are usually shown as an adjacent number, and if several resistors are present in a circuit, they will be labeled with a unique identiﬂer number such as R1, R2, R3, etc. As you can see, resistor symbols can be shown either horizontally or vertically: R1 150 This is resistor "R1" with a resistance value of 150 ohms. R2 25 This is resistor "R2" with a resistance value of 25 ohms. In keeping more with their physical appearance, an alternative schematic symbol for a resistor looks like a small, rectangular box: Resistors can also be shown to have varying rather than ﬂxed resistances. This might be for the purpose of describing an actual physical device designed for the purpose of providing an adjustable resistance, or it could be to show some component that just happens to have an unstable resistance: 266 variable resistance . . . or . . . In fact, any time you see a component symbol drawn with a diagonal arrow through it, that component has a variable rather than a ﬂxed value. This symbol "modiﬂer" (the diagonal arrow) is standard electronic symbol convention. In practice, resistors are not only rated in terms of their resistance in Aside: ohms, but also in term the amount of power they can dissipate in watts. Resistors dissipate heat as the electric currents through them overcome the "friction" of their resistance and can in fact become quite hot in actual applications. Most resistors found in small electronic devices such as portable radios are rated at 1/4 (0.25) watt or less. The power rating of any resistor is roughly proportional to its physical size. Also note how resistances (in ohms) have nothing to do with size! 14.13 Nonlinear conduction Ohm’s Law is a powerful tool for analyzing electric circuits, but it has a practical limitation. In the application of Ohm’s Law, we alwasy assume that the restistance does not change as a function of voltage and current. For most conductors, this is a reasonable approximation as long ads the temperature does not change too much. In a normal lightbulb, the resistance of the ﬂlament wire will increase dramatically as it warms from room temperature to operating temperature. If we increase the supply voltage in a real lamp circuit, the resulting increase in current causes the ﬂlament to increase in temperature, which increases its resistance. This eﬁectively limits the increase in current. Consequently, voltage and current do not follow the simple equation I = V =R, with a constant R (of 3 › ion our example). The lamp’s ﬂlament resistance does not remain stable for diﬁerent currents. The phenomenon of resistance changing with variations in temperature is one shared by almost all metals, of which most wires are made. For most applications, these changes in resistance are small enough to be ignored. In the application of metal lamp ﬂlaments, which increase a lot in temperature (up to about 1000oC, and starting from room temperature) the change is quite large. A more realistic analysis of a lamp circuit over several diﬁerent values of battery voltage would generate a plot of this shape: 267 I (current) E (voltage) The plot is no longer a straight line. It rises sharply on the left, as voltage increases from zero to a low level. As it progresses to the right we see the line attening out, the circuit requiring greater and greater increases in voltage to achieve equal increases in current. If we apply Ohm’s Law to ﬂnd the resistance of this lamp circuit with the voltage and current values plotted above, the calculated values will change with voltage or curreny. We could say that the resistance here is nonlinear, increasing with increasing current and voltage. The nonlinearity is caused by the eﬁects of high temperature on the metal wire of the lamp ﬂlament. 14.14 Circuit wiring So far, we’ve been analyzing single-battery, single-resistor circuits with no regard for the connecting wires between the components, so long as a complete circuit is formed. Does the wire length or circuit "shape" matter to our calculations? Let’s look at a couple of circuit conﬂgurations and ﬂnd out: Battery 10 V 1 4 2 Resistor 5 W 3 Battery 10 V 1 4 2 Resistor 5 W 3 268 When we draw wires connecting points in a circuit, we usually assume those wires have negligible resistance. As such, they contribute no appreciable eﬁect to the overall resistance of the circuit, and so the only resistance we have to contend with is the resistance in the components. In the above circuits, the only resistance comes from the 5 › resistors, so that is all we will consider in our calculations. In real life, metal wires actually do have resistance (and so do power sources!), but those resistances are generally so much smaller than the resistance present in the other circuit components that they can be safely ignored. If connecting wire resistance is very little or none, we can regard the connected points in a circuit as being electrically common. That is, points 1 and 2 in the above circuits may be physically joined close together or far apart, and it doesn’t matter for any voltage or resistance measurements relative to those points. The same goes for points 3 and 4. It is as if the ends of the resistor were attached directly across the terminals of the battery, so far as our Ohm’s Law calculations and voltage measurements are concerned. This is useful to know, because it means you can re-draw a circuit diagram or re-wire a circuit, shortening or lengthening the wires as desired without appreciably impacting the circuit’s function. All that matters is that the components attach to each other in the same sequence. It also means that voltage measurements between sets of "electrically common" points will be the same. That is, the voltage between points 1 and 4 (directly across the battery) will be the same as the voltage between points 2 and 3 (directly across the resistor). Take a close look at the following circuit, and try to determine which points are common to each other: 1 Battery 10 V 4 2 3 Resistor 5 W 6 5 Here, we only have 2 components excluding the wires: the battery and the resistor. Though the connecting wires take a convoluted path in forming a complete circuit, there are several electrically common points in the electrons’ path. Points 1, 2, and 3 are all common to each other, because they’re directly connected together by wire. The same goes for points 4, 5, and 6. The voltage between points 1 and 6 is 10 volts, coming straight from the battery. However, since points 5 and 4 are common to 6, and points 2 and
3 common to 1, that same 10 volts also exists between these other pairs of points: Between points 1 and 4 = 10 volts Between points 2 and 4 = 10 volts Between points 3 and 4 = 10 volts (directly across the resistor) Between points 1 and 5 = 10 volts Between points 2 and 5 = 10 volts Between points 3 and 5 = 10 volts Between points 1 and 6 = 10 volts (directly across the battery) Between points 2 and 6 = 10 volts Between points 3 and 6 = 10 volts Since electrically common points are connected together by (zero resistance) wire, there is no signiﬂcant voltage drop between them regardless of the amount of current conducted from one 269 to the next through that connecting wire. Thus, if we were to read voltages between common points, we should show (practically) zero: Between points 1 and 2 = 0 volts Between points 2 and 3 = 0 volts Between points 1 and 3 = 0 volts Between points 4 and 5 = 0 volts Between points 5 and 6 = 0 volts Between points 4 and 6 = 0 volts Points 1, 2, and 3 are electrically common Points 4, 5, and 6 are electrically common This makes sense mathematically, too. With a 10 volt battery and a 5 › resistor, the circuit current will be 2 amps. With wire resistance being zero, the voltage drop across any continuous stretch of wire can be determined through Ohm’s Law as such: E = I R E = (2 A)(0 W) E = 0 V It should be obvious that the calculated voltage drop across any uninterrupted length of wire in a circuit where wire is assumed to have zero resistance will always be zero, no matter what the magnitude of current, since zero multiplied by anything equals zero. Because common points in a circuit will exhibit the same relative voltage and resistance measurements, wires connecting common points are often labeled with the same designation. This is not to say that the terminal connection points are labeled the same, just the connecting wires. Take this circuit as an example: 1 wire #2 2 wire #2 Battery 10 V 4 wire #1 3 Resistor 5 W 6 5 wire #1 wire #1 Points 1, 2, and 3 are all common to each other, so the wire connecting point 1 to 2 is labeled the same (wire 2) as the wire connecting point 2 to 3 (wire 2). In a real circuit, the wire stretching from point 1 to 2 may not even be the same color or size as the wire connecting point 2 to 3, but they should bear the exact same label. The same goes for the wires connecting points 6, 5, and 4. Knowing that electrically common points have zero voltage drop between them is a valuable troubleshooting principle. If I measure for voltage between points in a circuit that are supposed to be common to each other, I should read zero. If, however, I read substantial voltage between those two points, then I know with certainty that they cannot be directly connected together. 270 If those points are supposed to be electrically common but they register otherwise, then I know that there is an "open failure" between those points. One ﬂnal note: for most practical purposes, wire conductors can be assumed to possess zero resistance from end to end. In reality, however, there will always be some small amount of resistance encountered along the length of a wire, unless it’s a superconducting wire. Knowing this, we need to bear in mind that the principles learned here about electrically common points are all valid to a large degree, but not to an absolute degree. That is, the rule that electrically common points are guaranteed to have zero voltage between them is more accurately stated as such: electrically common points will have very little voltage dropped between them. That small, virtually unavoidable trace of resistance found in any piece of connecting wire is bound to create a small voltage across the length of it as current is conducted through. So long as you understand that these rules are based upon ideal conditions, you won’t be perplexed when you come across some condition appearing to be an exception to the rule. 14.15 Polarity of voltage drops We can trace the direction that electrons will ow in the same circuit by starting at the negative (-) terminal and following through to the positive (+) terminal of the battery, the only source of voltage in the circuit. From this we can see that the electrons are moving counter-clockwise, from point 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. As the current encounters the 5 › resistance, voltage is dropped across the resistor’s ends. The signs of this voltage drop is negative (-) at point 4 with respect to positive (+) at point 3. We can mark the polarity of the resistor’s voltage drop with these negative and positive symbols, in accordance with the direction of current (whichever end of the resistor the current is entering is negative with respect to the end of the resistor it is exiting: Battery 10 V 1 + - 6 2 3 current current - + Resistor 5 W 4 5 We could make our table of voltages a little more complete by marking the polarity of the voltage for each pair of points in this circuit: Between points 1 (+) and 4 (-) = 10 volts Between points 2 (+) and 4 (-) = 10 volts Between points 3 (+) and 4 (-) = 10 volts Between points 1 (+) and 5 (-) = 10 volts Between points 2 (+) and 5 (-) = 10 volts Between points 3 (+) and 5 (-) = 10 volts Between points 1 (+) and 6 (-) = 10 volts Between points 2 (+) and 6 (-) = 10 volts Between points 3 (+) and 6 (-) = 10 volts 271 While it might seem a little silly to document polarity of voltage drop in this circuit, it is an important concept to master. It will be critically important in the analysis of more complex circuits involving multiple resistors and/or batteries. It should be understood that polarity has nothing to do with Ohm’s Law: there will never be negative voltages, currents, or resistance entered into any Ohm’s Law equations! There are other mathematical principles of electricity that do take polarity into account through the use of signs (+ or -), but not Ohm’s Law. 14.16 What are "series" and "parallel" circuits? Circuits consisting of just one battery and one load resistance are very simple to analyze, but they are not often found in practical applications. Usually, we ﬂnd circuits where more than two components are connected together. There are two basic ways in which to connect more than two circuit components: series and parallel. First, an example of a series circuit: Series R1 R3 1 + - 4 2 R2 3 Here, we have three resistors (labeled R1, R2, and R3), connected in a long chain from one terminal of the battery to the other. (It should be noted that the subscript labeling { those little numbers to the lower-right of the letter "R" { are unrelated to the resistor values in ohms. They serve only to identify one resistor from another.) The deﬂning characteristic of a series circuit is that there is only one path for electrons to ow. In this circuit the electrons ow in a counter-clockwise direction, from point 4 to point 3 to point 2 to point 1 and back around to 4. Now, let’s look at the other type of circuit, a parallel conﬂguration: 1 + - 8 Parallel 2 3 4 R1 R2 R3 7 6 5 Again, we have three resistors, but this time they form more than one continuous path for electrons to ow. There’s one path from 8 to 7 to 2 to 1 and back to 8 again. There’s another from 8 to 7 to 6 to 3 to 2 to 1 and back to 8 again. And then there’s a third path from 8 to 7 to 272 6 to 5 to 4 to 3 to 2 to 1 and back to 8 again. Each individual path (through R1, R2, and R3) is called a branch. The deﬂning characteristic of a parallel circuit is that all components are connected between the same set of electrically common points. Looking at the schematic diagram, we see that points 1, 2, 3, and 4 are all electrically common. So are points 8, 7, 6, and 5. Note that all resistors as well as the battery are connected between these two sets of points. And, of course, the complexity doesn’t stop at simple series and parallel either! We can have circuits that are a combination of series and parallel, too: Series-parallel R1 1 + - 6 2 5 R2 3 4 R3 In this circuit, we have two loops for electrons to ow through: one from 6 to 5 to 2 to 1 and back to 6 again, and another from 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. Notice how both current paths go through R1 (from point 2 to point 1). In this conﬂguration, we’d say that R2 and R3 are in parallel with each other, while R1 is in series with the parallel combination of R2 and R3. This is just a preview of things to come. Don’t worry! We’ll explore all these circuit conﬂg- urations in detail, one at a time! The basic idea of a "series" connection is that components are connected end-to-end in a line to form a single path for electrons to ow: Series connection R1 R2 R3 R4 only one path for electrons to flow! The basic idea of a "parallel" connection, on the other hand, is that all components are connected across each other’s leads. In a purely parallel circuit, there are never more than two sets of electrically common points, no matter how many components are connected. There are many paths for electrons to ow, but only one voltage across all components: 273 Parallel connection These points are electrically common R1 R2 R3 R4 These points are electrically common Series and parallel resistor conﬂgurations have very diﬁerent electrical properties. We’ll ex- plore the properties of each conﬂguration in the sections to come. 14.17 Simple series circuits Let’s start with a series circuit consisting of three resistors and a single battery: 9 V 1 + - 4 R1 3 kW 5 kW R3 2 10 kW R2 3 The ﬂrst principle to understand about series circuits is that the amount of current is the same through any component in the circuit. This is because there is only one path for electrons to ow in a series circuit, and because free electrons ow through conductors like marbles in a tube, the rate of ow (marble speed) at any point in the circuit (tube) at any speciﬂc point in time must be equal. From the way that the 9 volt battery is arranged, we can tell that the electrons in this circu
it will ow in a counter-clockwise direction, from point 4 to 3 to 2 to 1 and back to 4. However, we have one source of voltage and three resistances. How do we use Ohm’s Law here? An important caveat to Ohm’s Law is that all quantities (voltage, current, resistance, and power) must relate to each other in terms of the same two points in a circuit. For instance, with a single-battery, single-resistor circuit, we could easily calculate any quantity because they all applied to the same two points in the circuit: 274 kW 3 =I 9 volts 3 kW = 3 mA Since points 1 and 2 are connected together with wire of negligible resistance, as are points 3 and 4, we can say that point 1 is electrically common to point 2, and that point 3 is electrically common to point 4. Since we know we have 9 volts of electromotive force between points 1 and 4 (directly across the battery), and since point 2 is common to point 1 and point 3 common to point 4, we must also have 9 volts between points 2 and 3 (directly across the resistor). Therefore, we can apply Ohm’s Law (I = E/R) to the current through the resistor, because we know the voltage (E) across the resistor and the resistance (R) of that resistor. All terms (E, I, R) apply to the same two points in the circuit, to that same resistor, so we can use the Ohm’s Law formula with no reservation. However, in circuits containing more than one resistor, we must be careful in how we apply Ohm’s Law. In the three-resistor example circuit below, we know that we have 9 volts between points 1 and 4, which is the amount of electromotive force trying to push electrons through the series combination of R1, R2, and R3. However, we cannot take the value of 9 volts and divide it by 3k, 10k or 5k › to try to ﬂnd a current value, because we don’t know how much voltage is across any one of those resistors, individually. 9 V 1 + - 4 R1 3 kW 5 kW R3 2 10 kW R2 3 The ﬂgure of 9 volts is a total quantity for the whole circuit, whereas the ﬂgures of 3k, 10k, and 5k › are individual quantities for individual resistors. If we were to plug a ﬂgure for total voltage into an Ohm’s Law equation with a ﬂgure for individual resistance, the result would not relate accurately to any quantity in the real circuit. For R1, Ohm’s Law will relate the amount of voltage across R1 with the current through R1, given R1’s resistance, 3k›: 275 IR1 = ER1 3 kW ER1 = IR1(3 kW) But, since we don’t know the voltage across R1 (only the total voltage supplied by the battery across the three-resistor series combination) and we don’t know the current through R1, we can’t do any calculations with either formula. The same goes for R2 and R3: we can apply the Ohm’s Law equations if and only if all terms are representative of their respective quantities between the same two points in the circuit. So what can we do? We know the voltage of the source (9 volts) applied across the series combination of R1, R2, and R3, and we know the resistances of each resistor, but since those quantities aren’t in the same context, we can’t use Ohm’s Law to determine the circuit current. If only we knew what the total resistance was for the circuit: then we could calculate total current with our ﬂgure for total voltage (I=E/R). This brings us to the second principle of series circuits: the total resistance of any series circuit is equal to the sum of the individual resistances. This should make intuitive sense: the more resistors in series that the electrons must ow through, the more di–cult it will be for those electrons to ow. In the example problem, we had a 3 k›, 10 k›, and 5 k› resistor in series, giving us a total resistance of 18 k›: Rtotal = R1 + R2 + R3 + Rtotal = 3 kW 10 kW 5 kW + Rtotal = 18 kW In essence, we’ve calculated the equivalent resistance of R1, R2, and R3 combined. Knowing this, we could re-draw the circuit with a single equivalent resistor representing the series combination of R1, R2, and R3: 9 V 1 + - 4 R1 + R2 + R3 = 18 kW Now we have all the necessary information to calculate circuit current, because we have the voltage between points 1 and 4 (9 volts) and the resistance between points 1 and 4 (18 k›): Itotal= Etotal Rtotal Itotal = 9 volts 18 kW = 500 mA Knowing that current is equal through all components of a series circuit (and we just determined the current through the battery), we can go back to our original circuit schematic and note the current through each component: 276 9 V 1 + - 4 R1 3 kW I = 500 mA I = 500 mA 2 R2 10 kW R3 5 kW 3 Now that we know the amount of current through each resistor, we can use Ohm’s Law to determine the voltage drop across each one (applying Ohm’s Law in its proper context): ER1 = IR1 R1 ER2 = IR2 R2 ER3 = IR3 R3 ER1 = (500 mA)(3 kW) = 1.5 V ER2 = (500 mA)(10 kW) = 5 V ER3 = (500 mA)(5 kW) = 2.5 V Notice the voltage drops across each resistor, and how the sum of the voltage drops (1.5 + 5 + 2.5) is equal to the battery (supply) voltage: 9 volts. This is the third principle of series circuits: that the supply voltage is equal to the sum of the individual voltage drops. However, the method we just used to analyze this simple series circuit can be streamlined for better understanding. By using a table to list all voltages, currents, and resistances in the circuit, it becomes very easy to see which of those quantities can be properly related in any Ohm’s Law equation: R1 R2 R3 Total E I R Volts Amps Ohms Ohm’s Law Ohm’s Law Ohm’s Law Ohm’s Law The rule with such a table is to apply Ohm’s Law only to the values within each vertical column. For instance, ER1 only with IR1 and R1; ER2 only with IR2 and R2; etc. You begin your analysis by ﬂlling in those elements of the table that are given to you from the beginning: R1 R2 R3 3k 10k 5k E I R Total 9 Volts Amps Ohms As you can see from the arrangement of the data, we can’t apply the 9 volts of ET (total 277 voltage) to any of the resistances (R1, R2, or R3) in any Ohm’s Law formula because they’re in diﬁerent columns. The 9 volts of battery voltage is not applied directly across R1, R2, or R3. However, we can use our "rules" of series circuits to ﬂll in blank spots on a horizontal row. In this case, we can use the series rule of resistances to determine a total resistance from the sum of individual resistances: R1 R2 R3 Total 9 3k 10k 5k 18k E I R Volts Amps Ohms Rule of series circuits RT = R1 + R2 + R3 Now, with a value for total resistance inserted into the rightmost ("Total") column, we can apply Ohm’s Law of I=E/R to total voltage and total resistance to arrive at a total current of 500 „A: R1 R2 R3 3k 10k 5k E I R Volts Amps Ohms Total 9 500m 18k Ohm’s Law Then, knowing that the current is shared equally by all components of a series circuit (another "rule" of series circuits), we can ﬂll in the currents for each resistor from the current ﬂgure just calculated: R1 R2 R3 E I R 500m 3k 500m 10k 500m 5k Total 9 500m 18k Volts Amps Ohms Rule of series circuits IT = I1 = I2 = I3 Finally, we can use Ohm’s Law to determine the voltage drop across each resistor, one column at a time: 278 R1 1.5 500m 3k E I R R2 5 500m 10k R3 2.5 500m 5k Total 9 500m 18k Volts Amps Ohms Ohm’s Law Ohm’s Law Ohm’s Law 14.18 Simple parallel circuits Let’s start with a parallel circuit consisting of three resistors and a single battery: 1 + - 9 V 2 3 4 R1 10 kW R2 2 kW R3 1 kW 8 7 6 5 The ﬂrst principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrically common points in a parallel circuit, and voltage measured between sets of common points must always be the same at any given time. Therefore, in the above circuit, the voltage across R1 is equal to the voltage across R2 which is equal to the voltage across R3 which is equal to the voltage across the battery. This equality of voltages can be represented in another table for our starting values: R1 9 10k E I R R2 9 2k R3 9 1k Total 9 Volts Amps Ohms Just as in the case of series circuits, the same caveat for Ohm’s Law applies: values for voltage, current, and resistance must be in the same context in order for the calculations to work correctly. However, in the above example circuit, we can immediately apply Ohm’s Law to each resistor to ﬂnd its current because we know the voltage across each resistor (9 volts) and the resistance of each resistor: 279 IR1 = ER1 R1 IR2 = ER2 R2 IR3 = ER3 R3 IR1 = 9 V 10 kW = 0.9 mA IR2 = IR3 = E I R 9 V 2 kW 9 V 1 kW R1 9 0.9m 10k = 4.5 mA = 9 mA R2 9 4.5m 2k R3 9 9m 1k Total 9 Volts Amps Ohms Ohm’s Law Ohm’s Law Ohm’s Law At this point we still don’t know what the total current or total resistance for this parallel circuit is, so we can’t apply Ohm’s Law to the rightmost ("Total") column. However, if we think carefully about what is happening it should become apparent that the total current must equal the sum of all individual resistor ("branch") currents: 1 + - 9 V IR1 IT IT 2 3 4 IR2 IR3 R1 10 kW 2 kW R2 R3 1 kW 8 6 7 5 As the total current exits the negative (-) battery terminal at point 8 and travels through the circuit, some of the ow splits oﬁ at point 7 to go up through R1, some more splits oﬁ at point 6 to go up through R2, and the remainder goes up through R3. Like a river branching into several smaller streams, the combined ow rates of all streams must equal the ow rate of the whole river. The same thing is encountered where the currents through R1, R2, and R3 join to ow back to the positive terminal of the battery (+) toward point 1: the ow of electrons from point 2 to point 1 must equal the sum of the (branch) currents through R1, R2, and R3. This is the second principle of parallel circuits: the total circuit current is equal to the sum of the individual branch currents. Using this principle, we can ﬂll in the IT spot on our table with the sum of IR1, IR2, and IR3: 280 R1 9 0.9m 10k E I R R2 9 4.5m 2k R3 9 9m 1k Total 9 14.4m Volts Amps Ohms Rule of parallel circuits Itotal = I1 +
I2 + I3 Finally, applying Ohm’s Law to the rightmost ("Total") column, we can calculate the total circuit resistance: R1 9 0.9m 10k E I R R2 9 4.5m 2k R3 9 9m 1k Total 9 Volts 14.4m Amps 625 Ohms Rtotal = Etotal Itotal = 9 V 14.4 mA = 625 W Ohm’s Law Please note something very important here. The total circuit resistance is only 625 ›: less than any one of the individual resistors. In the series circuit, where the total resistance was the sum of the individual resistances, the total was bound to be greater than any one of the resistors individually. Here in the parallel circuit, however, the opposite is true: we say that the individual resistances diminish rather than add to make the total. This principle completes our triad of "rules" for parallel circuits, just as series circuits were found to have three rules for voltage, current, and resistance. Mathematically, the relationship between total resistance and individual resistances in a parallel circuit looks like this: Rtotal = 1 1 R2 + 1 R3 1 R1 + The same basic form of equation works for any number of resistors connected together in parallel, just add as many 1/R terms on the denominator of the fraction as needed to accommodate all parallel resistors in the circuit. 14.19 Power calculations When calculating the power dissipation of resistive components, use any one of the three power equations to derive and answer from values of voltage, current, and/or resistance pertaining to each component: 281 Power equations P = IE P = E2 E R P = I2R This is easily managed by adding another row to our familiar table of voltages, currents, and resistances: E I R P R1 R2 R3 Total Volts Amps Ohms Watts Power for any particular table column can be found by the appropriate Ohm’s Law equation (appropriate based on what ﬂgures are present for E, I, and R in that column). An interesting rule for total power versus individual power is that it is additive for any conﬂguration of circuit: series, parallel, series/parallel, or otherwise. Power is a measure of rate of work, and since power dissipated must equal the total power applied by the source(s) (as per the Law of Conservation of Energy in physics), circuit conﬂguration has no eﬁect on the mathematics. 14.20 Correct use of Ohm’s Law When working through worked examples it is important to try to ﬂgure out what you are doing correctly as well as what you are doing wrong. Make sure you don’t stop doing the good things and try to correct the mistakes. Circuit questions form a large part of high school and early university courses and it is important to understand the concepts properly. One common mistake which students make we’ll discuss here so you know to look out for it when you are working through examples and studying. When applying Ohm’s Laws students often mix up the contexts of voltage, current, and resistance. This means a student might mistakenly use a value for I through one resistor and the value for V across another resistor or a set of connected resistors. Remember this important rule: The variables used in Ohm’s Law equations must be common to the same two points in the circuit under consideration. This is especially important in series-parallel combination circuits where nearby components may have diﬁerent values for both voltage drop and current. When using Ohm’s Law to calculate a variable for a single component: be sure the voltage you’re using is solely across that single component and the current you’re referencing is solely through that single component and † † the resistance you’re referencing is solely for that single component. † When calculating a variable for a set of components in a circuit, be sure that the voltage, current, and resistance values are speciﬂc to that complete set of components only! 282 A good way to remember this is to pay close attention to the two points on either side of the component or set of components. Making sure that the voltage in question is across those two points, that the current in question is the electron ow from one of those points all the way to the other point, that the resistance in question is the equivalent of a single resistor between those two points, and that the power in question is the total power dissipated by all components between those two points. The "table" method presented for both series and parallel circuits in this chapter is a way to keep the components correct when using Ohm’s Law. In a table like the one shown below, you are only allowed to apply an Ohm’s Law equation for the values of a single vertical column at a time: R1 R2 R3 Total E I R P Volts Amps Ohms Watts Ohm’s Law Ohm’s Law Ohm’s Law Ohm’s Law Deriving values horizontally across columns is allowable as per the principles of series and parallel circuits: For series circuits: R1 R2 R3 Total E I R P Add Volts Equal Amps Add Add Ohms Watts Etotal = E1 + E2 + E3 Itotal = I1 = I2 = I3 Rtotal = R1 + R2 + R3 Ptotal = P1 + P2 + P3 283 For parallel circuits: R1 R2 R3 Total E I R P Equal Volts Add Diminish Add Amps Ohms Watts Etotal = E1 = E2 = E3 Itotal = I1 + I2 + I3 Rtotal = 1 1 R2 + 1 R3 1 R1 + Ptotal = P1 + P2 + P3 The "table" method helps to keep track of all relevant quantities. It also facilitates crosschecking of answers by making it easy to solve for the original unknown variables through other methods, or by working backwards to solve for the initially given values from your solutions. For example, if you have just solved for all unknown voltages, currents, and resistances in a circuit, you can check your work by adding a row at the bottom for power calculations on each resistor, seeing whether or not all the individual power values add up to the total power. If not, then you must have made a mistake somewhere! While this technique of "cross-checking" your work is nothing new, using the table to arrange all the data for the cross-check(s) results in a minimum of confusion. Aside: Although checking your work might not be fun when you have just worked hard on the problem the beneﬂts are great. Coming back to a problem after a small break (trying another problem) often helps to ﬂnd simple mistakes. If you have done all the work then ﬂnding a simple mistake will be quick to ﬂx because you know exactly what you need to do. Also if you start ﬂnding mistakes while checking you’ll build a mental list and ﬂnd that you’ll stop making them after a while. Do it and you’ll ﬂnd it will pay oﬁ ! 14.21 Conductor size The width of a conductor aﬁects the ow of electrons through it. The broader the cross-sectional area (thickness or area of a sl) of the conductor, the more room for electrons to ow, and consequently, the easier it is for ow to occur (less resistance). Electrical wire is usually round in cross-section (although there are some unique exceptions to this rule), and comes in two basic varieties: solid and stranded. Solid copper wire is just as it sounds: a single, solid strand of copper the whole length of the wire. Stranded wire is composed of smaller strands of solid copper wire twisted together to form a single, larger conductor. The greatest beneﬂt of stranded wire is its mechanical exibility, being able to withstand repeated 284 bending and twisting much better than solid copper (which tends to fatigue and break after time). 14.22 Fuses Normally, the ampacity rating of a conductor is a circuit design limit never to be intentionally exceeded, but there is an application where ampacity exceedence is expected: in the case of fuses. A fuse is nothing more than a short length of wire designed to melt and separate in the event of excessive current. Fuses are always connected in series with the component(s) to be protected from overcurrent, so that when the fuse blows (opens) it will open the entire circuit and stop current through the component(s). A fuse connected in one branch of a parallel circuit, of course, would not aﬁect current through any of the other branches. Normally, the thin piece of fuse wire is contained within a safety sheath to minimize hazards of arc blast if the wire burns open with violent force, as can happen in the case of severe overcurrents. In the case of small automotive fuses, the sheath is transparent so that the fusible element can be visually inspected. Residential wiring used to commonly employ screw-in fuses with glass bodies and a thin, narrow metal foil strip in the middle. 14.23 Important Equations and Quantities Quantity Symbol Unit S.I. Units Direction Units or Table 14.1: Units used in Electricity and Magnetism † † † † † † † † REVIEW: A circuit is an unbroken loop of conductive material that allows electrons to ow through continuously without beginning or end. If a circuit is "broken," that means it’s conductive elements no longer form a complete path, and continuous electron ow cannot occur in it. The location of a break in a circuit is irrelevant to its inability to sustain continuous electron ow. Any break, anywhere in a circuit prevents electron ow throughout the circuit. REVIEW: Electrons can be motivated to ow through a conductor by a the same force manifested in static electricity. Voltage is the measure of speciﬂc potential energy (potential energy per unit charge) between two locations. In layman’s terms, it is the measure of "push" available to motivate electrons. Voltage, as an expression of potential energy, is always relative between two locations, or points. Sometimes it is called a voltage "drop." 285 † † † † † † † † † † † † † † † † † † † † † When a voltage source is connected to a circuit, the voltage will cause a uniform ow of electrons through that circuit called a current. In a single (one loop) circuit, the amount current of current at any point is the same as the amount of current at any other point. If a circuit containing a voltage source is broken, the full voltage of that source will appear across the points of the break. The +/- orientation a voltage drop is called the polarity. It is also relative between two points. REVIEW:
Resistance is the measure of opposition to electric current. A short circuit is an electric circuit oﬁering little or no resistance to the ow of electrons. Short circuits are dangerous with high voltage power sources because the high currents encountered can cause large amounts of heat energy to be released. An open circuit is one where the continuity has been broken by an interruption in the path for electrons to ow. A closed circuit is one that is complete, with good continuity throughout. A device designed to open or close a circuit under controlled conditions is called a switch. The terms "open" and "closed" refer to switches as well as entire circuits. An open switch is one without continuity: electrons cannot ow through it. A closed switch is one that provides a direct (low resistance) path for electrons to ow through. REVIEW: Connecting wires in a circuit are assumed to have zero resistance unless otherwise stated. Wires in a circuit can be shortened or lengthened without impacting the circuit’s function { all that matters is that the components are attached to one another in the same sequence. Points directly connected together in a circuit by zero resistance (wire) are considered to be electrically common. Electrically common points, with zero resistance between them, will have zero voltage dropped between them, regardless of the magnitude of current (ideally). The voltage or resistance readings referenced between sets of electrically common points will be the same. These rules apply to ideal conditions, where connecting wires are assumed to possess absolutely zero resistance. In real life this will probably not be the case, but wire resistances should be low enough so that the general principles stated here still hold. REVIEW: Power is additive in any conﬂguration of resistive circuit: PT otal = P1 + P2 + . . . Pn REVIEW: 286 † † † † When electrons ow through a conductor, a magnetic ﬂeld will be produced around that conductor. The left-hand rule states that the magnetic ux lines produced by a current-carrying wire will be oriented the same direction as the curled ﬂngers of a person’s left hand (in the "hitchhiking" position), with the thumb pointing in the direction of electron ow. The magnetic ﬂeld force produced by a current-carrying wire can be greatly increased by shaping the wire into a coil instead of a straight line. If wound in a coil shape, the magnetic ﬂeld will be oriented along the axis of the coil’s length. The magnetic ﬂeld force produced by an electromagnet (called the magnetomotive force, or mmf), is proportional to the product (multiplication) of the current through the electromagnet and the number of complete coil "turns" formed by the wire. 287 Chapter 15 Magnets and Electromagnetism sectionPermanent magnets Magnetism has been known to mankind for many thousands of years. Lodestone, a magnetized form of the iron oxide mineral magnetite which has the property of attracting iron objects, is referred to in old European and Asian historical records, around 800 BC in Europe and earlier in the East, around 2600 BC. The root of the English word magnet is the Greek word magnes, thought to be derived from Magnesia in Asia Minor, once an important source of lodestone. Lodestone was used as a navigational compass as it was found to orient itself in a north-south direction if left free to rotate by suspension on a string or on a oat in water. Interesting Fact: A compass is a navigational instrument for ﬂnding directions. It consists of a magnetised pointer free to align itself accurately with Earth’s magnetic ﬂeld. A compass provides a known reference direction which is of great assistance in navigation. The cardinal points are north, south, east and west. A compass can be used in conjunction with a clock and a sextant to provide a very accurate navigation capability. This device greatly improved maritime trade by making travel safer and more e–cient. A compass can be any magnetic device using a needle to indicate the direction of the magnetic north of a planet’s magnetosphere. Any instrument with a magnetized bar or needle turning freely upon a pivot and pointing in a northerly and southerly direction can be considered a compass. Aside: In 1269, Frenchmen Peter Peregrinus and Pierre de Maricourt, using a compass and a lodestone, found that the magnetic force of the lodestone was diﬁerent at the opposite ends, which they deﬂned to be the poles of the magnet. Like poles of magnets repel one another whilst unlike poles attract. These poles always occur in pairs. It is impossible to isolate a single pole. Breaking a piece of magnet in half results in two pieces, each with it’s own pair of poles. 288 N magnet S . . . after breaking in half . . . N magnet S N magnet S The Earth itself is a magnet. Its magnetic poles are approximately aligned along the Earth’s axis of rotation. The magnitude of forces between the poles of magnets follows an inverse square law; i. e. it varies inversely as the square of the distance of separation. Magnetic forces are a result of magnetic ﬂelds. By placing a magnet underneath a piece of paper and sprinkling iron ﬂlings on top one can map the magnetic ﬂeld. The ﬂlings align themselves parallel to the ﬂeld. Magnetic ﬂelds can be represented by magnetic ﬂeld lines which are parallel to the magnetic ﬂeld and whose spacing represents the relative strength of the magnetic ﬂeld. The strength of the magnetic ﬂeld is referred to as the magnetic ux. Magnetic ﬂeld lines form closed loops. In a bar magnet magnetic ﬂeld lines emerge at one pole and then curve around to the other pole with the rest of the loop being inside the magnet. magnetic field N magnet S As already said, opposite poles of a magnet attract each other and bringing them together results in their magnetic ﬂeld lines converging. Like poles of a magnet repel each other and bringing them together results in their magnetic ﬂeld lines diverging. 289 Ferromagnetism is a phenomenon exhibited by materials like iron, nickel or cobalt. These materials are known as permanent magnets. They always magnetize so as to be attracted to a magnet, regardless of which magnetic pole is brought toward the unmagnetized iron: N iron S N magnet S attraction Interesting Fact: The cause of Earth’s magnetic ﬂeld is not known for certain, but is possibly explained by the dynamo theory. The magnetic ﬂeld extends several tens of thousands of kilometers into space. The ﬂeld is approximately a magnetic dipole, with one pole near the geographic north pole and the other near the geographic south pole. An imaginary line joining the magnetic poles would be inclined by approximately 11.3 from the planet’s axis of rotation. The location of the magnetic poles is not static but wanders as much as several kilometers a year. The two poles wander independently of each other and are not at exact opposite positions on the globe. The Earth’s magnetic ﬂeld reverses at intervals, ranging from tens of thousands to many millions of years, with an average interval of 250,000 years. It is believed that this last occurred some 780,000 years ago. The mechanism responsible for geomagnetic reversals is not well understood. When the North reappears in the opposite direction, we would interpret this as a reversal, whereas turning oﬁ and returning in the same direction is called a geomagnetic excursion. At present, the overall geomagnetic ﬂeld is becoming weaker at a rate which would, if it continues, cause the ﬂeld to disappear, albeit temporarily, by about around 3000-4000 AD. The deterioration began roughly 150 years ago and has accelerated in the past several years. So far the strength of the earth’s ﬂeld has decreased by 10 to 15 percent. The ability of a ferromagnetic material tends to retain its magnetization after an external ﬂeld is removed is called it’s retentivity. 290 Paramagnetic materials are materials like aluminum or platinum which become magnetized in an external magnetic ﬂeld in a similar way to ferromagnetic materials but lose their magnetism when the external magnetic ﬂeld is removed. Diamagnetism is exhibited by materials like copper or bismuth which become magnetized in a magnetic ﬂeld with a polarity opposite to the external magnetic ﬂeld. Unlike iron, they are slightly repelled by a magnet S diamagnetic material N N magnet S repulsion The cause of Earth’s magnetic ﬂeld is not known for certain, but is possibly explained by the dynamo theory. The magnetic ﬂeld extends several tens of thousands of kilometers into space. The ﬂeld is approximately a magnetic dipole, with one pole near the geographic north pole and the other near the geographic south pole. An imaginary line joining the magnetic poles would be inclined by approximately 11.3 from the planet’s axis of rotation. The location of the magnetic poles is not static but wanders as much as several kilometers a year. The two poles wander independently of each other and are not at exact opposite positions on the globe. Currently the south magnetic pole is further from the geographic south pole than than north magnetic pole is from the north geographic pole. The strength of the ﬂeld at the Earth’s surface at this time ranges from less than 30 microtesla (0.3 gauss) in an area including most of South America and South Africa to over 60 microtesla (0.6 gauss) around the magnetic poles in northern Canada and south of Australia, and in part of Siberia. The ﬂeld is similar to that of a bar magnet, but this similarity is superﬂcial. The magnetic ﬂeld of a bar magnet, or any other type of permanent magnet, is created by the coordinated motions of electrons (negatively charged particles) within iron atoms. The Earth’s core, however, is hotter than 1043 K, the temperature at which the orientations of electron orbits within iron become randomized. Therefore the Earth’s magnetic ﬂeld is not caused by magnetised iron deposits, but mostly by electric currents (known as telluric currents). Another feature that d
istinguishes the Earth magnetically from a bar magnet is its magnetosphere. A magnetosphere is the region around an astronomical object, in which phenomena are dominated by its magnetic ﬂeld. Earth is surrounded by a magnetosphere, as are the magnetized planets Jupiter, Saturn, Uranus and Neptune. Mercury is magnetized, but too weakly to trap plasma. Mars has patchy surface magnetization. The distant ﬂeld of Earth is greatly modiﬂed by the solar wind, a hot outow from the sun, consisting of solar ions (mainly hydrogen) moving at about 400 km/s . Earth’s magnetic ﬂeld forms an obstacle to the solar wind. The Earth’s magnetic ﬂeld reverses at intervals, ranging from tens of thousands to many It is believed that this last oc- millions of years, with an average interval of 250,000 years. 291 curred some 780,000 years ago. The mechanism responsible for geomagnetic reversals is not well understood. When the North reappears in the opposite direction, we would interpret this as a reversal, whereas turning oﬁ and returning in the same direction is called a geomagnetic excursion. At present, the overall geomagnetic ﬂeld is becoming weaker at a rate which would, if it continues, cause the ﬂeld to disappear, albeit temporarily, by about around 3000-4000 AD. The deterioration began roughly 150 years ago and has accelerated in the past several years. So far the strength of the earth’s ﬂeld has decreased by 10 to 15 percent. 15.1 Electromagnetism The discovery of the relationship between magnetism and electricity was, like so many other scientiﬂc discoveries, stumbled upon almost by accident. The Danish physicist Hans Christian Oersted was lecturing one day in 1820 on the possibility of electricity and magnetism being related to one another, and in the process demonstrated it conclusively by experiment in front of his whole class! By passing an electric current through a metal wire suspended above a magnetic compass, Oersted was able to produce a deﬂnite motion of the compass needle in response to the current. What began as conjecture at the start of the class session was conﬂrmed as fact at the end. Needless to say, Oersted had to revise his lecture notes for future classes! His serendipitous discovery paved the way for a whole new branch of science: electromagnetics. Detailed experiments showed that the magnetic ﬂeld produced by an electric current is always oriented perpendicular to the direction of ow. A simple method of showing this relationship is called the left-hand rule. Simply stated, the left-hand rule says that the magnetic ux lines produced by a current-carrying wire will be oriented the same direction as the curled ﬂngers of a person’s left hand (in the "hitchhiking" position), with the thumb pointing in the direction of electron ow: The "left-hand" rule I I I The magnetic ﬂeld encircles this straight piece of current-carrying wire, the magnetic ux I lines having no deﬂnite "north" or "south’ poles. (NOTE TO SELF: Need to add wires attracting or wires repelling) While the magnetic ﬂeld surrounding a current-carrying wire is indeed interesting, it is quite weak for common amounts of current, able to deect a compass needle and not much more. To 292 create a stronger magnetic ﬂeld force (and consequently, more ﬂeld ux) with the same amount of electric current, we can wrap the wire into a coil shape, where the circling magnetic ﬂelds around the wire will join to create a larger ﬂeld with a deﬂnite magnetic (north and south) polarity: S N magnetic field The amount of magnetic ﬂeld force generated by a coiled wire is proportional to the current through the wire multiplied by the number of "turns" or "wraps" of wire in the coil. This ﬂeld force is called magnetomotive force (mmf), and is very much analogous to electromotive force (E) in an electric circuit. An electromagnet is a piece of wire intended to generate a magnetic ﬂeld with the passage of electric current through it. Though all current-carrying conductors produce magnetic ﬂelds, an electromagnet is usually constructed in such a way as to maximize the strength of the magnetic ﬂeld it produces for a special purpose. Electromagnets ﬂnd frequent application in research, industry, medical, and consumer products. As an electrically-controllable magnet, electromagnets ﬂnd application in a wide variety of "electromechanical" devices: machines that eﬁect mechanical force or motion through electrical power. Perhaps the most obvious example of such a machine is the electric motor. Relay Applying current through the coil causes the switch to close. Relays can be constructed to actuate multiple switch contacts, or operate them in "reverse" (energizing the coil will open the switch contact, and unpowering the coil will allow it to spring closed again). 293 Multiple-contact relay Relay with "normallyclosed" contact 15.2 Magnetic units of measurement If the burden of two systems of measurement for common quantities (English vs. metric) throws your mind into confusion, this is not the place for you! Due to an early lack of standardization in the science of magnetism, we have been plagued with no less than three complete systems of measurement for magnetic quantities. First, we need to become acquainted with the various quantities associated with magnetism. There are quite a few more quantities to be dealt with in magnetic systems than for electrical systems. With electricity, the basic quantities are Voltage (E), Current (I), Resistance (R), and Power (P). The ﬂrst three are related to one another by Ohm’s Law (E=IR ; I=E/R ; R=E/I), while Power is related to voltage, current, and resistance by Joule’s Law (P=IE ; P=I2R ; P=E2/R). With magnetism, we have the following quantities to deal with: Magnetomotive Force { The quantity of magnetic ﬂeld force, or "push." Analogous to electric voltage (electromotive force). Field Flux { The quantity of total ﬂeld eﬁect, or "substance" of the ﬂeld. Analogous to electric current. Field Intensity { The amount of ﬂeld force (mmf) distributed over the length of the elec- tromagnet. Sometimes referred to as Magnetizing Force. Flux Density { The amount of magnetic ﬂeld ux concentrated in a given area. Reluctance { The opposition to magnetic ﬂeld ux through a given volume of space or material. Analogous to electrical resistance. Permeability { The speciﬂc measure of a material’s acceptance of magnetic ux, analogous to the speciﬂc resistance of a conductive material (‰), except inverse (greater permeability means easier passage of magnetic ux, whereas greater speciﬂc resistance means more di–cult passage of electric current). . . But wait . the fun is just beginning! Not only do we have more quantities to keep track of with magnetism than with electricity, but we have several diﬁerent systems of unit measurement for each of these quantities. As with common quantities of length, weight, volume, and temperature, we have both English and metric systems. However, there is actually more than 294 one metric system of units, and multiple metric systems are used in magnetic ﬂeld measurements! One is called the cgs, which stands for Centimeter-Gram-Second, denoting the root measures upon which the whole system is based. The other was originally known as the mks system, which stood for Meter-Kilogram-Second, which was later revised into another system, called rmks, standing for Rationalized Meter-Kilogram-Second. This ended up being adopted as an international standard and renamed SI (Systeme International). Quantity Symbol Unit of Measurement and abbreviation Field Force mmf Gilbert (Gb) Amp-turn Amp-turn CGS SI English Field Flux Field Intensity Flux Density Reluctance F H B ´ Permeability m Maxwell (Mx) Weber (Wb) Line Oersted (Oe) Amp-turns per meter Amp-turns per inch Gauss (G) Tesla (T) Lines per square inch Gilberts per Maxwell Amp-turns per Weber Amp-turns per line Gauss per Oersted Tesla-meters per Amp-turn Lines per inch-Ampturn And yes, the „ symbol is really the same as the metric preﬂx "micro." I ﬂnd this especially confusing, using the exact same alphabetical character to symbolize both a speciﬂc quantity and a general metric preﬂx! As you might have guessed already, the relationship between ﬂeld force, ﬂeld ux, and reluctance is much the same as that between the electrical quantities of electromotive force (E), current (I), and resistance (R). This provides something akin to an Ohm’s Law for magnetic circuits: A comparison of "Ohm’s Law" for electric and magnetic circuits: E = IR Electrical mmf = F´ Magnetic And, given that permeability is inversely analogous to speciﬂc resistance, the equation for ﬂnding the reluctance of a magnetic material is very similar to that for ﬂnding the resistance of a conductor: A comparison of electrical and magnetic opposition: R = r l A Electrical ´ = l mA Magnetic 295 In either case, a longer piece of material provides a greater opposition, all other factors being equal. Also, a larger cross-sectional area makes for less opposition, all other factors being equal. 15.3 Electromagnetic induction While Oersted’s surprising discovery of electromagnetism paved the way for more practical applications of electricity, it was Michael Faraday who gave us the key to the practical generation of electricity: electromagnetic induction. Faraday discovered that a voltage would be generated across a length of wire if that wire was exposed to a perpendicular magnetic ﬂeld ux of changing intensity. An easy way to create a magnetic ﬂeld of changing intensity is to move a permanent magnet next to a wire or coil of wire. Remember: the magnetic ﬂeld must increase or decrease in intensity perpendicular to the wire (so that the lines of ux "cut across" the conductor), or else no voltage will be induced: Electromagnetic induction current changes direction with change in magnet motion voltage changes polarity with change in magnet motion - + V + - N S magnet moved back and forth Faraday was able to mathematically relate the rate of
change of the magnetic ﬂeld ux with induced voltage (note the use of a lower-case letter "e" for voltage. This refers to instantaneous voltage, or voltage at a speciﬂc point in time, rather than a steady, stable voltage.): e = N dF dt Where, e = (Instantaneous) induced voltage in volts N = F = t = Number of turns in wire coil (straight wire = 1) Magnetic flux in Webers Time in seconds The "d" terms are standard calculus notation, representing rate-of-change of ux over time. 296 "N" stands for the number of turns, or wraps, in the wire coil (assuming that the wire is formed in the shape of a coil for maximum electromagnetic e–ciency). This phenomenon is put into obvious practical use in the construction of electrical generators, which use mechanical power to move a magnetic ﬂeld past coils of wire to generate voltage. However, this is by no means the only practical use for this principle. If we recall that the magnetic ﬂeld produced by a current-carrying wire was always perpendicular to that wire, and that the ux intensity of that magnetic ﬂeld varied with the amount of current through it, we can see that a wire is capable of inducing a voltage along its own length simply due to a change in current through it. This eﬁect is called self-induction: a changing magnetic ﬂeld produced by changes in current through a wire inducing voltage along the length of that same wire. If the magnetic ﬂeld ux is enhanced by bending the wire into the shape of a coil, and/or wrapping that coil around a material of high permeability, this eﬁect of self-induced voltage will be more intense. A device constructed to take advantage of this eﬁect is called an inductor, and will be discussed in greater detail in the next chapter. A device speciﬂcally designed to produce the eﬁect of mutual inductance between two or more coils is called a transformer. Because magnetically-induced voltage only happens when the magnetic ﬂeld ux is changing in strength relative to the wire, mutual inductance between two coils can only happen with alternating (changing { AC) voltage, and not with direct (steady { DC) voltage. The only applications for mutual inductance in a DC system is where some means is available to switch power on and oﬁ to the coil (thus creating a pulsing DC voltage), the induced voltage peaking at every pulse. A very useful property of transformers is the ability to transform voltage and current levels If the according to a simple ratio, determined by the ratio of input and output coil turns. energized coil of a transformer is energized by an AC voltage, the amount of AC voltage induced in the unpowered coil will be equal to the input voltage multiplied by the ratio of output to input wire turns in the coils. Conversely, the current through the windings of the output coil compared to the input coil will follow the opposite ratio: if the voltage is increased from input coil to output coil, the current will be decreased by the same proportion. This action of the transformer is analogous to that of mechanical gear, belt sheave, or chain sprocket ratios: 297 Torque-reducing geartrain Large gear (many teeth) Small gear (few teeth) + + high torque, low speed low torque, high speed "Step-down" transformer high voltage AC voltage source many turns low voltage few turns Load high current low current A transformer designed to output more voltage than it takes in across the input coil is called a "step-up" transformer, while one designed to do the opposite is called a "step-down," in reference to the transformation of voltage that takes place. The current through each respective coil, of course, follows the exact opposite proportion. 15.4 AC Most students of electricity begin their study with what is known as direct current (DC), which is electricity owing in a constant direction, and/or possessing a voltage with constant polarity. DC is the kind of electricity made by a battery (with deﬂnite positive and negative terminals), or the kind of charge generated by rubbing certain types of materials against each other. As useful and as easy to understand as DC is, it is not the only "kind" of electricity in use. Certain sources of electricity (most notably, rotary electro-mechanical generators) naturally produce voltages alternating in polarity, reversing positive and negative over time. Either as a voltage switching polarity or as a current switching direction back and forth, this "kind" of electricity is known as Alternating Current (AC): 298 DIRECT CURRENT (DC) ALTERNATING CURRENT (AC) I I I I Whereas the familiar battery symbol is used as a generic symbol for any DC voltage source, the circle with the wavy line inside is the generic symbol for any AC voltage source. One might wonder why anyone would bother with such a thing as AC. It is true that in some cases AC holds no practical advantage over DC. In applications where electricity is used to dissipate energy in the form of heat, the polarity or direction of current is irrelevant, so long as there is enough voltage and current to the load to produce the desired heat (power dissipation). However, with AC it is possible to build electric generators, motors and power distribution systems that are far more e–cient than DC, and so we ﬂnd AC used predominately across the world in high power applications. To explain the details of why this is so, a bit of background knowledge about AC is necessary. If a machine is constructed to rotate a magnetic ﬂeld around a set of stationary wire coils with the turning of a shaft, AC voltage will be produced across the wire coils as that shaft is rotated, in accordance with Faraday’s Law of electromagnetic induction. This is the basic operating principle of an AC generator, also known as an alternator : Alternator operation Step #1 Step #2 S N no current! Load Step #3 N S no current! Load N S + I - I Load Step #4 S N - I I Load + Notice how the polarity of the voltage across the wire coils reverses as the opposite poles of 299 the rotating magnet pass by. Connected to a load, this reversing voltage polarity will create a reversing current direction in the circuit. The faster the alternator’s shaft is turned, the faster the magnet will spin, resulting in an alternating voltage and current that switches directions more often in a given amount of time. While DC generators work on the same general principle of electromagnetic induction, their construction is not as simple as their AC counterparts. With a DC generator, the coil of wire is mounted in the shaft where the magnet is on the AC alternator, and electrical connections are made to this spinning coil via stationary carbon "brushes" contacting copper strips on the rotating shaft. All this is necessary to switch the coil’s changing output polarity to the external circuit so the external circuit sees a constant polarity: (DC) Generator operation Step #1 Step #2 N S SN Load Step #3 N S SN - + N S + Load Step #4 - + SN + N S - I N S - I Load Load The generator shown above will produce two pulses of voltage per revolution of the shaft, both pulses in the same direction (polarity). In order for a DC generator to produce constant voltage, rather than brief pulses of voltage once every 1/2 revolution, there are multiple sets of coils making intermittent contact with the brushes. The diagram shown above is a bit more simpliﬂed than what you would see in real life. The problems involved with making and breaking electrical contact with a moving coil should be obvious (sparking and heat), especially if the shaft of the generator is revolving at high speed. If the atmosphere surrounding the machine contains ammable or explosive vapors, the practical problems of spark-producing brush contacts are even greater. An AC generator (alternator) does not require brushes and commutators to work, and so is immune to these problems experienced by DC generators. The beneﬂts of AC over DC with regard to generator design is also reected in electric motors. While DC motors require the use of brushes to make electrical contact with moving coils of wire, AC motors do not. In fact, AC and DC motor designs are very similar to their 300 generator counterparts (identical for the sake of this tutorial), the AC motor being dependent upon the reversing magnetic ﬂeld produced by alternating current through its stationary coils of wire to rotate the rotating magnet around on its shaft, and the DC motor being dependent on the brush contacts making and breaking connections to reverse current through the rotating coil every 1/2 rotation (180 degrees). So we know that AC generators and AC motors tend to be simpler than DC generators and DC motors. This relative simplicity translates into greater reliability and lower cost of manufacture. But what else is AC good for? Surely there must be more to it than design details of generators and motors! Indeed there is. There is an eﬁect of electromagnetism known as mutual induction, whereby two or more coils of wire placed so that the changing magnetic ﬂeld created by one induces a voltage in the other. If we have two mutually inductive coils and we energize one coil with AC, we will create an AC voltage in the other coil. When used as such, this device is known as a transformer : Transformer AC voltage source Induced AC voltage The fundamental signiﬂcance of a transformer is its ability to step voltage up or down from the powered coil to the unpowered coil. The AC voltage induced in the unpowered ("secondary") coil is equal to the AC voltage across the powered ("primary") coil multiplied by the ratio of secondary coil turns to primary coil turns. If the secondary coil is powering a load, the current through the secondary coil is just the opposite: primary coil current multiplied by the ratio of primary to secondary turns. This relationship has a very close mechanical analogy, using torque and speed to represent voltage and current, respectively: 301 Speed multiplication geartrain Large gear (many teeth) Small gear
(few teeth) high torque low speed + + low torque high speed "Step-down" transformer high voltage AC voltage source many turns low voltage few turns Load high current low current If the winding ratio is reversed so that the primary coil has less turns than the secondary coil, the transformer "steps up" the voltage from the source level to a higher level at the load: Speed reduction geartrain Large gear (many teeth) Small gear (few teeth) low torque high speed + + high torque low speed "Step-up" transformer high voltage AC voltage source low voltage few turns high current many turns Load low current The transformer’s ability to step AC voltage up or down with ease gives AC an advantage unmatched by DC in the realm of power distribution. When transmitting electrical power over long distances, it is far more e–cient to do so with stepped-up voltages and stepped-down currents 302 (smaller-diameter wire with less resistive power losses), then step the voltage back down and the current back up for industry, business, or consumer use use. high voltage Power Plant Step-up low voltage . . . to other customers Step-down Home or Business low voltage Transformer technology has made long-range electric power distribution practical. Without the ability to e–ciently step voltage up and down, it would be cost-prohibitive to construct power systems for anything but close-range (within a few miles at most) use. As useful as transformers are, they only work with AC, not DC. Because the phenomenon of mutual inductance relies on changing magnetic ﬂelds, and direct current (DC) can only produce steady magnetic ﬂelds, transformers simply will not work with direct current. Of course, direct current may be interrupted (pulsed) through the primary winding of a transformer to create a changing magnetic ﬂeld (as is done in automotive ignition systems to produce high-voltage spark plug power from a low-voltage DC battery), but pulsed DC is not that diﬁerent from AC. Perhaps more than any other reason, this is why AC ﬂnds such widespread application in power systems. If we were to follow the changing voltage produced by a coil in an alternator from any point on the sine wave graph to that point when the wave shape begins to repeat itself, we would have marked exactly one cycle of that wave. This is most easily shown by spanning the distance between identical peaks, but may be measured between any corresponding points on the graph. The degree marks on the horizontal axis of the graph represent the domain of the trigonometric sine function, and also the angular position of our simple two-pole alternator shaft as it rotates: one wave cycle 0 90 180 270 Alternator shaft position (degrees) 180 90 360 (0) one wave cycle 270 360 (0) Since the horizontal axis of this graph can mark the passage of time as well as shaft position in degrees, the dimension marked for one cycle is often measured in a unit of time, most often seconds or fractions of a second. When expressed as a measurement, this is often called the period of a wave. The period of a wave in degrees is always 360, but the amount of time one period occupies depends on the rate voltage oscillates back and forth. A more popular measure for describing the alternating rate of an AC voltage or current wave than period is the rate of that back-and-forth oscillation. This is called frequency. The modern 303 unit for frequency is the Hertz (abbreviated Hz), which represents the number of wave cycles completed during one second of time. In the United States of America, the standard power-line frequency is 60 Hz, meaning that the AC voltage oscillates at a rate of 60 complete back-andforth cycles every second. In Europe, where the power system frequency is 50 Hz, the AC voltage only completes 50 cycles every second. A radio station transmitter broadcasting at a frequency of 100 MHz generates an AC voltage oscillating at a rate of 100 million cycles every second. Prior to the canonization of the Hertz unit, frequency was simply expressed as "cycles per second." Older meters and electronic equipment often bore frequency units of "CPS" (Cycles Per Second) instead of Hz. Many people believe the change from self-explanatory units like CPS to Hertz constitutes a step backward in clarity. A similar change occurred when the unit of "Celsius" replaced that of "Centigrade" for metric temperature measurement. The name Centigrade was based on a 100-count ("Centi-") scale ("-grade") representing the melting and boiling points of H2O, respectively. The name Celsius, on the other hand, gives no hint as to the unit’s origin or meaning. Period and frequency are mathematical reciprocals of one another. That is to say, if a wave has a period of 10 seconds, its frequency will be 0.1 Hz, or 1/10 of a cycle per second: Frequency in Hertz = 1 Period in seconds An instrument called an oscilloscope is used to display a changing voltage over time on a graphical screen. You may be familiar with the appearance of an ECG or EKG (electrocardiograph) machine, used by physicians to graph the oscillations of a patient’s heart over time. The ECG is a special-purpose oscilloscope expressly designed for medical use. General-purpose oscilloscopes have the ability to display voltage from virtually any voltage source, plotted as a graph with time as the independent variable. The relationship between period and frequency is very useful to know when displaying an AC voltage or current waveform on an oscilloscope screen. By measuring the period of the wave on the horizontal axis of the oscilloscope screen and reciprocating that time value (in seconds), you can determine the frequency in Hertz. OSCILLOSCOPE vertical Y DC GND AC V/div trigger timebase 1m X s/div DC GND AC 16 divisions @ 1ms/div = a period of 16 ms Frequency = 1 period = 1 16 ms = 62.5 Hz Voltage and current are by no means the only physical variables subject to variation over time. Much more common to our everyday experience is sound, which is nothing more than 304 the alternating compression and decompression (pressure waves) of air molecules, interpreted by our ears as a physical sensation. Because alternating current is a wave phenomenon, it shares many of the properties of other wave phenomena, like sound. For this reason, sound (especially structured music) provides an excellent analogy for relating AC concepts. In musical terms, frequency is equivalent to pitch. Low-pitch notes such as those produced by a tuba or bassoon consist of air molecule vibrations that are relatively slow (low frequency). High-pitch notes such as those produced by a ute or whistle consist of the same type of vibrations in the air, only vibrating at a much faster rate (higher frequency). Here is a table showing the actual frequencies for a range of common musical notes: Note A A sharp (or B flat) B C (middle) C sharp (or D flat) D D sharp (or E flat) E F F sharp (or G flat) G G sharp (or A flat) A A sharp (or B flat) B C Musical designation Frequency (in hertz) A1 A# or Bb B1 C C# or Db D D# or Eb E F F# or Gb G G# or Ab A A# or Bb B C1 220.00 233.08 246.94 261.63 277.18 293.66 311.13 329.63 349.23 369.99 392.00 415.30 440.00 466.16 493.88 523.25 Astute observers will notice that all notes on the table bearing the same letter designation are related by a frequency ratio of 2:1. For example, the ﬂrst frequency shown (designated with the letter "A") is 220 Hz. The next highest "A" note has a frequency of 440 Hz { exactly twice as many sound wave cycles per second. The same 2:1 ratio holds true for the ﬂrst A sharp (233.08 Hz) and the next A sharp (466.16 Hz), and for all note pairs found in the table. Audibly, two notes whose frequencies are exactly double each other sound remarkably similar. This similarity in sound is musically recognized, the shortest span on a musical scale separating such note pairs being called an octave. Following this rule, the next highest "A" note (one octave above 440 Hz) will be 880 Hz, the next lowest "A" (one octave below 220 Hz) will be 110 Hz. A view of a piano keyboard helps to put this scale into perspective: 305 C# Db D# Eb F# Gb G# Ab A# Bb C# Db D# Eb F# Gb G# Ab A# Bb C# Db D# Eb F# Gb G# Ab A# Bb one octave As you can see, one octave is equal to eight white keys’ worth of distance on a piano keyboard. The familiar musical mnemonic (doe-ray-mee-fah-so-lah-tee-doe) { yes, the same pattern immortalized in the whimsical Rodgers and Hammerstein song sung in The Sound of Music { covers one octave from C to C. While electromechanical alternators and many other physical phenomena naturally produce sine waves, this is not the only kind of alternating wave in existence. Other "waveforms" of AC are commonly produced within electronic circuitry. Here are but a few sample waveforms and their common designations: Square wave Triangle wave one wave cycle one wave cycle Sawtooth wave These waveforms are by no means the only kinds of waveforms in existence. They’re simply a few that are common enough to have been given distinct names. Even in circuits that are supposed to manifest "pure" sine, square, triangle, or sawtooth voltage/current waveforms, the real-life result is often a distorted version of the intended waveshape. Some waveforms are so complex that they defy classiﬂcation as a particular "type" (including waveforms associated with many kinds of musical instruments). Generally speaking, any waveshape bearing close resemblance to a perfect sine wave is termed sinusoidal, anything diﬁerent being labeled as nonsinusoidal. Being that the waveform of an AC voltage or current is crucial to its impact in a circuit, we need to be aware of the fact that AC waves come in a variety of shapes. 306 15.5 Measurements of AC magnitude So far we know that AC voltage alternates in polarity and AC current alternates in direction. We also know that AC can alternate in a variety of diﬁerent ways, and by tracing the alternation over time we can plot it as a "waveform." We can measure t
he rate of alternation by measuring the time it takes for a wave to evolve before it repeats itself (the "period"), and express this as cycles per unit time, or "frequency." In music, frequency is the same as pitch, which is the essential property distinguishing one note from another. However, we encounter a measurement problem if we try to express how large or small an AC quantity is. With DC, where quantities of voltage and current are generally stable, we have little trouble expressing how much voltage or current we have in any part of a circuit. But how do you grant a single measurement of magnitude to something that is constantly changing? One way to express the intensity, or magnitude (also called the amplitude), of an AC quantity is to measure its peak height on a waveform graph. This is known as the peak or crest value of an AC waveform: Peak Time Another way is to measure the total height between opposite peaks. This is known as the peak-to-peak (P-P) value of an AC waveform: Peak-to-Peak Time Unfortunately, either one of these expressions of waveform amplitude can be misleading when comparing two diﬁerent types of waves. For example, a square wave peaking at 10 volts is obviously a greater amount of voltage for a greater amount of time than a triangle wave peaking at 10 volts. The eﬁects of these two AC voltages powering a load would be quite diﬁerent: 307 10 V Time 10 V (peak) 10 V (peak) more heat energy dissipated (same load resistance) less heat energy dissipated One way of expressing the amplitude of diﬁerent waveshapes in a more equivalent fashion is to mathematically average the values of all the points on a waveform’s graph to a single, aggregate number. This amplitude measure is known simply as the average value of the waveform. If we average all the points on the waveform algebraically (that is, to consider their sign, either positive or negative), the average value for most waveforms is technically zero, because all the positive points cancel out all the negative points over a full cycle: + + + + + + + + + - - - - - - - - - True average value of all points (considering their signs) is zero! This, of course, will be true for any waveform having equal-area portions above and below the "zero" line of a plot. However, as a practical measure of a waveform’s aggregate value, "average" is usually deﬂned as the mathematical mean of all the points’ absolute values over a cycle. In other words, we calculate the practical average value of the waveform by considering all points on the wave as positive quantities, as if the waveform looked like this: 308 + + + + + + + + + + ++ + + + + + + Practical average of points, all values assumed to be positive. Polarity-insensitive mechanical meter movements (meters designed to respond equally to the positive and negative half-cycles of an alternating voltage or current) register in proportion to the waveform’s (practical) average value, because the inertia of the pointer against the tension of the spring naturally averages the force produced by the varying voltage/current values over time. Conversely, polarity-sensitive meter movements vibrate uselessly if exposed to AC voltage or current, their needles oscillating rapidly about the zero mark, indicating the true (algebraic) average value of zero for a symmetrical waveform. When the "average" value of a waveform is referenced in this text, it will be assumed that the "practical" deﬂnition of average is intended unless otherwise speciﬂed. Another method of deriving an aggregate value for waveform amplitude is based on the waveform’s ability to do useful work when applied to a load resistance. Unfortunately, an AC measurement based on work performed by a waveform is not the same as that waveform’s "average" value, because the power dissipated by a given load (work performed per unit time) is not directly proportional to the magnitude of either the voltage or current impressed upon it. Rather, power is proportional to the square of the voltage or current applied to a resistance (P = E2/R, and P = I2R). Although the mathematics of such an amplitude measurement might not be straightforward, the utility of it is. Consider a bandsaw and a jigsaw, two pieces of modern woodworking equipment. Both types of saws cut with a thin, toothed, motor-powered metal blade to cut wood. But while the bandsaw uses a continuous motion of the blade to cut, the jigsaw uses a back-and-forth motion. The comparison of alternating current (AC) to direct current (DC) may be likened to the comparison of these two saw types: Bandsaw blade motion wood Jigsaw wood blade motion (analogous to DC) (analogous to AC) The problem of trying to describe the changing quantities of AC voltage or current in a single, aggregate measurement is also present in this saw analogy: how might we express the speed of a jigsaw blade? A bandsaw blade moves with a constant speed, similar to the way DC voltage pushes or DC current moves with a constant magnitude. A jigsaw blade, on the other hand, moves back and forth, its blade speed constantly changing. What is more, the back-and-forth motion of any two jigsaws may not be of the same type, depending on the mechanical design 309 of the saws. One jigsaw might move its blade with a sine-wave motion, while another with a triangle-wave motion. To rate a jigsaw based on its peak blade speed would be quite misleading when comparing one jigsaw to another (or a jigsaw with a bandsaw!). Despite the fact that these diﬁerent saws move their blades in diﬁerent manners, they are equal in one respect: they all cut wood, and a quantitative comparison of this common function can serve as a common basis for which to rate blade speed. Picture a jigsaw and bandsaw side-by-side, equipped with identical blades (same tooth pitch, angle, etc.), equally capable of cutting the same thickness of the same type of wood at the same rate. We might say that the two saws were equivalent or equal in their cutting capacity. Might this comparison be used to assign a "bandsaw equivalent" blade speed to the jigsaw’s back-and-forth blade motion; to relate the wood-cutting eﬁectiveness of one to the other? This is the general idea used to assign a "DC equivalent" measurement to any AC voltage or current: whatever magnitude of DC voltage or current would produce the same amount of heat energy dissipation through an equal resistance: 10 V RMS 10 V 5 A RMS 5 A RMS 5 A 5 A 2 W 50 W power dissipated Equal power dissipated through equal resistance loads 2 W 50 W power dissipated Suppose we were to wrap a coil of insulated wire around a loop of ferromagnetic material and energize this coil with an AC voltage source: iron core wire coil As an inductor, we would expect this iron-core coil to oppose the applied voltage with its inductive reactance, limiting current through the coil as predicted by the equations XL = 2…fL and I=E/X (or I=E/Z). For the purposes of this example, though, we need to take a more 310 detailed look at the interactions of voltage, current, and magnetic ux in the device. Kirchhoﬁ’s voltage law describes how the algebraic sum of all voltages in a loop must equal zero. In this example, we could apply this fundamental law of electricity to describe the respective voltages of the source and of the inductor coil. Here, as in any one-source, one-load circuit, the voltage dropped across the load must equal the voltage supplied by the source, assuming zero voltage dropped along the resistance of any connecting wires. In other words, the load (inductor coil) must produce an opposing voltage equal in magnitude to the source, in order that it may balance against the source voltage and produce an algebraic loop voltage sum of zero. From where does this opposing voltage arise? If the load were a resistor, the opposing voltage would originate from the "friction" of electrons owing through the resistance of the resistor. With a perfect inductor (no resistance in the coil wire), the opposing voltage comes from another mechanism: the reaction to a changing magnetic ux in the iron core. Michael Faraday discovered the mathematical relationship between magnetic ux (') and induced voltage with this equation: e = N dF dt Where, e = (Instantaneous) induced voltage in volts N = F = t = Number of turns in wire coil (straight wire = 1) Magnetic flux in Webers Time in seconds The instantaneous voltage (voltage dropped at any instant in time) across a wire coil is equal to the number of turns of that coil around the core (N) multiplied by the instantaneous rate-ofchange in magnetic ux (d'/dt) linking with the coil. Graphed, this shows itself as a set of sine waves (assuming a sinusoidal voltage source), the ux wave 90o lagging behind the voltage wave: e = voltage F = magnetic flux e F Magnetic ux through a ferromagnetic material is analogous to current through a conductor: it must be motivated by some force in order to occur. In electric circuits, this motivating force is voltage (a.k.a. electromotive force, or EMF). In magnetic "circuits," this motivating force is magnetomotive force, or mmf. Magnetomotive force (mmf) and magnetic ux (') are related to each other by a property of magnetic materials known as reluctance (the latter quantity symbolized by a strange-looking letter "R"): 311 A comparison of "Ohm’s Law" for electric and magnetic circuits: E = IR Electrical mmf = F´ Magnetic In our example, the mmf required to produce this changing magnetic ux (') must be supplied by a changing current through the coil. Magnetomotive force generated by an electromagnet coil is equal to the amount of current through that coil (in amps) multiplied by the number of turns of that coil around the core (the SI unit for mmf is the amp-turn). Because the mathematical relationship between magnetic ux and mmf is directly proportional, and because the mathematical relationship between mmf and current is also directly proportional (no rates-of-change present in either equation), the cur
rent through the coil will be in-phase with the ux wave: e = voltage F = magnetic flux i = coil current e F i This is why alternating current through an inductor lags the applied voltage waveform by 90o: because that is what is required to produce a changing magnetic ux whose rate-of-change produces an opposing voltage in-phase with the applied voltage. Due to its function in providing magnetizing force (mmf) for the core, this current is sometimes referred to as the magnetizing current. It should be mentioned that the current through an iron-core inductor is not perfectly sinusoidal (sine-wave shaped), due to the nonlinear B/H magnetization curve of iron. In fact, if the inductor is cheaply built, using as little iron as possible, the magnetic ux density might reach high levels (approaching saturation), resulting in a magnetizing current waveform that looks something like this: e = voltage F = magnetic flux i = coil current e F i When a ferromagnetic material approaches magnetic ux saturation, disproportionately greater levels of magnetic ﬂeld force (mmf) are required to deliver equal increases in magnetic ﬂeld ux ('). Because mmf is proportional to current through the magnetizing coil (mmf = NI, where "N" 312 is the number of turns of wire in the coil and "I" is the current through it), the large increases of mmf required to supply the needed increases in ux results in large increases in coil current. Thus, coil current increases dramatically at the peaks in order to maintain a ux waveform that isn’t distorted, accounting for the bell-shaped half-cycles of the current waveform in the above plot. The situation is further complicated by energy losses within the iron core. The eﬁects of hysteresis and eddy currents conspire to further distort and complicate the current waveform, making it even less sinusoidal and altering its phase to be lagging slightly less than 90o behind the applied voltage waveform. This coil current resulting from the sum total of all magnetic eﬁects in the core (d'/dt magnetization plus hysteresis losses, eddy current losses, etc.) is called the exciting current. The distortion of an iron-core inductor’s exciting current may be minimized if it is designed for and operated at very low ux densities. Generally speaking, this requires a core with large cross-sectional area, which tends to make the inductor bulky and expensive. For the sake of simplicity, though, we’ll assume that our example core is far from saturation and free from all losses, resulting in a perfectly sinusoidal exciting current. As we’ve seen already in the inductors chapter, having a current waveform 90o out of phase with the voltage waveform creates a condition where power is alternately absorbed and returned to the circuit by the inductor. If the inductor is perfect (no wire resistance, no magnetic core losses, etc.), it will dissipate zero power. Let us now consider the same inductor device, except this time with a second coil wrapped around the same iron core. The ﬂrst coil will be labeled the primary coil, while the second will be labeled the secondary: iron core wire coil wire coil If this secondary coil experiences the same magnetic ux change as the primary (which it should, assuming perfect containment of the magnetic ux through the common core), and has the same number of turns around the core, a voltage of equal magnitude and phase to the applied voltage will be induced along its length. In the following graph, the induced voltage waveform is drawn slightly smaller than the source voltage waveform simply to distinguish one from the other: 313 ep = primary coil voltage es = secondary coil voltage F = magnetic flux ip = primary coil current ep es F ip 314 Chapter 16 Electronics (NOTE TO SELF: Mark: I have very little idea of how to make this ow, ﬂt in or even how best to explain any of it. All the content in here is just trawled from other GFDL projects: www.wikipedia.com www.wikibooks.com The syllabus document has NO meaningful information on this stuﬁ.) Electronics: 16.1 capacitive and inductive circuits 16.1.1 A capacitor (NOTE TO SELF: If we are going to talk of capacitive circuits we need a deﬂnition of capacitor.) A capacitor (historically known as a "condenser") is a device that stores energy in an electric ﬂeld, by accumulating an internal imbalance of electric charge. It is made of two conductors separated by a dielectric (insulator). The problem of two parallel plates with a uniform electric ﬂeld between them is a capacitor. When voltage exists one end of the capacitor is getting drained and the other end is getting ﬂlled with charge. This is known as charging. Charging creates a charge imbalance between the two plates and creates a reverse voltage that stops the capacitor from charging. This is why when capacitors are ﬂrst connected to voltage charge ows only to stop as the capacitor becomes charged. When a capacitor is charged current stops owing and it becomes an open circuit. It is as if the capacitor gained inﬂnite resistance. Just as the capacitor charges it can be discharged. 16.1.2 An inductor (NOTE TO SELF: If we are going to talk of inductive circuits we need a deﬂnition of a inductor) An inductor is a device which stores energy in a magnetic ﬂeld. Inductors are formed of a coil of conductive material. When current ows through the wire it creates a magnetic ﬂeld which exists inside the coil. When the current stops the magnetic ﬂeld gets less, but we have learnt that a changing magnetic ﬂeld induces a current in a wire. So when the current turns oﬁ the magnetic ﬂeld decreases inducing another current in the wire. As the ﬂeld decreases in strength so does the induced magnetic ﬂeld. Normally they are made of copper wire, but not always (Example: aluminum wire, or spiral pattern etched on circuit board). The material around and within the coil aﬁects its properties; 315 common types are air-core (only a coil of wire), iron-core, and ferrite core. Iron and ferrite types are more e–cient because they conduct the magnetic ﬂeld much better than air; of the two, ferrite is more e–cient because stray electricity cannot ow through it. Interesting Fact: Some inductors have more than a core, which is just a rod the coil is formed about. Some are formed like transformers, using two E-shaped pieces facing each other, the wires wound about the central leg of the E’s. The E’s are made of laminated iron/steel or ferrite. Important qualities of an inductor There are several important properties for an inductor. * Current carrying capacity is determined by wire thickness. * Q, or quality, is determined by the uniformity of the windings, as well as the core material and how thoroughly it surrounds the coil. * Last but not least, the inductance of the coil. The inductance is determined by several factors. * coil shape: short and squat is best * core material * windings: winding in opposite directions will cancel out the inductance eﬁect, and you will have only a resistor. 16.2 ﬂlters and signal tuning (NOTE TO SELF: I think this relies on an understanding of second order ODEs and thats beyond the scope of the maths syllabus - we can put something high level but there is no way they’ll understand it properly - surely we should teach as little phenomonology as possible - the waves chapter has a ton of it already) 16.3 active circuit elements, diode, LED and ﬂeld eﬁect transistor, operational ampliﬂer 16.3.1 Diode A diode functions as the electronic version of a one-way valve. By restricting the direction of movement of charge carriers, it allows an electric current to ow in one direction, but blocks it in the opposite direction. It is a one-way street for current. 316 Diode behavior is analogous to the behavior of a hydraulic device called a check valve. A check valve allows uid ow through it in one direction only: Check valves are essentially pressure-operated devices: they open and allow ow if the pressure across them is of the correct "polarity" to open the gate (in the analogy shown, greater uid pressure on the right than on the left). If the pressure is of the opposite "polarity," the pressure diﬁerence across the check valve will close and hold the gate so that no ow occurs. Like check valves, diodes are essentially "pressure-" operated (voltage-operated) devices. The essential diﬁerence between forward-bias and reverse-bias is the polarity of the voltage dropped across the diode. Let’s take a closer look at the simple battery-diode-lamp circuit shown earlier, this time investigating voltage drops across the various components: 317 When the diode is forward-biased and conducting current, there is a small voltage dropped across it, leaving most of the battery voltage dropped across the lamp. When the battery’s polarity is reversed and the diode becomes reverse-biased, it drops all of the battery’s voltage and leaves none for the lamp. If we consider the diode to be a sort of self-actuating switch (closed in the forward-bias mode and open in the reverse-bias mode), this behavior makes sense. The most substantial diﬁerence here is that the diode drops a lot more voltage when conducting than the average mechanical switch (0.7 volts versus tens of millivolts). This forward-bias voltage drop exhibited by the diode is due to the action of the depletion region formed by the P-N junction under the inuence of an applied voltage. When there is no voltage applied across a semiconductor diode, a thin depletion region exists around the region of the P-N junction, preventing current through it. The depletion region is for the most part devoid of available charge carriers and so acts as an insulator: 318 16.3.2 LED A light-emitting diode (LED) is a semiconductor device that emits light when charge ows in the correct direction through it. If you apply a voltage to force current to ow in the direction the LED allows it will light up. This notation of having two small arrows pointing away from the device is common to the schemati
c symbols of all light-emitting semiconductor devices. Conversely, if a device is lightactivated (meaning that incoming light stimulates it), then the symbol will have two small arrows pointing toward it. It is interesting to note, though, that LEDs are capable of acting as lightsensing devices: they will generate a small voltage when exposed to light, much like a solar cell on a small scale. This property can be gainfully applied in a variety of light-sensing circuits. The color depends on the semiconducting material used to construct the LED, and can be in the near-ultraviolet, visible or infrared part of the electromagnetic spectrum. Interesting Fact: Nick Holonyak Jr. (1928 ) of the University of Illinois at Urbana-Champaign developed the ﬂrst practical visible-spectrum LED in 1962. 319 Physical function Because LEDs are made of diﬁerent chemical substances than normal rectifying diodes, their forward voltage drops will be diﬁerent. Typically, LEDs have much larger forward voltage drops than rectifying diodes, anywhere from about 1.6 volts to over 3 volts, depending on the color. Typical operating current for a standard-sized LED is around 20 mA. When operating an LED from a DC voltage source greater than the LED’s forward voltage, a series-connected "dropping" resistor must be included to prevent full source voltage from damaging the LED. Consider this example circuit: With the LED dropping 1.6 volts, there will be 4.4 volts dropped across the resistor. Sizing the resistor for an LED current of 20 mA is as simple as taking its voltage drop (4.4 volts) and dividing by circuit current (20 mA), in accordance with Ohm’s Law (R=E/I). This gives us a ﬂgure of 220 ?. Calculating power dissipation for this resistor, we take its voltage drop and multiply by its current (P=IE), and end up with 88 mW, well within the rating of a 1/8 watt resistor. Higher battery voltages will require larger-value dropping resistors, and possibly higher-power rating resistors as well. Consider this example for a supply voltage of 24 volts: Here, the dropping resistor must be increased to a size of 1.12 k? in order to drop 22.4 volts at 20 mA so that the LED still receives only 1.6 volts. This also makes for a higher resistor power dissipation: 448 mW, nearly one-half a watt of power! Obviously, a resistor rated for 1/8 watt power dissipation or even 1/4 watt dissipation will overheat if used here. Dropping resistor values need not be precise for LED circuits. Suppose we were to use a 1 k? resistor instead of a 1.12 k? resistor in the circuit shown above. The result would be a slightly greater circuit current and LED voltage drop, resulting in a brighter light from the LED and slightly reduced service life. A dropping resistor with too much resistance (say, 1.5 k? instead of 1.12 k?) will result in less circuit current, less LED voltage, and a dimmer light. LEDs are quite tolerant of variation in applied power, so you need not strive for perfection in sizing the dropping resistor. Also because of their unique chemical makeup, LEDs have much, much lower peak-inverse voltage (PIV) ratings than ordinary rectifying diodes. A typical LED might only be rated at 5 volts in reverse-bias mode. Therefore, when using alternating current to power an LED, you should connect a protective rectifying diode in series with the LED to prevent reverse breakdown every other half-cycle: 320 Light emission The wavelength of the light emitted, and therefore its color, depends on the materials forming the pn junction. A normal diode, typically made of silicon or germanium, emits invisible far-infrared light (so it can’t be seen), but the materials used for an LED have emit light corresponding to near-infrared, visible or near-ultraviolet frequencies. Considerations in use Unlike incandescent light bulbs, which can operate with either AC or DC, LEDs require a DC supply of the correct electrical polarity. When the voltage across the pn junction is in the correct direction, a signiﬂcant current ows and the device is said to be forward biased. The voltage across the LED in this case is ﬂxed for a given LED and is proportional to the energy of the emitted photons. If the voltage is of the wrong polarity, the device is said to be reverse biased, very little current ows, and no light is emitted. Because the voltage versus current characteristics of an LED are much like any diode, they can be destroyed by connecting them to a voltage source much higher than their turn on voltage. The voltage drop across a forward biased LED increases as the amount of light emitted increases because of the optical power being radiated. One consequence is that LEDs of the same type can be readily operated in parallel. The turn-on voltage of an LED is a function of the color, a higher forward drop is associated with emitting higher energy (bluer) photons. The reverse voltage that most LEDs can sustain without damage is usually only a few volts. Some LED units contain two diodes, one in each direction and each a diﬁerent color (typically red and green), allowing two-color operation or a range of colors to be created by altering the percentage of time the voltage is in each polarity. LED materials LED development began with infrared and red devices made with gallium arsenide. Advances in materials science have made possible the production of devices with ever shorter wavelengths, producing light in a variety of colors. Conventional LEDs are made from a variety of inorganic minerals, producing the following colors: aluminium gallium arsenide (AlGaAs): red and infrared gallium arsenide/phosphide (GaAsP): red, orange-red, orange, and yellow gallium nitride (GaN): green, pure green (or emerald green), and blue gallium phosphide (GaP): red, yellow and green zinc selenide (ZnSe): blue † † † † † 321 indium gallium nitride (InGaN): bluish-green and blue silicon carbide (SiC): blue diamond (C): ultraviolet † † † silicon (Si) - under development † (NOTE TO SELF: The above list is taken from public sources, but at least one LED given as blue does not produce blue light. (There is a good chance that almost none do, because of the higher frequency of blue.) This is a common problem in daily life due to the majority of mankind being ignorant of colour theory and conating blue with light blue with cyan, the latter often called "sky blue". A cyan LED may be distinguished from a blue LED in that adding a yellow phosphor to the output makes green, rather than white light. And often aqua is called blue-green when in actuality the latter is cyan, and light cyan-green would be aqua. What adds to the confusion is that cyan LEDs are enclosed in blue plastic. A great amount of work is needed to dispel these intuitive myths of colour mixing before accurate descriptions of physical phenomena and their production can happen. - This needs to be sorted out) Blue and white LEDs and Other colors Commercially viable blue LEDs based invented by Shuji Nakamura while working in Japan at Nichia Corporation in 1993 and became widely available in the late 1990s. They can be added to existing red and green LEDs to produce white light. Most "white" LEDs in production today use a 450nm 470nm blue GaN (gallium nitride) LED covered by a yellowish phosphor coating usually made of cerium doped yttrium aluminium garnet (YAG:Ce) crystals which have been powdered and bound in a type of viscous adhesive. The LED chip emits blue light, part of which is converted to yellow by the YAG:Ce. The single crystal form of YAG:Ce is actually considered a scintillator rather than a phosphor. Since yellow light stimulates the red and green receptors of the eye, the resulting mix of blue and yellow light gives the appearance of white. The newest method used to produce white light LEDs uses no phosphors at all and is based on homoepitaxially grown zinc selenide (ZnSe) on a ZnSe substrate which simultaneously emits blue light from its active region and yellow light from the substrate. Other colors Recent color developments include pink and purple. They consist of one or two phosphor layers over a blue LED chip. The ﬂrst phosphor layer of a pink LED is a yellow glowing one, and the second phosphor layer is either red or orange glowing. Purple LEDs are blue LEDs with an orange glowing phosphor over the chip. Some pink LEDs have run into issues. For example, some are blue LEDs painted with uorescent paint or ﬂngernail polish that can wear oﬁ, and some are white LEDs with a pink phosphor or dye that unfortunately fades after a short tme. Ultraviolet, blue, pure green, white, pink and purple LEDs are relatively expensive compared to the more common reds, oranges, greens, yellows and infrareds and are thus less commonly used in commercial applications. The semiconducting chip is encased in a solid plastic lens, which is much tougher than the glass envelope of a traditional light bulb or tube. The plastic may be colored, but this is only for cosmetic reasons and does not aﬁect the color of the light emitted. 322 Operational parameters and e–ciency Most typical LEDs are designed to operate with no more than 30-60 milliwatts of electrical power. It is projected that by 2005, 10-watt units will be available. These devices will produce about as much light as a common 50-watt incandescent bulb, and will facilitate use of LEDs for general illumination needs. Interesting Fact: In September 2003 a new type of blue LED was demonstrated by the company Cree, Inc. to have 35% e–ciency at 20 mA. This produced a commercially packaged white light having 65 lumens per watt at 20 mA, becoming the brightest white LED commercially available at the time. Organic light-emitting diodes (OLEDs) If the emissive layer material of an LED is an organic compound, it is known as an Organic Light Emitting Diode (OLED). To function as a semiconductor, the organic emissive material must have conjugated pi bonds. The emissive material can be a small organic molecule in a crystalline phase,
or a polymer. Polymer materials can be exible; such LEDs are known as PLEDs or FLEDs. Compared with regular LEDs, OLEDs are lighter and polymer LEDs can have the added beneﬂt of being exible. Some possible future applications of OLEDs could be: Light sources Wall decorations Luminous cloth † † † LED applications Here is a list of known applications for LEDs, some of which are further elaborated upon in the following text: in general, commonly used as information indicators in various types of embedded systems (many of which are listed below) thin, lightweight message displays, e.g. in public information signs (at airports and railway stations, among other places) status indicators, e.g. on/oﬁ lights on professional instruments and consumers audio/video equipment infrared LEDs in remote controls (for TVs, VCRs, etc) clusters in tra–c signals, replacing ordinary bulbs behind colored glass car indicator lights and bicycle lighting; also for pedestrians to be seen by car tra–c † † † † † † 323 calculator and measurement instrument displays (seven segment displays), although now mostly replaced by LCDs red or yellow LEDs are used in indicator and [alpha]numeric displays in environments where night vision must be retained: aircraft cockpits, submarine and ship bridges, astronomy observatories, and in the ﬂeld, e.g. night time animal watching and military ﬂeld use red or yellow LEDs are also used in photographic darkrooms, for providing lighting which does not lead to unwanted exposure of the ﬂlm illumination, e.g. ashlights (a.k.a. torches, UK), and backlights for LCD screens signaling/emergency beacons and strobes movement sensors, e.g. in mechanical and optical computer mice and trackballs † † † † † † in LED printers, e.g. high-end color printers † LEDs oﬁer beneﬂts in terms of maintenance and safety. † † † † The typical working lifetime of a device, including the bulb, is ten years, which is much longer than the lifetimes of most other light sources. LEDs fail by dimming over time, rather than the abrupt burn-out of incandescent bulbs. LEDs give oﬁ less heat than incandescent light bulbs and are less fragile than uorescent lamps. Since an individual device is smaller than a centimetre in length, LED-based light sources used for illumination and outdoor signals are built using clusters of tens of devices. Because they are monochromatic, LED lights have great power advantages over white lights where a speciﬂc color is required. Unlike the white lights, the LED does not need a ﬂlter that absorbs most of the emitted white light. Colored uorescent lights are made, but they are not widely available. LED lights are inherently colored, and are available in a wide range of colors. One of the most recently introduced colors is the emerald green (bluish green, about 500 nm) that meets the legal requirements for tra–c signals and navigation lights. Interesting Fact: The largest LED display in the world is 36 metres high (118 feet), at Times Square, New York, U.S.A. There are applications that speciﬂcally require light that does not contain any blue component. Examples are photographic darkroom safe lights, illumination in laboratories where certain photo-sensitive chemicals are used, and situations where dark adaptation (night vision) must be preserved, such as cockpit and bridge illumination, observatories, etc. Yellow LED lights are a good choice to meet these special requirements because the human eye is more sensitive to yellow light. 324 16.3.3 Transistor The transistor is a solid state semiconductor device used for ampliﬂcation and switching, and has three terminals. The transistor itself does not amplify current though, which is a common misconception, but a small current or voltage applied to one terminal controls the current through the other two, hence the term transistor; a voltage- or current-controlled resistor. It is the key component in all modern electronics. In digital circuits, transistors are used as very fast electrical switches, and arrangements of transistors can function as logic gates, RAM-type memory and other devices. In analog circuits, transistors are essentially used as ampliﬂers. Transistor was also the common name in the sixties for a transistor radio, a pocket-sized portable radio that utilized transistors (rather than vacuum tubes) as its active electronics. This is still one of the dictionary deﬂnitions of transistor. The only functional diﬁerence between a PNP transistor and an NPN transistor is the proper biasing (polarity) of the junctions when operating. For any given state of operation, the current directions and voltage polarities for each type of transistor are exactly opposite each other. Bipolar transistors work as current-controlled current regulators. In other words, they restrict the amount of current that can go through them according to a smaller, controlling current. The main current that is controlled goes from collector to emitter, or from emitter to collector, depending on the type of transistor it is (PNP or NPN, respectively). The small current that controls the main current goes from base to emitter, or from emitter to base, once again depending on the type of transistor it is (PNP or NPN, respectively). According to the confusing standards of semiconductor symbology, the arrow always points against the direction of electron ow: 325 Bipolar transistors are called bipolar because the main ow of electrons through them takes place in two types of semiconductor material: P and N, as the main current goes from emitter to collector (or visa-versa). In other words, two types of charge carriers { electrons and holes { comprise this main current through the transistor. As you can see, the controlling current and the controlled current always mesh together through the emitter wire, and their electrons always ow against the direction of the transistor’s arrow. This is the ﬂrst and foremost rule in the use of transistors: all currents must be going in the proper directions for the device to work as a current regulator. The small, controlling current is usually referred to simply as the base current because it is the only current that goes through the base wire of the transistor. Conversely, the large, controlled current is referred to as the collector current because it is the only current that goes through the collector wire. The emitter current is the sum of the base and collector currents, in compliance with Kirchhoﬁ’s Current Law. If there is no current through the base of the transistor, it shuts oﬁ like an open switch and prevents current through the collector. If there is a base current, then the transistor turns on like a closed switch and allows a proportional amount of current through the collector. Collector current is primarily limited by the base current, regardless of the amount of voltage available to push it. The next section will explore in more detail the use of bipolar transistors as switching elements. Importance The transistor is considered by many to be one of the greatest discoveries or inventions in modern history, ranking with banking and the printing press. Key to the importance of the transistor in modern society is its ability to be produced in huge numbers using simple techniques, resulting in vanishingly small prices. Computer "chips" consist of millions of transistors and sell for rands, with per-transistor costs in the thousandths-of-cents. The low cost has meant that the transistor has become an almost universal tool for nonmechanical tasks. Whereas a common device, say a refrigerator, would have used a mechanical device for control, today it is often less expensive to simply use a few million transistors and the appropriate computer program to carry out the same task through "brute force". Today 326 transistors have replaced almost all electromechanical devices, most simple feedback systems, and appear in huge numbers in everything from computers to cars. Hand-in-hand with low cost has been the increasing move to "digitizing" all information. With transistorized computers oﬁering the ability to quickly ﬂnd (and sort) digital information, more and more eﬁort was put into making all information digital. Today almost all media in modern society is delivered in digital form, converted and presented by computers. Common "analog" forms of information such as television or newspapers spend the vast majority of their time as digital information, being converted to analog only for a small portion of the time. Interesting Fact: The transistor was invented at Bell Laboratories in December 1947 (ﬂrst demonstrated on December 23) by John Bardeen, Walter Houser Brattain, and William Bradford Shockley, who were awarded the Nobel Prize in physics in 1956. 16.3.4 The transistor as a switch Because a transistor’s collector current is proportionally limited by its base current, it can be used as a sort of current-controlled switch. A relatively small ow of electrons sent through the base of the transistor has the ability to exert control over a much larger ow of electrons through the collector. Suppose we had a lamp that we wanted to turn on and oﬁ by means of a switch. Such a circuit would be extremely simple: For the sake of illustration, let’s insert a transistor in place of the switch to show how it can control the ow of electrons through the lamp. Remember that the controlled current through a transistor must go between collector and emitter. Since it’s the current through the lamp that we want to control, we must position the collector and emitter of our transistor where the two contacts of the switch are now. We must also make sure that the lamp’s current will move against the direction of the emitter arrow symbol to ensure that the transistor’s junction bias will be correct: 327 In this example I happened to choose an NPN transistor. A PNP transistor could also have been chosen for the job, and its application would look like this: The choice between NPN and P
NP is really arbitrary. All that matters is that the proper current directions are maintained for the sake of correct junction biasing (electron ow going against the transistor symbol’s arrow). Going back to the NPN transistor in our example circuit, we are faced with the need to add something more so that we can have base current. Without a connection to the base wire of the transistor, base current will be zero, and the transistor cannot turn on, resulting in a lamp that is always oﬁ. Remember that for an NPN transistor, base current must consist of electrons owing from emitter to base (against the emitter arrow symbol, just like the lamp current). Perhaps the simplest thing to do would be to connect a switch between the base and collector wires of the transistor like this: If the switch is open, the base wire of the transistor will be left "oating" (not connected to anything) and there will be no current through it. In this state, the transistor is said to be cutoﬁ. If the switch is closed, however, electrons will be able to ow from the emitter through to the base of the transistor, through the switch and up to the left side of the lamp, back to the positive side of the battery. This base current will enable a much larger ow of electrons from the emitter through to the collector, thus lighting up the lamp. In this state of maximum circuit current, the transistor is said to be saturated. 328 Of course, it may seem pointless to use a transistor in this capacity to control the lamp. After all, we’re still using a switch in the circuit, aren’t we? If we’re still using a switch to control the lamp { if only indirectly { then what’s the point of having a transistor to control the current? Why not just go back to our original circuit and use the switch directly to control the lamp current? There are a couple of points to be made here, actually. First is the fact that when used in this manner, the switch contacts need only handle what little base current is necessary to turn the transistor on, while the transistor itself handles the majority of the lamp’s current. This may be an important advantage if the switch has a low current rating: a small switch may be used to control a relatively high-current load. Perhaps more importantly, though, is the fact that the current-controlling behavior of the transistor enables us to use something completely diﬁerent to turn the lamp on or oﬁ. Consider this example, where a solar cell is used to control the transistor, which in turn controls the lamp: Or, we could use a thermocouple to provide the necessary base current to turn the transistor on: 329 Even a microphone of su–cient voltage and current output could be used to turn the transistor on, provided its output is rectiﬂed from AC to DC so that the emitter-base PN junction within the transistor will always be forward-biased: The point should be quite apparent by now: any su–cient source of DC current may be used to turn the transistor on, and that source of current need only be a fraction of the amount of current needed to energize the lamp. Here we see the transistor functioning not only as a switch, but as a true ampliﬂer: using a relatively low-power signal to control a relatively large amount of power. Please note that the actual power for lighting up the lamp comes from the battery to the right of the schematic. It is not as though the small signal current from the solar cell, thermocouple, or microphone is being magically transformed into a greater amount of power. Rather, those small power sources are simply controlling the battery’s power to light up the lamp. Field-Eﬁect Transistor (FET) (NOTE TO SELF: Schematic can be found under GFDL on wikipedia) The schematic symbols for p- and n-channel MOSFETs. The symbols to the right include an extra terminal for the transistor body (allowing for a seldom-used channel bias) whereas in those to the left the body is implicitly connected to the source. The most common variety of ﬂeld-eﬁect transistors, the enhancement-mode MOSFET (metaloxide semiconductor ﬂeld-eﬁect transistor) consists of a unipolar conduction channel and a metal gate separated from the main conduction channel by a thin layer of (SiO2) glass. This is why an alternative name for the FET is ’unipolar transistor.’ When a potential diﬁerence (of the proper polarity) is impressed across gate and source, charge carriers are introduced to the channel, making it conductive. The amount of this current can be modulated, or (nearly) completely turned oﬁ, by varying the gate potential. Because the gate is insulated, no DC current ows to or from the gate electrode. This lack of a gate current and the ability of the MOSFET to act like a switch, allows particularly e–cient digital circuits to be created, with very low power consumption at low frequencies. The power consumption increases markedly with frequency, because the capacitive loading of the FET control terminal takes more energy to slew at higher frequencies, in direct proportion to the frequency. Hence, MOSFETs have become the dominant technology used in computing hardware such as microprocessors and memory devices such as RAM. Bipolar transistors are more rugged and hence more useful for low-impedance loads and inductively reactive (e.g. motor) loads. Power MOSFETs become less conductive with increasing temperature and can therefore be applied in shunt, to increase current capacity, unlike the bipolar transistor, which has a negative 330 temperature coe–cient of resistance, and is therefore prone to thermal runaway. The downside of this is that, while the power FET can protect itself from overheating by diminishing the current through it, high temperatures need to be avoided by using a larger heat sink than for an equivalent bipolar device. Macroscopic FET power transistors are actually composed of many little transistors. They are stacked (on-chip) to increase breakdown potential and paralleled to reduce Ron, i.e. allowing for more current, bussing the gates to provide a single control (gate) terminal. The depletion mode FET is a little diﬁerent. It uses a back-biased diode for the control terminal, which presents a capacitive load to the driving circuit in normal operation. With the gate tied to the source, a DFET is fully on. Changing the potential of a DFET (pulling an Nchannel gate downward, for example) will turn it oﬁ, i.e. ’deplete’ the channel (drain-source) of charge carriers. MOSFETs, formerly called IGFETs (for Insulated Gate Field-Eﬁect Transistor) can be depletion-mode, enhancement-mode, or mixed-mode, but are almost always enhancement mode in modern commercial practice. This means that, with the source and gate tied together (thus equipotential) the channel will be oﬁ (high impedance or non-conducting). The n-channel device (reverse for P-channel), like in the DFET, is turned on by raising the potential of the gate. Typically, the gate on a MOSFET will withstand +-20V, relative to the source terminal. If one were to raise the gate potential of an n-channel device without limiting the current to a few milliamps, one would destroy the gate diode, like any other small diode. Why do we typically think of n-channel devices as the default? In silicon devices, the ones that use majority carriers that are electrons, rather than holes, are slightly faster and can carry more current than their P-type counterparts. The same is true in GaAs devices. The FET is simpler in concept than the bipolar transistor and can be constructed from a wide range of materials. The most common use of MOSFET transistors today is the CMOS (complementary metallic oxide semiconductor) integrated circuit which is the basis for most digital electronic devices. These use a totem-pole arrangement where one transistor (either the pull-up or the pull-down) is on while the other is oﬁ. Hence, there is no DC drain, except during the transition from one state to the other, which is very short. As mentioned, the gates are capacitive, and the charging and discharging of the gates each time a transistor switches states is the primary cause of power drain. The C in CMOS stands for ’complementary.’ The pull-up is a P-channel device (using holes for the mobile carrier of charge) and the pull-down is N-channel (electron carriers). This allows busing of the control terminals, but limits the speed of the circuit to that of the slower P device (in silicon devices). The bipolar solutions to push-pull include ’cascode’ using a current source for the load. Circuits that utilize both unipolar and bipolar transistors are called Bi-Fet. A recent development is called ’vertical P.’ Formerly, BiFet chip users had to settle for relatively poor (horizontal) P-type FET devices. This is no longer the case and allows for quieter and faster analog circuits. A clever variant of the FET is the dual-gate device. This allows for two opportunities to turn the device oﬁ, as opposed to the dual-base (bipolar) transistor which presents two opportunities to turn the device on. FETs can switch signals of either polarity, if their amplitude is signiﬂcantly less than the gate swing, as the devices (especially the parasitic diode-free DFET) are basically symmetrical. This means that FETs are the most suitable type for analog multiplexing. With this concept, one can construct a solid-state mixing board, for example. The power MOSFET has a ’parasitic diode’ (back-biased) normally shunting the conduction channel that has half the current capacity of the conduction channel. Sometimes this is useful in driving dual-coil magnetic circuits (for spike protection), but in other cases it causes problems. 331 The high impedance of the FET gate makes it rather vulnerable to electrostatic damage, though this is not usually a problem after the device has been installed. A more recent device for power control is the insulated-gate bipolar transistor, or IGBT. This has a control structure akin to a MOSFET coupled with a bipolar-like main cond
uction channel. These have become quite popular. 16.4 principles of digital electronics logical gates, counting circuits 16.4.1 Electronic logic gates The simplest form of electronic logic is diode logic (DL). This allows AND and OR gates to be built, but not inverters, and so is an incomplete form of logic. To built a complete logic system, valves or transistors can be used. The simplest family of logic gates using bipolar transistors is called resistor-transistor logic, or RTL. Unlike diode logic gates, RTL gates can be cascaded indeﬂnitely to produce more complex logic functions. These gates were used in early integrated circuits. For higher speed, the resistors used in RTL were replaced by diodes, leading to diodetransistor logic, or DTL. It was then discovered that one transistor could do the job of two diodes in the space of one diode, so transistor-transistor logic, or TTL, was created. In some types of chip, to reduce size and power consumption still further, the bipolar transistors were replaced with complementary ﬂeld-eﬁect transistors (MOSFETs), resulting in complementary metal-oxide-semiconductor (CMOS) logic. For small-scale logic, designers now use prefabricated logic gates from families of devices such as the TTL 7400 series invented by Texas Instruments and the CMOS 4000 series invented by RCA, and their more recent descendants. These devices usually contain transistors with multiple emitters, used to implement the AND function, which are not available as separate components. Increasingly, these ﬂxed-function logic gates are being replaced by programmable logic devices, which allow designers to pack a huge number of mixed logic gates into a single integrated circuit. Electronic logic gates diﬁer signiﬂcantly from their relay-and-switch equivalents. They are much faster, consume much less power, and are much smaller (all by a factor of a million or more in most cases). Also, there is a fundamental structural diﬁerence. The switch circuit creates a continuous metallic path for current to ow (in either direction) between its input and its output. The semiconductor logic gate, on the other hand, acts as a high-gain voltage ampliﬂer, which sinks a tiny current at its input and produces a low-impedance voltage at its output. It is not possible for current to ow between the output and the input of a semiconductor logic gate. Another important advantage of standardised semiconductor logic gates, such as the 7400 and 4000 families, is that they are cascadable. This means that the output of one gate can be wired to the inputs of one or several other gates, and so on ad inﬂnitum, enabling the construction of circuits of arbitrary complexity without requiring the designer to understand the internal workings of the gates. In practice, the output of one gate can only drive a ﬂnite number of inputs to other gates, a number called the ’fanout limit’, but this limit is rarely reached in the newer CMOS logic circuits, as compared to TTL circuits. Also, there is always a delay, called the ’propagation delay’, from a change an input of a gate to the corresponding change in its output. When gates are cascaded, the total propagation delay is approximately the sum of the individual delays, an eﬁect which can become a problem in high-speed circuits. The US symbol for an AND gate is: AND symbol and the IEC symbol is AND symbol. The US circuit symbol for an OR gate is: OR symbol and the IEC symbol is: OR symbol. 332 The US circuit symbol for a NOT gate is: NOT symbol and the IEC symbol is: NOT symbol. In electronics a NOT gate is more commonly called an inverter. The circle on the symbol is called a bubble, and is generally used in circuit diagrams to indicate an inverted input or output. The US circuit symbol for a NAND gate is: NAND symbol and the IEC symbol is: NAND symbol. The US circuit symbol for a NOR gate is: NOR symbol and the IEC symbol is: NOR symbol. In practice, the cheapest gate to manufacture is usually the NAND gate. Additionally, Charles Peirce showed that NAND gates alone (as well as NOR gates alone) can be used to reproduce all the other logic gates. Two more gates are the exclusive-OR or XOR function and its inverse, exclusive-NOR or XNOR. Exclusive-OR is true only when exactly one of its inputs is true. In practice, these gates are built from combinations of simpler logic gates. The US circuit symbol for an XOR gate is: XOR symbol and the IEC symbol is: XOR symbol. 16.5 Counting circuits An arithmetic and logical unit (ALU) adder provides the basic functionality of arithmetic operations within a computer, and is a signiﬂcant component of the arithmetic and logical unit. Adders are composed of half adders and full adders, which add two-bit binary pairs, and ripple carry adders and carry look ahead adders which do addition operations to a series of binary numbers. (NOTE TO SELF: Pictures on wikipedia under GFDL) 16.5.1 Half Adder A half adder is a logical circuit that performs an addition operation on two binary digits. The half adder produces a sum and a carry value which are both binary digits. Sum(s) = A xor B Cot(c) = A and B Half adder circuit diagram Half adder circuit diagram Following is the logic table for a half adder: A B Sum Cot 16.5.2 Full adder A full adder is a logical circuit that performs an addition operation on three binary digits. The full adder produces a sum and carry value, which are both binary digits. Sum = (A xor B) xor Cin Cot = (A nand B) nand (Cin nand (A xor B)) Full adder circuit diagram Full adder circuit diagram 333 A B Cin Sum Cot Quantity Symbol Unit S.I. Units Direction Units or Table 16.1: Units used in Electronics 334 Chapter 17 The Atom Atoms are the building blocks of matter. They are the basis of all the structures and organisms in the universe. The planets, the sun, grass and trees, the air we breathe, and people are all made up of atoms. 17.1 Models of the Atom 17.2 Structure of the Atom Atoms are very small and cannot be seen with the naked eye. They consist of two main parts: the positively charged nucleus at the centre and the negatively charged elementary particles called electrons which surround the nucleus in their orbitals. (Elementary particle means that the electron cannot be broken down to anything smaller and can be thought of as a point particle.) The nucleus of an atom is made up of a collection of positively charged protons and neutral particles called neutrons. interesting fact: the neutrons and protons are not elementary particles. They are actually made up of even smaller particles called quarks. Both protons and neutrons are made of three quarks each. There are all sorts of other particles composed of quarks which nuclear physicists study using huge detectors - you can ﬂnd out more about this by reading the essay in Chapter ??. (NOTE TO SELF: Insert diagram of atomic structure - see lab posters) Atoms are electrically neutral which means that they have the same number of negative electrons as positive protons. The number of protons in an atom is called the atomic number which is sometimes also called Z. (NOTE TO SELF: check A and Z) The atomic number is what distinguishes the diﬁerent chemical elements in the Periodic table from each other. In fact, the elements are listed on the Periodic table in order of their atomic numbers. For example, the ﬂrst element, hydrogen (H), has one proton whereas the sixth element, carbon (C) has 6 protons. Atoms with the same number of protons (atomic number) share physical properties and show similar chemical behaviour. The number of neutrons plus protons in the nucleus is called the atomic mass of the atom. 335 17.3 Isotopes Two atoms are considered to be the same element if they have the same number of protons (atomic number). However, they do not have to have the same number of neutrons or overall atomic mass. Atoms which have the same number of protons but diﬁerent numbers of neutrons are called isotopes. For example, the hydrogen atom has one proton and no neutrons. Therefore its atomic number is Z=1 and atomic mass is A=1. If a neutron is added to the hydrogen nucleus, then a new atom is formed with atomic mass A=2 but atomic number is still Z=1. This atom is called deuterium and is an isotope of hydrogen. 17.4 Energy quantization and electron conﬂguration 17.5 Periodicity of ionization energy to support atom ar- rangement in Periodic Table 17.6 Successive ionisation energies to provide evidence for arrangement of electrons into core and valence [Brink and Jones sections: de Broglie - matter shows particle and wave characteristics, proved by Davisson and Germer. Shroedinger and Heisenberg developed this model into quantum mechanics] The nucleus (atomic nucleus) is the center of an atom. It is composed of one or more protons and usually some neutrons as well. The number of protons in an atom’s nucleus is called the atomic number, and determines which element the atom is (for example hydrogen, carbon, oxygen, etc.). Though the positively charged protons exert a repulsive electromagnetic force on each other, the distances between nuclear particles are small enough that the strong interaction (which is stronger than the electromagnetic force but decreases more rapidly with distance) predominates. (The gravitational attraction is negligible, being a factor 1036 weaker than this electromagnetic repulsion.) The discovery of the electron was the ﬂrst indication that the atom had internal structure. This structure was initially imagined according to the "raisin cookie" or "plum pudding" model, in which the small, negatively charged electrons were embedded in a large sphere containing all the positive charge. Ernest Rutherford and Marsden, however, discovered in 1911 that alpha particles from a radium source were sometimes scattered backwards from a gold foil, which led to the acceptance of a planetary model, in which the electrons orbited a tiny nucleus in the same way that the planets orbit the sun. Interesti
ng Fact: The word atom is derived from the Greek atomos, indivisible, from a-, not, and tomos, a cut. An atom is the smallest portion into which a chemical element can be divided while still retaining its properties. Atoms are the basic constituents of molecules and ordinary matter. Atoms are composed of subatomic particles. Atoms are composed mostly of empty space, but also of smaller subatomic particles. At the center of the atom is a tiny positive nucleus composed of nucleons (protons and neutrons). The rest of the atom contains only the fairly exible electron shells. Usually atoms are electrically neutral with as many electrons as protons. 336 Atoms are generally classiﬂed by their atomic number, which corresponds to the number of protons in the atom. For example, carbon atoms are those atoms containing 6 protons. All atoms with the same atomic number share a wide variety of physical properties and exhibit the same chemical behavior. The various kinds of atoms are listed in the Periodic table. Atoms having the same atomic number, but diﬁerent atomic masses (due to their diﬁerent numbers of neutrons), are called isotopes. The simplest atom is the hydrogen atom, having atomic number 1 and consisting of one proton and one electron. It has been the subject of much interest in science, particularly in the early development of quantum theory. In The chemical behavior of atoms is largely due to interactions between the electrons. particular the electrons in the outermost shell, called the valence electrons, have the greatest inuence on chemical behavior. Core electrons (those not in the outer shell) play a role, but it is usually in terms of a secondary eﬁect due to screening of the positive charge in the atomic nucleus. There is a strong tendency for atoms to completely ﬂll (or empty) the outer electron shell, which in hydrogen and helium has space for two electrons, and in all other atoms has space for eight. This is achieved either by sharing electrons with neighboring atoms or by completely removing electrons from other atoms. When electrons are shared a covalent bond is formed between the two atoms. Covalent bonds are the strongest type of atomic bond. When one or more electrons are completely removed from one atom by another, ions are formed. Ions are atoms that possess a net charge due to an imbalance in the number of protons and electrons. The ion that stole the electron(s) is called an anion and is negatively charged. The atom that lost the electron(s) is called a cation and is positively charged. Cations and anions are attracted to each other due to coulombic forces between the positive and negative charges. This attraction is called ionic bonding and is weaker than covalent bonding. As mentioned above covalent bonding implies a state in which electrons are shared equally between atoms, while ionic bonding implies that the electrons are completely conﬂned to the anion. Except for a limited number of extreme cases, neither of these pictures is completely accurate. In most cases of covalent bonding, the electron is unequally shared, spending more time around the more electronegative atom, resulting in the covalent bond having some ionic character. Similarly, in ionic bonding the electrons often spend a small fraction of time around the more electropositive atom, resulting in some covalent character for the ionic bond. [edit] Models of the atom * Democritus’ shaped-atom model (for want of a better name) * The plum pudding model * Cubical atom * The Bohr model * The quantum mechanical model The Plum pudding model of the atom was made after the discovery of the electron but before the discovery of the proton or neutron. In it, the atom is envisioned as electrons surrounded by a soup of positive charge, like plums surrounded by pudding. This model was disproved by an experiment by Ernest Rutherford when he discovered the nucleus of the atom. The Bohr Model is a physical model that depicts the atom as a small positively charged nucleus with electrons in orbit at diﬁerent levels, similar in structure to the solar system. Because of its simplicity, the Bohr model is still commonly used and taught today. In the early part of the 20th century, experiments by Ernest Rutherford and others had established that atoms consisted of a small dense positively charged nucleus surrounded by orbiting negatively charged electrons. However classical physics at that time was unable to explain why the orbiting electrons did not spiral into the nucleus. The simplest possible atom is hydrogen, which consists of a nucleus and one orbiting electron. Since the nucleus is positive and the electron are oppositely charged they will attract one another by coulomb force, in much the same way that the sun attracts the earth by gravitational force. 337 However, if the electron orbits the nucleus in a classical orbit, it ought to emit electromagnetic radiation (light) according to well established theories of electromagnetism. If the orbiting electron emits light, it must lose energy and spiral into the nucleus, so why do atoms even exist? What’s more, the spectra of atoms show that the orbiting electrons can emit light but only at certain frequencies. This made no sense at all to the scientists of the time. These di–culties were resolved in 1913 by Niels Bohr who proposed that: * (1) The orbiting electrons existed in orbits that had discrete quantized energies. That is, not every orbit is possible but only certain speciﬂc ones. The exact energies of the allowed orbits depends on the atom in question. * (2) The laws of classical mechanics do not apply when electrons make the jump from one allowed orbit to another. * (3) When an electron makes a jump from one orbit to another the energy diﬁerence is carried oﬁ (or supplied) by a single quantum of light (called a photon) which has a frequency that directly depends on the energy diﬁerence between the two orbitals. f = E / h where f is the frequency of the photon, E the energy diﬁerence, and h is a constant of proportionality known as Planck’s constant. Deﬂning we can write where ? is the angular frequency of the photon. * (4) The allowed orbits depend on quantized (discrete) values of orbital angular momentum, L according to the equation Where n = 1,2,3, and is called the angular momentum quantum number. These assumptions explained many of the observations seen at the time, such as why spectra consist of discrete lines. Assumption 4) states that the lowest value of n is 1. This corresponds to a smallest possible radius (for the mathematics see Ohanian-principles of physics or any of the large, usually American, college introductory physics textbooks) of 0.0529 nm. This is known as the Bohr radius, and explains why atoms are stable. Once an electron is in the lowest orbit, it can go no further. It cannot emit any more light because it would need to go into a lower orbit, but it can’t do that if it is already in the lowest allowed orbit. The Bohr model is sometimes known as the semiclassical model because although it does include some ideas of quantum mechanics it is not a full quantum mechanical description of the atom. Assumption 2) states that the laws of classical mechanics don’t apply during a quantum jump but doesn’t state what laws should replace classical mechanics. Assumption 4) states that angular momentum is quantised but does not explain why. In order to fully describe an atom we need to use the full theory of quantum mechanics, which was worked out by a number of people in the years following the Bohr model. This theory treats the electrons as waves, which create 3D standing wave patterns in the atom. (This is why quantum mechanics is sometimes called wave mechanics.) This theory considers that idea of electrons as being little billiard ball like particles that travel round in orbits as absurdly wrong; instead electrons form probability clouds. You might ﬂnd the electron here with a certain probability; you might ﬂnd it over there with a diﬁerent probability. However it is interesting to note that if you work out the average radius of an electron in the lowest possible energy state it turns out to be exactly equal to the Bohr radius (although it takes many more pages of mathematics to work it out). The full quantum mechanics theory is a beautiful theory that has been experimentally tested and found to be incredibly accurate, however it is mathematically much more advanced, and often using the much simpler Bohr model will get you the results with much less hassle. The thing to remember is that it is only a model, an aid to understanding. Atoms are not really little solar systems. * See also: Hydrogen atom, quantum mechanics, Schrdinger equation, Niels Bohr. * An interactive demonstration (http://webphysics.davidson.edu/faculty/dmb/hydrogen/) of the prob- 338 ability clouds of electron in Hydrogen atorm according to the full QM solution. 17.7 Bohr orbits Brink and Jones sections: Standing waves (quantisation). Atom seen as positive nucleus with vibrating electron waves surrounding it. Shrodinger’s equation calucaltes the energy of these waves and their shape and position{ most probable region of movement of electrons called orbitals (talk about n=1,2 energy levels and spdf orbitals). 17.8 Heisenberg uncertainty Principle Quantum mechanics is a physical theory that describes the behavior of physical systems at short distances. Quantum mechanics provides a mathematical framework derived from a small set of basic principles capable of producing experimental predictions for three types of phenomena that classical mechanics and classical electrodynamics cannot account for: quantization, wave-particle duality, and quantum entanglement. The related terms quantum physics and quantum theory are sometimes used as synonyms of quantum mechanics, but also to denote a superset of theories, including pre-quantum mechanics old quantum theory, or, when the term quantum mechanics is used in a more restr
icted sense, to include theories like quantum ﬂeld theory. Quantum mechanics is the underlying theory of many ﬂelds of physics and chemistry, includ- ing condensed matter physics, quantum chemistry, and particle physics. 17.9 Pauli exclusion principle The Pauli exclusion principle is a quantum mechanical principle which states that no two identical fermions may occupy the same quantum state. Formulated by Wolfgang Pauli in 1925, it is also referred to as the "exclusion principle" or "Pauli principle." The Pauli principle only applies to fermions, particles which form antisymmetric quantum states and have half-integer spin. Fermions include protons, neutrons, and electrons, the three types of elementary particles which constitute ordinary matter. The Pauli exclusion principle governs many of the distinctive characteristics of matter. Particles like the photon and graviton do not obey the Pauli exclusion principle, because they are bosons (i.e. they form symmetric quantum states and have integer spin) rather than fermions. The Pauli exclusion principle plays a role in a huge number of physical phenomena. One of the most important, and the one for which it was originally formulated, is the electron shell structure of atoms. An electrically neutral atom contains bound electrons equal in number to the protons in the nucleus. Since electrons are fermions, the Pauli exclusion principle forbids them from occupying the same quantum state. For example, consider a neutral helium atom, which has two bound electrons. Both of these electrons can occupy the lowest-energy (1s) states by acquiring opposite spin. This does not violate the Pauli principle because spin is part of the quantum state of the electron, so the two electrons are occupying diﬁerent quantum states. However, the spin can take only two diﬁerent In a lithium atom, which contains three bound electrons, the third values (or eigenvalues.) electron cannot ﬂt into a 1s state, and has to occupy one of the higher-energy 2s states instead. Similarly, successive elements produce successively higher-energy shells. The chemical properties of an element largely depends on the number of electrons in the outermost shell, which gives rise to the periodic table of the elements. 339 The Pauli principle is also responsible for the large-scale stability of matter. Molecules cannot be pushed arbitrarily close together, because the bound electrons in each molecule are forbidden from entering the same state as the electrons in the other molecules - this is the reason for the repulsive r-12 term in the Lennard-Jones potential. The Pauli principle is the reason you do not fall through the oor. Astronomy provides the most spectacular demonstrations of this eﬁect, in the form of white dwarf stars and neutron stars. In both types of objects, the usual atomic structures are disrupted by large gravitational forces, leaving the constituents supported only by a "degeneracy pressure" produced by the Pauli exclusion principle. This exotic form of matter is known as degenerate matter. In white dwarfs, the atoms are held apart by the degeneracy pressure of the electrons. In neutron stars, which exhibit even larger gravitational forces, the electrons have merged with the protons to form neutrons, which produce a larger degeneracy pressure. Another physical phenomenon for which the Pauli principle is responsible is ferromagnetism, in which the exclusion eﬁect implies an exchange energy that induces neigboring electron spins to align (whereas classically they would anti-align). 17.10 Ionization Energy (ﬂrst, second etc.) 17.11 Electron conﬂguration i.e. ﬂlling the orbitals starting from 1s..... Aufbau principle unpaired and paired electrons Hund’s rule: 1 e- in each orbital before pairing in p orbitals shorthand: 1s22s22p1 etc 17.12 Valency Capacity for bonding Covalent bonding is a form of chemical bonding characterized by the sharing of one or more pairs of electrons, by two atoms, in order to produce a mutual attraction; atoms tend to share electrons, so as to ﬂll their outer electron shells. Such bonds are always stronger than the intermolecular hydrogen bond and similar in strength or stronger than the ionic bond. Commonly covalent bond implies the sharing of just a single pair of electrons. The sharing of two pairs is called a double bond and three pairs is called a triple bond. Aromatic rings of atoms and other resonant structures are held together by covalent bonds that are intermediate between single and double. The triple bond is relatively rare in nature, and two atoms are not observed to bond more than triply. Covalent bonding most frequently occurs between atoms with similar electronegativities, where neither atom can provide su–cient energy to completely remove an electron from the other atom. Covalent bonds are more common between non-metals, whereas ionic bonding is more common between two metal atoms or a metal and a non-metal atom. Covalent bonding tends to be stronger than other types of bonding, such as ionic bonding. In addition unlike ionic bonding, where ions are held together by a non-directional coulombic attraction, covalent bonds are highly directional. As a result, covalently bonded molecules tend to form in a relatively small number of characteristic shapes, exhibiting speciﬂc bonding angles. 340 17.13 341 Chapter 18 Modern Physics 18.1 Introduction to the idea of a quantum Imagine that a beam of light is actually made up of little "packets" or "bundles" of energy, called quanta. It’s like looking at a crowd of people from above. At ﬂrst, it seems as though they are one huge patch, without any spaces between them. You would never suspect that they were people. But as you move closer, you slowly begin to see that they are individuals, and when you get even closer, you may even recognize a few. Light seems like a continuous wave at ﬂrst, but when we zoom in at the subatomic level, we notice that a beam of light actually consists of little "packets" of energy, or quanta. This idea introduces the concept of the quantum (particle) nature of light, which is demonstrated by the photoelectric eﬁect. When a metal surface is illuminated with light, electrons can be emitted from the surface. This is known as the photoelectric eﬁect. 18.2 The wave-particle duality The wave nature of light is demonstrated by diﬁraction, interference, and polarization of light; and the particle nature of light is demonstrated by the photoelectric eﬁect. So light has both wave-like and particle-like properties, but only shows one or the other, depending on the kind of experiment we perform. A wave-type experiment shows the wave nature, and a particle-type experiment shows particle nature. When you’re watching a cricketer on the ﬂeld, you see only that side of his personality. So to you, he is just a good cricketer. You do not see his golﬂng side, for example. Only when he is playing golf, will that side be revealed to you. The same applies to light. Now, we consider light to behave not as a wave, but as particles. But what do we call a ’particle’ of light? Photon : A photon is a quantum (energy packet) of light. Imagine a sheet of metal. On the surface, there are electrons that are waiting to be set free. If a photon comes along and strikes the surface of the metal, then it will give its entire energy packet to one electron. This means that the electron now has some energy, and it may escape (leave the surface) if this energy Ek is greater than the minimum energy required to free an electron Emin. Now, suppose the electron needs 5eV of kinetic energy to escape. And suppose this little 342 photon has just 2eV of energy in its energy packet. Then the electron will not leave the surface of the metal. But suppose the photon has 8eV of energy. This means that the electron will emerge with 3eV of kinetic energy. Note that this does not mean the photon can give 5eV of energy to one electron and 3eV to another. A photon will give all of its energy to just one electron. The minimum amount of energy needed for an electron to escape (electrons do not normally leave a metal whenever they please), is called the work function of the metal. In our example, the work function is 5eV. The work function has a diﬁerent value for each metal: 4.70eV for copper and 2.28eV for sodium. It is worth mentioning that the best conductors are those with the smallest work functions. The frequency of the radiation is very important, because if it is below a certain threshold value, no electrons will be emitted. Even if the intensity of the light is increased, and the light is allowed to fall on the surface for a long period of time, if the frequency of the radiation is below the threshold frequency, electrons will not be emitted. We therefore reason that E = h? Where E is the energy of the photon, h = 6:57X10¡34Js is Plancks constant, and ? is the frequency of the radiation. This means that the kinetic energy acquired by the electron is equal to the energy of the photon minus the work function, ?, i.e. Ek = h? - ? The electrons emerge with a range of velocities from zero up to a maximum vmax. The maximum kinetic energy, (1/2)mvmax2, depends (linearly) on the frequency of the radiation, and is independent of its intensity. For incident radiation of a given frequency, the number of electrons emitted per unit time is proportional to the intensity of the radiation. Electron emission takes place from the instant the light shines on the surface, i.e. there is no detectable time delay. What are the uses of the photoelectric eﬁect? For this work, Einstein received the Nobel prize in 1905. 18.3 Practical Applications of Waves: Electromagnetic Waves In physics, wave-particle duality holds that light and matter simultaneously exhibit properties of waves and of particles. This concept is a consequence of quantum mechanics. In 1905, Einstein reconciled Huygens’ view with that of Newton; he explained the photoelectric eﬁect (an eﬁect in which
light did not seem to act as a wave) by postulating the existence of photons, quanta of energy with particulate qualities. Einstein postulated that light’s frequency, ?, is related to the energy, E, of its photons: E = hf 10¡34Js). (18.1) where h is Planck’s constant (6:626 £ In 1924, De Broglie claimed that all matter has a wave-like nature; he related wavelength, ?, and momentum, p: ‚ = h p (18.2) This is a generalization of Einstein’s equation above since the momentum of a photon is given by, p = ; (18.3) E c where c is the speed of light in vacuum, and ? = c / ?. 343 De Broglie’s formula was conﬂrmed three years later by guiding a beam of electrons (which have rest mass) through a crystalline grid and observing the predicted interference patterns. Similar experiments have since been conducted with neutrons and protons. Authors of similar recent experiments with atoms and molecules claim that these larger particles also act like waves. This is still a contoversial subject because these experimenters have assumed arguments of waveparticle duality and have assumed the validity of deBroglie’s equation in their argument. The Planck constant h is extremely small and that explains why we don’t perceive a wave-like quality of everyday objects: their wavelengths are exceedingly small. The fact that matter can have very short wavelengths is exploited in electron microscopy. In quantum mechanics, the wave-particle duality is explained as follows: every system and particle is described by state functions which encode the probability distributions of all measurable variables. The position of the particle is one such variable. Before an observation is made the position of the particle is described in terms of probability waves which can interfere with each other. 344 Chapter 19 Inside atomic nucleus Amazingly enough, human mind that is kind of contained inside a couple of liters of human’s brain, is able to deal with extremely large as well as extremely small objects such as the whole universe and its smallest building blocks. So, what are these building blocks? As we already know, the universe consists of galaxies, which consist of stars with planets moving around. The planets are made of molecules, which are bound groups (chemical compounds) of atoms. There are more than 1020 stars in the universe. Currently, scientists know over 12 million chemical compounds i.e. 12 million diﬁerent molecules. All this variety of molecules is made of only a hundred of diﬁerent atoms. For those who believe in beauty and harmony of nature, this number is still too large. They would expect to have just few diﬁerent things from which all other substances are made. In this chapter, we are going to ﬂnd out what these elementary things are. 19.1 What the atom is made of The Greek word ﬁ¿ o„o” (atom) means indivisible. The discovery of the fact that an atom is actually a complex system and can be broken in pieces was the most important step and pivoting point in the development of modern physics. It was discovered (by Rutherford in 1911) that an atom consists of a positively charged nucleus and negative electrons moving around it. At ﬂrst, people tried to visualize an atom as a microscopic analog of our solar system where planets move around the sun. This naive planetary model assumes that in the world of very small objects the Newton laws of classical mechanics are valid. This, however, is not the case. The microscopic world is governed by quantum mechanics which does not have such notion as trajectory. Instead, it describes the dynamics of particles in terms of quantum states that are characterized by probability distributions of various observable quantities. For example, an electron in the atom is not moving along a certain trajectory but rather along all imaginable trajectories with diﬁerent probabilities. If we were trying to catch this electron, after many such attempts we would discover that the electron can be found anywere around the nucleus, even very close to and very far from it. However, the probabilities of ﬂnding the electron 345 P (r) RBohr r Figure 19.1: Probability density P (r) for ﬂnding the electron at a distance r from the proton in the ground state of hydrogen atom. at diﬁerent distances from the nucleus would be diﬁerent. What is amazing: the most probable distance corresponds to the classical trajectory! You can visualize the electron inside an atom as moving around the nucleus chaotically and extremely fast so that for our \mental eyes" it forms a cloud. In some places this cloud is more dense while in other places more thin. The density of the cloud corresponds to the probability of ﬂnding the electron in a particular place. Space distribution of this density (probability) is what we can calculate using quantum mechanics. Results of such calculation for hydrogen atom are shown in Fig. 19.1. As was mentioned above, the most probable distance (maximum of the curve) coincides with the Bohr radius. Quantum mechanical equation for any bound system (like an atom) can have solutions only at a discrete set of energies E1; E2; E3 : : : , etc. There are simply no solutions for the energies E in between these values, such as, for instance, E1 < E < E2. This is why a bound system of microscopic particles cannot have an arbitrary energy and can only be in one of the quantum states. Each of such states has certain energy and certain space conﬂguration, i.e. distribution of the probability. A bound quantum system can make transitions from one quantum state to another either spontaneously or as a result of interaction with other systems. The energy conservation law is one of the most fundamental and is valid in quantum world as well as in classical world. This means that any transition between the states with energies Ei and Ej is accompanied with either Ei ¡ emission or absorption of the energy ¢E = j . This is how an atom emits light. Ejj Electron is a very light particle. Its mass is negligible as compared to the total mass of the atom. For example, in the lightest of all atoms, hydrogen, the electron constitutes only 0.054% of the atomic mass. In the silicon atoms that are the main component of the rocks around us, all 14 electrons make up only 0.027% of the mass. Thus, when holding a heavy rock in your hand, you actually feel the collective weight of all the nuclei that are inside it. 346 19.2 Nucleus Is the nucleus a solid body? Is it an elementary building block of nature? No and no! Although it is very small, a nucleus consists of something even smaller. 19.2.1 Proton The only way to do experiments with such small objects as atoms and nuclei, is to collide them with each other and watch what happens. Perhaps you think that this is a barbaric way, like colliding a \Mercedes" and \Toyota" in order to learn what is under their bonnets. But with microscopic particles nothing else can be done. In the early 1920’s Rutherford and other physicists made many experiments, changing one element into another by striking them with energetic helium nuclei. They noticed that all the time hydrogen nuclei were emitted in the process. It was apparent that the hydrogen nucleus played a fundamental role in nuclear structure and was a constituent part of all other nuclei. By the late 1920’s physicists were regularly referring to hydrogen nucleus as proton. The term \proton" seems to have been coined by Rutherford, and ﬂrst appears in print in 1920. 19.2.2 Neutron Thus it was established that atomic nuclei consist of protons. Number of protons in a nucleus is such that makes up its positive charge. This number, therefore, coincides with the atomic number of the element in the Mendeleev’s Periodic table. This sounded nice and logical, but serious questions remained. Indeed, how can positively charged protons stay together in a nucleus? Repelling each other by electric force, they should y away in diﬁerent directions. Who keeps them together? Furthermore, the proton mass is not enough to account for the nuclear masses. For example, if the protons were the only particles in the nucleus, then a helium nucleus (atomic number 2) would have two protons and therefore only twice the mass of hydrogen. However, it actually is four times heavier than hydrogen. This suggests that it must be something else inside nuclei in addition to protons. These additional particles that kind of \glue" the protons and make up the nuclear mass, apparently, are electrically neutral. They were therefore called neutrons. Rutherford predicted the existence of the neutron in 1920. Twelve years later, in 1932, his assistant James Chadwick found it and measured its mass, which turned out to be almost the same but slightly larger than that of the proton. 19.2.3 Isotopes Thus, in the early 1930’s it was ﬂnally proved that atomic nucleus consists of two types of particles, the protons and neutrons. The protons are positively charged while the neutrons are electrically neutral. The proton charge is exactly equal but opposite to that of the electron. The masses of proton and neutron are almost the same, approximately 1836 and 1839 electron masses, respectively. 347 Apart from the electric charge, the proton and neutron have almost the same properties. This is why there is a common name for them: nucleon. Both the proton and neutron are nucleons, like a man and a woman are both humans. In physics literature, the proton is denoted by letter p and the neutron by n. Sometimes, when the diﬁerence between them is unimportant, it is used the letter N meaning nucleon (in the same sense as using the word \person" instead of man or woman). Chemical properties of an element are determined by the charge of its atomic nucleus, i.e. by the number of protons. This number is called the atomic number and is denoted by letter Z. The mass of an atom depends on how many nucleons its nucleus contains. The number of nucleons, i.e. total number of protons and neutrons, is called the atomic mass number and
is denoted by letter A. Standard nuclear notation shows the chemical symbol, the mass number and the atomic number of the isotope. number of nucleons number of protons A ZX chemical symbol For example, the iron nucleus (26-th place in the Mendeleev’s periodic table of the elements) with 26 protons and 30 neutrons is denoted as 56 26Fe ; where the total nuclear charge is Z = 26 and the mass number A = 56. The number of neutrons Z (here, it is used the same letter N , as for nucleon, but this is simply the diﬁerence N = A should not cause any confusion). Chemical symbol is inseparably linked with Z. This is why the lower index is sometimes omitted and you may encounter the simpliﬂed notation like 56Fe. ¡ If we add or remove a few neutrons from a nucleus, the chemical properties of the atom remain the same because its charge is the same. This means that such atom should remain in the same place of the Periodic table. In Greek, \same place" reads ¶¶o& ¿ ¶o…o& (isos topos). The nuclei, having the same number of protons, but diﬁerent number of neutrons, are called therefore isotopes. Diﬁerent isotopes of a given element have the same atomic number Z, but diﬁerent mass numbers A since they have diﬁerent numbers of neutrons N . Chemical properties of diﬁerent isotopes of an element are identical, but they will often have great diﬁerences in nuclear stability. For stable isotopes of the light elements, the number of neutrons will be almost equal to the number of protons, but for heavier elements, the number of neutrons is always greater than Z and the neutron excess tends to grow when Z increases. This is because neutrons are kind of glue that keeps repelling protons together. The greater the repelling charge, the more glue you need. 348 19.3 Nuclear force Since atomic nuclei are very stable, the protons and neutrons must be kept inside them by some force and this force must be rather strong. What is this force? All of modern particle physics was discovered in the eﬁort to understand this force! Trying to answer this question, at the beginning of the XX-th century, physicists found that all they knew before, was inadequate. Actually, by that time they knew only gravitational and electromagnetic forces. It was clear that the forces holding nucleons were not electromagnetic. Indeed, the protons, being positively charged, repel each other and all nuclei would decay in a split of a second if some other forces would not hold them together. On the other hand, it was also clear that they were not gravitational, which would be too weak for the task. The simple conclusion was that nucleons are able to attract each other by yet unknown nuclear forces, which are stronger than the electromagnetic ones. Further studies proved that this hypothesis was correct. Nuclear force has rather unusual properties. Firstly, it is charge independent. This means 10¡13 cm, that in all pairs nn, pp, and np nuclear forces are the same. Secondly, at distances 100 times stronger than the electromagnetic the nuclear force is attractive and very strong, » repulsion. Thirdly, the nuclear force is of a very short range. If the nucleons move away from each other for more than few fermi (1 fm=10¡13 cm) the nuclear attraction practically disappears. Therefore the nuclear force looks like a \strong man with very short hands". » 19.4 Binding energy and nuclear masses 19.4.1 Binding energy When a system of particles is bound, you have to spend certain energy to disintegrate it, i.e. to separate the particles. The easiest way to do it is to strike the system with a moving particle that carries kinetic energy, like we can destroy a glass bottle with a bullet or a stone. If our bulletparticle moves too slow (i.e. does not have enough kinetic energy) it cannot disintegrate the system. On the other hand, if its kinetic energy is too high, the system is not only disintegrated but the separated particles acquire some kinetic energy, i.e. move away with some speed. There is an intermediate value of the energy which is just enough to destroy the system without giving its fragments any speed. This minimal energy needed to break up a bound system is called binding energy of this system. It is usually denoted by letter B. 19.4.2 Nuclear energy units The standart unit of energy, Joule, is too large to measure the energies associated with individual nuclei. This is why in nuclear physics it is more convenient to use a much smaller unit called Mega-electron-Volt (MeV). This is the amount of energy that an electron acquires after passing between two charged plates with the potential diﬁerence (voltage) of one million Volts. Sounds very huge, isn’t it? But look at this relation and think again. In the units of MeV, most of the energies in nuclear world can be expressed by values with only few digits before decimal point and without ten to the power of something. For 1 MeV = 1:602 10¡13 J £ 349 example, the binding energy of proton and neutron (which is the simplest nuclear system and is called deuteron) is Bpn = 2:225 MeV : The simplicity of the numbers is not the only advantage of using the unit MeV. Another, more important advantage, comes from the fact that most of experiments in nuclear physics are collision experiments, where particles are accelerated by electric ﬂeld and collide with other particles. From the above value of Bpn, for instance, we immediately know that in order to break up deuterons, we need to bombard them with a ux of electrons accelerated through a voltage not less than 2.225 million Volts. No calculation is needed! On the other hand, if we know that a charged particle (with a unit charge) passes through a voltage, say, 5 million Volts, we can, without any calculation, say that it acqures the energy of 5 MeV. It is very convenient. Isn’t it? 19.4.3 Mass defect Comparing the masses of atomic nuclei with the masses of the nucleons that constitute them, we encounter a surprising fact: Total mass of the nucleons is greater than mass of the nucleus! For example, for the deuteron we have md < mp + mn ; where md, mp, and mn are the masses of deuteron, proton, and neutron, respectively. The diﬁerence is rather small, but on the nuclear scale is noticeable since the mass of proton, for example, (mp + mn) ¡ md = 3:968 £ 10¡30 kg ; mp = 1672:623 10¡30 kg £ is also very small. This phenomenon is called \mass defect". Where the mass disappears to, when nucleons are bound? To answer this question, we notice that the energy of a bound state is lower than the energy of free particles. Indeed, to liberate them from a bound complex, we have to give them some energy. Thinking in the opposite direction, we conclude that, when forming a bound state, the particles have to get rid of the energy excess, which is exactly equal to the binding energy. This is observed experimentally: When a proton captures a neutron to form a deuteron, the excess energy of 2.225 MeV is emitted via electromagnetic radiation. A logical conclusion from the above comes by itself: When proton and neutron are bounding, some part of their mass disappears together with the energy that is carried away by the radiation. And in the opposite process, when we break up the deutron, we give it the energy, some part of which makes up the lost mass. Albert Einstein came to the idea of the equivalence between the mass and energy long before any experimental evidences were found. In his theory of relativity, he showed that total energy E of a moving body with mass m is ; v2 c2 mc2 E = 1 ¡ r 350 (19.1) where v is its velocity and c the speed of light. Applying this equation to a non-moving body (v = 0), we conclude that it possesses the rest energy E0 = mc2 (19.2) simply because it has mass. As you will see, this very formula is the basis for making nuclear bombs and nuclear power stations! All the development of physics and chemistry, preceding the theory of relativity, was based on the assumption that the mass and energy of a closed system are conserving in all possible processes and they are conserved separately. In reality, it turned out that the conserving quantity is the mass-energy, Ekin + Epot + Erad + mc2 = const ; i.e. the sum of kinetic energy, potential energy, the energy of radiation, and the mass of the system. In chemical reactions the fraction of the mass that is transformed into other forms of energy (and vise versa), is so small that it is not detectable even in most precise measurements. In nuclear processes, however, the energy release is very often millions times higher and therefore is observable. You should not think that mutual transformations of mass and energy are the features of only nuclear and atomic processes. If you break up a piece of rubber or chewing gum, for example, in two parts, then the sum of masses of these parts will be slightly larger than the mass of the whole piece. Of course we will not be able to detect this \mass defect" with our scales. But we can calculate it, using the Einstein formula (19.2). For this, we would need to measure somehow the mechanical work A used to break up the whole piece (i.e. the amount of energy supplied to it). This can be done by measuring the force and displacement in the breaking process. Then, according to Eq. (19.2), the mass defect is ¢m = A c2 : To estimate possible eﬁect, let us assume that we need to stretch a piece of rubber in 10 cm before it breaks, and the average force needed for this is 10 N (approximately 1 kg). Then and hence A = 10 N £ 0:1 m = 1 J ; ¢m = 1 J (299792458 m=s)2 … 10¡17 kg: 1:1 £ This is very small value for measuring with a scale, but huge as compared to typical masses of atoms and nuclei. 19.4.4 Nuclear masses Apparently, an individual nucleus cannot be put on a scale to measure its mass. Then how can nuclear masses be measured? This is done with the help of the devices called mass spectrometers. In them, a ux of identical nuclei, accelerated to a certain energy, is directed to a screen where it makes a visible
mark. 351 Before striking the screen, this ux passes through magnetic ﬂeld, which is perpendicular to velocity of the nuclei. As a result, the ux is deected to certain angle. The greater the mass, the smaller is the angle (because of inertia). Thus, measuring the displacement of the mark from the center of the screen, we can ﬂnd the deection angle and then calculate the mass. Since mass and energy are equivalent, in nuclear physics it is customary to measure masses of all particles in the units of energy, namely, in MeV. Examples of masses of subatomic particles are given in Table 19.1. The values given in this table, are the energies to which the nuclear particle number of protons number of neutrons mass (MeV) e p n 2 1H 3 1H 3 2He 4 2He 7 3Li 9 4Be 12 6C 16 8O .511 938.272 939.566 1875.613 2808.920 2808.391 3727.378 6533.832 8392.748 11174.860 14895.077 238 92U 92 146 221695.831 Table 19.1: Masses of electron, nucleons, and some nuclei. masses are equivalent via the Einstein formula (19.2). £ There are several advantages of using the units of MeV to measure particle masses. First of all, like with nuclear energies, we avoid handling very small numbers that involve ten to the power of something. For example, if we were measuring masses in kg, the electron mass would 10¡31 kg. When masses are given in the equivalent energy units, it is very be me = 9:1093897 easy to calculate the mass defect. Indeed, adding the masses of proton and neutron, given in the second and third rows of Table 19.1, and subtracting the mass of 2 1H, we obtain the binding energy 2.225 MeV of the deuteron without further ado. One more advantage comes from particle physics. In collisions of very fast moving particles new particles (like electrons) can be created from vacuum, i.e. kinetic energy is directly transformed into mass. If the mass is expressed in the energy units, we know how much energy is needed to create this or that particle, without calculations. 352 19.5 Radioactivity As was said before, the nucleus experiences the intense struggle between the electric repulsion of protons and nuclear attraction of the nucleons to each other. It therefore should not be surprising that there are many nuclei that are unstable. They can spontaneously (i.e. without an external push) break in pieces. When the fragments reach the distances where the short range nuclear attraction disappears, they ﬂercely push each other away by the electric forces. Thus accelerated, they move in diﬁerent directions like small bullets making destruction on their way. This is an example of nuclear radioactivity but there are several other varieties of radioactive decay. 19.5.1 Discovery of radioactivity Nuclear radioactivity was discovered by Antoine Henri Becquerel in 1896. Following Wilhelm Roentgen who discovered the X-rays, Becquerel pursued his own investigations of these mysterious rays. The material Becquerel chose to work with contained uranium. He found that the crystals containing uranium and exposed to sunlight, made images on photographic plates even wrapped in black paper. He mistakingly concluded that the sun’s energy was being absorbed by the uranium which then emitted X-rays. The truth was revealed thanks to bad weather. On the 26th and 27th of February 1896 the skies over Paris were overcast and the uranium crystals Becquerel intended to expose to the sun were returned to a drawer and put over (by chance) the photographic plates. On the ﬂrst of March, Becquerel developed the plates and to his surprise, found that the images on them were clear and strong. Therefore the uranium emitted radiation without an external source of energy such as the sun. This was the ﬂrst observation of the nuclear radioactivity. Later, Becquerel demonstrated that the uranium radiation was similar to the X-rays but, unlike them, could be deected by a magnetic ﬂeld and therefore must consist of charged particles. For his discovery of radioactivity, Becquerel was awarded the 1903 Nobel Prize for physics. 19.5.2 Nuclear ﬁ, ﬂ, and rays Classical experiment that revealed complex content of the nuclear radiation, was done as follows. The radium crystals (another radioactive element) were put at the bottom of a narrow straight channel made in a thick piece of lead and open at one side. The lead absorbed everything except the particles moving along the channel. This device therefore produced a ux of particles moving in one direction like bullets from a machine gun. In front of the channel was a photoplate that could register the particles. Without the magnetic ﬂeld, the image on the plate was in the form of one single dot. When the device was immersed into a perpendicular magnetic ﬂeld, the ux of particles was split in three uxes, which was reected by three dots on the photographic plate. One of the three uxes was stright, while two others were deected in opposite directions. This showed that the initial ux contained positive, negative, and neutral particles. They were 353 named respectively the ﬁ, ﬂ, and particles. The ﬁ-rays were found to be the 4He nuclei, two protons and two neutrons bound together. They have weak penetrating ability, a few centimeters of air or a few sheets of paper can eﬁectively block them. The ﬂ-rays proved to be electrons. They have a greater penetrating power than the ﬁ-particles and can penetrate 3 mm of aluminum. The -rays are not deected because they are high energy photons. They have the same nature as the radio waves, visible light, and the X-rays, but have much shorter wavelength and therefore are much more energetic. Among the three, the -rays have the greatest penetrating power being able to pass through several centimeters of lead and still be detected on the other side. 19.5.3 Danger of the ionizing radiation The ﬁ, ﬂ, and particles moving through matter, collide with atoms and knock out electrons from them, i.e. make positive ions out of the atoms. This is why these rays are called ionizing radiation. Apart from ionizing the atoms, this radiation destroys molecules. For humans and all other organisms, this is the most dangerous feature of the radiation. Imagine thousands of tiny tiny bullets passing through your body and making destruction on their way. Although people do not feel any pain when exposed to nuclear radiation, it harms the cells of the body and thus can make people sick or even kill them. Illness can strike people years after their exposure to nuclear radiation. For example, the ionizing particles can randomly modify the DNA (long organic molecules that store all the information on how a particular cell should function in the body). As a result, some cells with wrong DNA may become cancer cells. Fortunately, our body is able to repair some damages caused by radiation. Indeed, we are constantly bombarded by the radiation coming from the outer space as well as from the inner parts of our own planet and still survive. However, if the number of damages becomes too large, the body will not cope with them anymore. There are established norms and acceptable limits for the radiation that are considered safe for human body. If you are going to work in contact with radioactive materials or near them, make sure that the exposure dose is monitored and the limits are adhered to. You should understand that no costume can protect you from -rays! Only a thick wall of concrete or metal can stop them. The special costumes and masks that people wear when handling radioactive materials, protect them not from the rays but from contamination with that materials. Imagine if few specks of radioactive dirt stain your everyday clothes or if you inhale radioactive atoms. They will remain with you all the time and will shoot the \bullets" at you even when you are sleeping. In many cases, a very eﬁective way of protecting yourself from the radiation is to keep certain distance. Radiation from nuclear sources is distributed equally in all directions. Therefore the number n of dangerous particles passing every second through a unit area (say 1 cm2) is the total number N of particles emitted during 1 second, divided by the surface of a sphere n = N 4…r2 ; 354 where r is the distance at which we make the observation. From this simple formula, it is seen that the radiation intensity falls down with incresing distance quadratically. In other words, if you increase the distance by a factor of 2, your exposure to the radiation will be decreased by a factor of 4. 19.5.4 Decay law Unstable nuclei decay spontaneously. A given nucleus can decay next moment, next day or even next century. Nobody can predict when it is going to happen. Despite this seemingly chaotic and \unscientiﬂc" situation, there is a strict order in all this. Atomic nuclei, being microscopic objects, are ruled by quantum probabilistic laws. Although we cannot predict the exact moment of its decay, we can calculate the probability that a nucleus will decay within this or that time interval. Nuclei decay because of their internal dynamics and not because they become \old" or somehow \rotten". To illustrate this, let us imagine that yesterday morning we found that a certain nucleus was going to decay within 24 hours with the probability of 50%. However, this morning we found that it is still \alive". This fact does not mean that the decay probability for another 24 hours increased. Not at all! It remains the same, 50%, because the nucleus remains the same, nothing wrong happened to it. This can go on and on for centuries. Actually, we never deal with individual nuclei but rather with huge numbers of identical nuclei. For such collections (ensembles) of quantum objects, the probabilistic laws become statictical laws. Let us assume that in the above example we had 1 million identical nuclei instead of only one. Then by this morning only half of these nuclei would survive because the decay probability for 24 hours was 50%. Among the remaining 500000 nuclei, 250000 will decay by tomorrow morning, then aft
er another 24 hours only 125000 will remain and so on. The number of unstable nuclei that are still \alive" continuously decreases with time according to the curve shown in Fig. 19.2. If initially, at time t = 0, their number is N0, then after certain time interval T1=2 only half of these nuclei will remain, namely, 1 2 N0. Another one half of the remaining half will decay during another such interval. So, after the time 2T1=2, we will have only one quarter of the initial amount, and so on. The time interval T1=2, during which one half of unstable nuclei decay, is called their half-life time. It is speciﬂc for each unstable nucleus and vary from a fraction of a second to thousands and millions of years. A few examples of such lifetimes are given in Table 19.2 19.5.5 Radioactive dating Examining the amounts of the decay products makes possible radioactive dating. The most famous is the Carbon dating, a variety of radioactive dating which is applicable only to matter which was once living and presumed to be in equilibrium with the atmosphere, taking in carbon dioxide from the air for photosynthesis. Cosmic ray protons blast nuclei in the upper atmosphere, producing neutrons which in turn bombard nitrogen, the major constituent of the atmosphere. This neutron bombardment produces the radioactive isotope 14 6C. The radioactive carbon-14 combines with oxygen to form 355 N (t) N0 1 2 N0 1 4 N0 1 8 N0 0 T1=2 2T1=2 3T1=2 4T1=2 t Figure 19.2: The time T1=2 during which one half of the initial amount of unstable particles decay, is called their half-life time. isotope T1=2 decay mode 214 84Po 89 36Kr 222 86Rn 90 38Sr 226 88Ra 14 6C 238 92U 115 49In 1:64 10¡4 s £ 3:16 min 3.83 days 28:5 years 103 years 1:6 £ 5:73 4:47 4:41 £ £ £ 103 years 109 years 1014 years ﬁ; ﬂ¡; ﬁ; ﬂ¡ ﬁ; ﬂ¡ ﬁ; ﬂ¡ Table 19.2: Half-life times of several unstable isotopes. carbon dioxide and is incorporated into the cycle of living things. The isotope 14 6C decays (see Table 19.2) inside living bodies but is replenished from the air 356 and food. Therefore, while an organism is alive, the concentration of this isotope in the body remains constant. After death, the replenishment from the breath and food stops, but the isotopes that are in the dead body continue to decay. As a result the concentration of 14 6C in it gradually decreases according to the curve shown in Fig. 19.2. The time t = 0 on this Figure corresponds to the moment of death, and N0 is the equilibrium concentration of 14 6C in living organisms. Therefore, by measuring the radioactive emissions from once-living matter and comparing its activity with the equilibrium level of emissions from things living today, an estimation of the time elapsed can be made. For example, if the rate of the radioactive emissions from a piece of wood, caused by the decay of 14 6C, is one-half lower than from living trees, then we can conclude it is elapsed exactly one half-life-time that we are at the point t = T1=2 on the curve 19.2, i.e. period. According to the Table 19.2), this means that the tree, from which this piece of wood was made, was cut approximately 5730 years ago. This is how physicists help archaeologists to assign dates to various organic materials. 19.6 Nuclear reactions Those of you who studied chemistry, are familiar with the notion of chemical reaction, which, in essence, is just regrouping of atoms that constitute molecules. As a result, reagent chemical compounds are transformed into product compounds. In the world of nuclear particles, similar processes are possible. When nuclei are close to each other, nucleons from one nucleus can \jump" into another one. This happens because there are attractive and repulsive forces between the nucleons. The complicated interplay of these forces may cause their regrouping. As a result, the reagent particles are transformed into product particles. Such processes are called nuclear reactions. For example, when two isotopes 3 in such a way that the isotope 4 reactions, this process is denoted as 2He collide, the six nucleons constituting them, can rearrange 2He is formed and two protons are liberated. Similarly to chemical 2He + 3 3 2He ¡! 4 2He + p + p + 12:86 MeV : (19.3) The same as in chemical reactions, nuclear reactions can also be either exothermic (i.e. releasing energy) or endothermic (i.e. requiring an energy input). The above reaction releases 12.86 MeV of energy. This is because the total mass on the left hand side of Eq. (19.3) is in 12.86 MeV greater than the total mass of the products on the right hand side (you can check this using Table 19.1). Thus, when considering a particular nuclear reaction, we can always learn if it releases or absorbs energy. For this, we only need to compare total masses on the left and right hand sides of the equation. Now, you can understand why it is very convenient to express masses in the units of energy. Composing equations like (19.3), we should always check the superscripts and subscripts of the nuclei in order to have the same number of nucleons and the same charge on both sides of the equation. In the above example, we have six nucleons and the charge +4 in both the initial and ﬂnal states of the reaction. To make the checking of nucleon number and charge conservation easier, sometimes the proton and neutron are denoted with superscripts and subscripts as well, 357 namely, 1 subscripts are the same on both sides of the equation. 1p and 1 0n. In this case, all we need is to check that sum of superscripts and sum of 19.7 Detectors How can we observe such tiny tiny things as protons and ﬁ-particles? There is no microscope that would be able to discern them. From the very beginning of the sub-atomic era, scientists have been working on the development of special instruments that are called particle detectors. These devices enable us either to register the mere fact that certain particle has passed through certain point in space or to observe the trace of its path (the trajectory). Actually, this is as good as watching the particle. Although the particle sizes are awfully small, when passing through some substances, they leave behind visible traces of tens of centimeters in length. By measuring the curvature of the trajectory of a particle deected in electric or magnetic ﬂeld, a physicist can determine the charge and mass of the particle and thus can identify it. 19.7.1 Geiger counter The most familiar device for registering charged particles is the Geiger counter. It cannot tell you anything about the particle except the fact that it has passed through the counter. The counter consists of a thin metal cylinder ﬂlled with gas. A wire electrode runs along the center of the tube and is kept at a high voltage ( 2000 V) relative to the cylinder. When a particle passes through the tube, it causes ionization of the gas atoms and thus an electric discharge between the cylinder and the wire. The electric pulse can be counted by a computer or made to produce a \click" in a loudspeaker. The number of counts per second tells us about intensity of the radiation. » 19.7.2 Fluorescent screen The very ﬂrst detector was the uorescent screen. When a charged particle hits the screen, a human eye can discern a ash of light at the point of impact. In fact, we all use this kind of detectors every day when watching TV of looking at a computer (if it does not have an LCD screen of course). Indeed, the images on the screens of their electron-ray tubes are formed by the accelerated electrons. 19.7.3 Photo-emulsion Another type of particle detector, dating back to Becquerel, is the nuclear photographic emulsion. Passage of charged particles is recorded in the emulsion in the same way that ordinary black and white photographic ﬂlm records a picture. The only diﬁerence is that nuclear photoemulsion is made rather thick in order to catch a signiﬂcant part of the particle path. After the developing, a permanent record of the charged particle trajectory is available. 19.7.4 Wilson’s chamber In the ﬂelds of sub-atomic physics and nuclear physics, Wilson’s cloud chamber is the most fundamental device to observe the trajectories of particles. Its basic principle was discovered by C. T. R. Wilson in 1897, and it was put to the practical use in 1911. 358 The top and the side of the chamber are covered with round glasses of several centimeters in diameter. At the bottom of the chamber, a piston is placed. The air ﬂlled in the chamber is saturated with vapor of water. When pulling down the piston quickly, the volume of the chamber would be expanded and the temperature goes down. As a result, the air inside would be supersaturated with the vapor. If a fast moving charged particle enters the chamber when it is in such a supersaturated state, the vapor of water would condense along the line of the ions generated by the particle, which is the path of the particle. Thus we can observe the trace, and also take a photograph. To make clear the trace, a light is sometimes illuminated from the side. When placing the cloud chamber in a magnetic ﬂeld, we can obtain various informations about the charged particle by measuring the curvature of the trace and other data. The bubble chamber and the spark chamber have taken place of the cloud chamber which is nowadays used only for the educational purposes. Wilson’s cloud chamber has however played a very important role in the history of physics. 19.7.5 Bubble chamber Bubble chamber is a particle detector of major importance during the initial years of high-energy physics. The bubble chamber has produced a wealth of physics from about 1955 well into the 1970s. It is based on the principle of bubble formation in a liquid heated above its boiling point, which is then suddenly expanded, starting boiling where passing charged particles have ionized the atoms of the liquid. The technique was honoured by the Nobel prize award to D. Glaser in 1960. Even today, bubble chamber photographs provide the aesthetically most a
ppealing visualization of subnuclear collisions. 19.7.6 Spark chamber Spark chamber is a historic device using electric discharges over a gap between two electrodes with large potential diﬁerence, to render passing particles visible. Sparks occurred where the gas had been ionized. Most often, multiple short gaps were used, but wide-gap chambers with gaps up to 40 cm were also built. The spark chamber is still of great scientiﬂc value in that it remains relatively simple and cheap to build as well as enabling an observer to view the paths of charged particles. 19.8 Nuclear energy Nuclei can produce energy via two diﬁerent types of reactions, namely, ﬂssion and fusion reactions. Fission is a break up of a nucleus in two or more pieces (smaller nuclei). Fusion is the opposite process: Formation of a bigger nucleus from two small nuclei. A question may arise: How two opposite processes can both produce energy? Can we make an inexhaustible souce of energy by breaking up and then fusing the same nuclei? Of cousre not! The energy conservation law cannot be circumvented in no way. When speaking about fusion and ﬂssion, we speak about diﬁerent ranges of nuclei. Energy can only be released when either light nuclei fuse or heavy nuclei ﬂssion. To understand why this is so, let us recollect that for releasing energy the mass of initial nuclei must be greater than the mass of the products of a nuclear reaction. The mass diﬁerence is transformed into the released energy. And why the product nuclei can loose some mass as compared to the initial nuclei? Because they are more tightly bound, i.e. their binding energies 359 are lager. Fig. 19.3 shows the dependence of the binding energy B per nucleon on the number A of 9 MeV nucleons constituting a nucleus. As you see, the curve reaches the maximum value of per nucleon at around A 50. The nuclei with such number of nucleons cannot produce energy neither through fusion nor through ﬂssion. They are kind of \ashes" and cannot serve as a fuel. In contrast to them, very light nuclei, when fused with each other, make more tightly bound products as well as very heavy nuclei do when split up in lighter fragments. » » B=A, (MeV) 10 8 6 4 2 0 \| {z } "fusion % \| {z } - ˆ ﬂssion \| {z } . ˆ 10 8 6 4 2 0 0 50 100 150 200 250 number of nucleons, A Figure 19.3: Binding energy per nucleon. In ﬂssion processes, which were discovered and used ﬂrst, a heavy nucleus like, for example, uranium or plutonium, splits up in two fragments which are both positively charged. These fragments repel each other by an electric force and move apart at a high speed, distributing their kinetic energy in the surrounding material. In fusion reactions everything goes in the opposite direction. Very light nuclei, like hydrogen or helium isotopes, when approaching each other to a distance of a few fm (1 fm = 10¡13 cm), experience strong attraction which overpowers their Coulomb (that is electric) repulsion. As a result the two nuclei fuse into a single nucleus. They collapse with extremely high speeds towards each other. To form a stable nucleus they must get rid of the excessive energy. This energy is emitted by ejecting a neutron or a photon. 19.8.1 Nuclear reactors Since the discovery of radioactivity it was known that heavy nuclei release energy in the processes of spontaneous decay. This process, however, is rather slow and cannot be inuenced (speed up or slow down) by humans and therefore could not be eﬁectively used for large-scale energy production. Nonetheless, it is ideal for feeding the devices that must work autonomously in remote 360 places for a long time and do not require much energy. For this, heat from the spontaneousdecays can be converted into electric power in a radioisotope thermoelectric generator. These generators have been used to power space probes and some lighthouses built by Russian engineers. Much more eﬁective way of using nuclear energy is based on another type of nuclear decay which is considered next. Chain reaction The discovery that opened up the era of nuclear energy was made in 1939 by German physicists O. Hahn, L. Meitner, F Strassmann, and O. Frisch. They found that a uranium nucleus, after absorbing a neutron, splits into two fragments. This was not a spontaneous but induced ﬂssion n + 235 92U ¡! 54Xe + 94 140 38Sr + n + n + 185 MeV (19.4) » that released 185 MeV of energy as well as two neutrons which could cause similar reactions on surrounding nuclei. The fact that instead of one initial neutron, in the reaction (19.4) we obtain two neutrons, is crucial. This gives us the possibility to make the so-called chain reaction schematically shown in Fig. 19.4 Figure 19.4: Chain reaction on uranium nuclei. In such process, one neutron breaks one heavy nucleus, the two released neutrons break two more heavy nuclei and produce four neutrons which, in turn, can break another four nuclei and so on. This process develops extremely fast. In a split of a second a huge amount of energy can be released, which means explosion. In fact, this is how the so-called atomic bomb works. Can we control the development of the chain reaction? Yes we can! This is done in nuclear reactors that produce energy for our use. How can it be done? Critical mass First of all, if the piece of material containing ﬂssile nuclei is too small, some neutrons may reach its surface and escape without causing further ﬂssions. For each type of ﬂssile material there is therefore a minimal mass of a sample that can support explosive chain reaction. It is called the critical mass. For example, the critical mass of 235 92U is approximately 50 kg. If the mass is 361 below the critical value, nuclear explosion is not possible, but the energy is still released and the sample becomes hot. The closer mass is to its critical value, the more energy is released and more intensive is the neutron radiation from the sample. The criticality of a sample (i.e. its closeness to the critical state) can be reduced by changing its geometry (making its surface bigger) or by putting inside it some other material (boron or cadmium) that is able to absorb neutrons. On the other hand, the criticality can be increased by putting neutron reectors around the sample. These reectors work like mirrors from which the escaped neutrons bounce back into the sample. Thus, moving in and out the absorbing material and reectors, we can keep the sample close to the critical state. How a nuclear reactor works In a typical nuclear reactor, the fuel is not in one piece, but in the form of several hundred vertical rods, like a brush. Another system of rods that contain a neutron absorbing material (control rods) can move up and down in between the fuel rods. When totally in, the control rods absorb so many neutrons, that the reactor is shut down. To start the reactor, operator gradually moves the control rods up. In an emergency situation they are dropped down automatically. To collect the energy, water ows through the reactor core. It becomes extremely hot and goes to a steam generator. There, the heat passes to water in a secondary circuit that becomes steam for use outside the reactor enclosure for rotating turbines that generate electricity. Nuclear power in South Africa By 2004 South Africa had only one commercial nuclear reactor supplying power into the national grid. It works in Koeberg located 30 km north of Cape Town. A small research reactor was also operated at Pelindaba as part of the nuclear weapons program, but was dismantled. Koeberg Nuclear Power station is a uranium Pressurized Water Reactor (PWR). In such a reactor, the primary coolant loop is pressurised so the water does not boil, and heat exchangers, called steam generators, are used to transmit heat to a secondary coolant which is allowed to boil to produce steam. To remove as much heat as possible, the water temperature in the primary loop is allowed to rise up to about 300 –C which requires the pressure of 150 atmospheres (to keep water from boiling). The Koeberg power station has the largest turbine generators in the southern hemisphere and produces 10000 MWh of electric energy. Construction of Koeberg began in 1976 and two of its Units were commissioned in 1984-1985. Since then, the plant has been in more or less continuous operation and there have been no serious incidents. » Eskom that operates this power station, may be the current technology leader. It is developing a new type of nuclear reactor, a modular pebble-bed reactor (PBMR). In contrast to traditional nuclear reactors, in this new type of reactors the fuel is not assembled in the form of rods. The uranium, thorium or plutonium fuels are in oxides (ceramic form) contained within spherical pebbles made of pyrolitic graphite. The pebbles, having a size of a tennis ball, are in a bin or can. An inert gas, helium, nitrogen or carbon dioxide, circulates through the spaces between the fuel pebbles. This carries heat away from the reactor. 362 Ideally, the heated gas is run directly through a turbine. However since the gas from the primary coolant can be made radioactive by the neutrons in the reactor, usually it is brought to a heat exchanger, where it heats another gas, or steam. The primary advantage of pebble-bed reactors is that they can be designed to be inherently safe. When a pebble-bed reactor gets hotter, the more rapid motion of the atoms in the fuel increases the probability of neutron capture by 238 92U isotopes through an eﬁect known as Doppler broadening. This isotope does not split up after capturing a neutron. This reduces the number of neutrons available to cause 235 92U ﬂssion, reducing the power output by the reactor. This natural negative feedback places an inherent upper limit on the temperature of the fuel without any operator intervention. The reactor is cooled by an inert, ﬂreproof gas, so it cannot have a steam explosion as a water reactor can. A pebble-bed reactor thus can have all of its supporting machinery fa
il, and the reactor will not crack, melt, explode or spew hazardous wastes. It simply goes up to a designed "idle" temperature, and stays there. In that state, the reactor vessel radiates heat, but the vessel and fuel spheres remain intact and undamaged. The machinery can be repaired or the fuel can be removed. A large advantage of the pebble bed reactor over a conventional water reactor is that they operate at higher temperatures. The reactor can directly heat uids for low pressure gas turbines. The high temperatures permit systems to get more mechanical energy from the same amount of thermal energy. Another advantage is that fuel pebbles for diﬁerent fuels might be used in the same basic design of reactor (though perhaps not at the same time). Proponents claim that some kinds of pebble-bed reactors should be able to use thorium, plutonium and natural unenriched Uranium, as well as the customary enriched uranium. One of the projects in progress is to develop pebbles and reactors that use the plutonium from surplus or expired nuclear explosives. On June 25, 2003, the South African Republic’s Department of Environmental Aﬁairs and Tourism approved ESKOM’s prototype 110 MW pebble-bed modular reactor for Koeberg. Eskom also has approval for a pebble-bed fuel production plant in Pelindaba. The uranium for this fuel is to be imported from Russia. If the trial is successful, Eskom says it will build up to ten local PBMR plants on South Africa’s seacoast. Eskom also wants to export up to 20 PBMR plants per year. The estimated export revenue is 8 billion rand a year, and could employ about 57000 people. 19.8.2 Fusion energy For a given mass of fuel, a fusion reaction like 1H + 3 2 1H ¡! 4 2He + n + 17:59 MeV : (19.5) yield several times more energy than a ﬂssion reaction. This is clear from the curve given in Fig. 19.3. Indeed, a change of the binding energy (per nucleon) is much more signiﬂcant for a fusion reaction than for a ﬂssion reaction. Fusion is, therefore, a much more powerful source of energy. For example, 10 g of Deuterium which can be extracted from 500 litres of water and 15 g of Tritium produced from 30 g of Lithium would give enough fuel for the lifetime electricity 363 needs of an average person in an industrialised country. But this is not the only reason why fusion attracted so much attention from physicists. Another, more fundamental, reason is that the fusion reactions were responsible for the synthesis of the initial amount of light elements at primordial times when the universe was created. Furthermore, the synthesis of nuclei continues inside the stars where the fusion reactions produce all the energy which reaches us in the form of light. Thermonuclear reactions If fusion is so advantageous, why is it not used instead of ﬂssion reactors? The problem is in the electric repulsion of the nuclei. Before the nuclei on the left hand side of Eq. (19.5) can fuse, 10¡13 cm. This is not an we have to bring them somehow close to each other to a distance of easy task! They both are positively charged and \refuse" to approach each other. » What we can do is to make a mixture of the atoms containing such nuclei and heat it up. At high temperatures the atoms move very fast. They ﬂercely collide and loose all the electrons. The mixture becomes plasma, i.e. a mixture of bare nuclei and free moving electrons. If the temperature is high enough, the colliding nuclei can overcome the electric repulsion and approach each other to a fusion distance. When the nuclei fuse, they release much more energy than was spent to heat up the plasma. Thus the initial energy \investment" pays oﬁ. The typical temperature needed to ignite the reaction of the type (19.5) is extremely high. In fact, it is the same temperature that our sun has in its center, namely, 15 million degrees. This is why the reactions (19.3), (19.5), and the like are called thermonuclear reactions. » Human-made thermonuclear reactions The same as with ﬂssion reactions, the ﬂrst application of thermonuclear reactions was in weapons, namely, in the hydrogen bomb, where fusion is ignited by the explosion of an ordinary (ﬂssion) plutonium bomb which heats up the fuel to solar temperatures. In an attempt to make a controllable fusion, people encounter the problem of holding the plasma. It is relatively easy to achieve a high temperature (with laser pulses, for example). But as soon as plasma touches the walls of the container, it immediately cools down. To keep it from touching the walls, various ingenious methods are tried, such as strong magnetic ﬂeld and laser beams directed to plasma from all sides. In spite of all eﬁorts and ingenious tricks, all such attempts till now have failed. Most probably this straightforward approach to controllable fusion is doomed because one has to hold in hands a \piece of burning sun". Cold fusion To visualize the struggle of the nuclei approaching each other, imagine yourself pushing a metallic ball towards the top of a slope shown in Fig. 19.5. The more kinetic energy you give to the ball, the higher it can climb. Your purpose is to make it fall into the narrow well that is behind the barrier. 364 Coulomb barrier Veﬁ ¡ ¡“ projectile x R Figure 19.5: Eﬁective nucleus{nucleus potential as a function of the separation between the nuclei. In fact, the curve in Fig. 19.5 shows the dependence of relative potential energy Veﬁ between two nuclei on the distance R separating them. The deep narrow well corresponds to the strong short-range attraction, and the 1=R barrier represents the Coulomb (electric) repulsion. The nuclei need to overcome this barrier in order to \touch" each other and fuse, i.e. to fall into the narrow and deep potential well. One way to achieve this is to give them enough kinetic energy, which means to rise the temperature. However, there is another way based on the quantum laws. » As you remember, when discussing the motion of the electron inside an atom (see Sec. 19.1), we said that it formed a \cloud" of probability around the nucleus. The density of this cloud diminishes at very short and very long distances but never disappears completely. This means that we can ﬂnd the electron even inside the nucleus though with a rather small probability. The nuclei moving towards each other, being microscopic objects, obey the quantum laws as well. The probability density for ﬂnding one nucleus at a distance R from another one also forms a cloud. This density is non-zero even under the barrier and on the other side of the barrier. This means that, in contrast to classical objects, quantum particles, like nuclei, can penetrate through potential barriers even if they do not have enough energy to go over it! This is called the tunneling eﬁect. The tunneling probability strongly depends on thickness of the barrier. Therefore, instead of lifting the nuclei against the barrier (which means rising the temperature), we can try to make the barrier itself thinner or to keep them close to the barrier for such a long time that even a low penetration probability would be realized. How can this be done? The idea is to put the nuclei we want to fuse, inside a molecule where they can stay close to each other for a long time. Furthermore, in a molecule, the Coulomb barrier becomes thinner because of electron screening. In this way fusion may proceed even at 365 room temperature. This idea of cold fusion was originally (in 1947) discussed by F. C. Frank and (in 1948) put forward by A. D. Sakharov, the \father" of Russian hydrogen bomb, who at the latest stages of his career was worldwide known as a prominent human rights activist and a winner of the Nobel Prize for Peace. When working on the bomb project, he initiated research into peaceful applications of nuclear energy and suggested the fusion of two hydrogen isotopes via the reaction (19.5) by forming a molecule of them where one of the electrons is replaced by a muon. The muon is an elementary particle (see Sec. 19.9), which has the same characteristics as an electron. The only diﬁerence between them is that the muon is 200 times heavier than the electron. In other words, a muon is a heavy electron. What will happen if we make a muonic atom of hydrogen, that is a bound state of a proton and a muon? Due to its large mass the muon would be very close to the proton and the size of such atom would be 200 times smaller than that of an ordinary atom. This is clearly seen from the formula for the atomic Bohr radius where the mass is in the denominator. RBohr = ~2 me2 ; Now, what happens if we make a muonic molecule? It will also be 200 times smaller than an ordinary molecule. The Coulomb barrier will be 200 times thinner and the nuclei 200 times closer to each other. This is just what we need! Speaking in terms of the eﬁective nucleus{nucleus potential shown in Fig. 19.5, we can say that the muon modiﬂes this potential in such a way that a second minimum appears. Such a modiﬂed potential is (schematically) shown in Fig. 19.6. Veﬁ probability density ¡ ¡“ x R Figure 19.6: Eﬁective nucleus{nucleus potential (thick curve) for nuclei conﬂned in a molecule. Thin curve shows the corresponding distribution of the probability for ﬂnding the nuclei at a given distance from each other. The molecule is a bound state in the shallow but wide minimum of this potential. Most of 366 the time, the nuclei are at the distance corresponding to the maximum of the probability density distribution (shown by the thin curve). Observe that this density is not zero under the barrier (though is rather small) and even at R = 0. This means that the system can (with a small probability) jump from the shallow well into the deep well through the barrier, i.e. can tunnel and fuse. Unfortunately, the muon is not a stable particle. Its lifetime is only 10¡6 sec. This means that a muonic molecule cannot exist longer than 1 microsecond. As a matter of fact, from a quantum mechanical point of view, this is quite a long interval.
» The quantum mechanical wave function (that describes the probability density) oscillates with a frequency which is proportional to the energy of the system. With a typical binding 1017 s¡1. This means that the particle energy of a muonic molecule of 300 eV this frequency is hits the barrier with this frequency and during 1 microsecond it makes 1011 attempts to jump 10¡7. Therefore, during through it. The calculations show that the penetration probability is 1 microsecond nuclei can penetrate through the barrier 10000 times and fusion can happen much faster than the decay rate of the muon. » » Cold fusion via the formation of muonic molecules was done in many laboratories, but unfortunately, it cannot solve the problem of energy production for our needs. The obstacle is the negative e–ciency, i.e. to make muonic cold fusion we have to spend more energy than it produces. The reason is that muons do not exist like protons or electrons. We have to produce them in accelerators. This takes a lot of energy. Actually, the muon serves as a catalyst for the fusion reaction. After helping one pair of nuclei to fuse, the muon is liberated from the molecule and can form another molecule, and so on. It was estimated that the e–ciency of the energy production would be positive only if each muon ignited at least 1000 fusion events. Experimentalists tried their best, but by now the record number is only 150 fusion events per muon. This is too few. The main reason why the muon does not catalyze more reactions is that it is eventually trapped by a 4He nucleus which is a by-product of fusion. Helium captures the muon into an atomic orbit with large binding energy, and it cannot escape. Nonetheless, the research in the ﬂeld of cold fusion continues. There are some other ideas of how to keep nuclei close to each other. One of them is to put the nuclei inside a crystal. Another way out is to increase the penetration probability by using molecules with special properties, namely, those that have quantum states with almost the same energies as the excited states on the compound nucleus. Scientists try all possibilities since the energy demands of mankind grow continuously and therefore the stakes in this quest are high. 19.9 Elementary particles In our quest for the elementary building blocks of the universe, we delved inside atomic nucleus and found that it is composed of protons and neutrons. Are the three particles, e, p, and n, the blocks we are looking for? The answer is \no". Even before the structure of the atom was understood, Becquerel discovered the redioactivity (see Sec. 19.5.1) that afterwards puzzled physicists and forced them to look deeper, i.e. inside protons and neutrons. 367 19.9.1 ﬂ decay Among the three types of radioactivity, the ﬁ and rays were easily explained. The emission of ﬁ particle is kind of ﬂssion reaction, when an initial nucleus spontaneously decays in two fragments one of which is the nucleus 4 2He (i.e. ﬁ particle). The rays are just electromagnetic quanta emitted by a nuclear system when it transits from one quantum state to another (the same like an atom emits light). The ﬂ rays posed the puzzle. On the one hand, they are just electrons and you may think that it looks simple. But on the other hand, they are not the electrons from the atomic shell. It was found that they come from inside the nucleus! After the ﬂ-decay, the charge of the nucleus increases in one unit, A Z (parent nucleus) ¡! A Z+1 (daughter nucleus) + e ; which is in accordance with the charge conservation law. There was another puzzle associated with the ﬂ decay: The emitted electrons did not have a certain energy. Measuring their kinetic energies, you could ﬂnd very fast and very slow electrons as well as the electrons with all intermediate speeds. How could identical parent nuclei, after loosing diﬁerent amount of energy, become identical daughter nuclei. May be energy is not conserving in the quantum world? The fact was so astonishing that even Niels Bohr put forward the idea of statistical nature of the energy conservation law. To explain the ﬂrst puzzle, it was naively suggested that neutron is a bound state of proton and electron. At that time, physicists believed that if something is emitted from an object, this something must be present inside that object before the emission. They could not imagine that a particle could be created from vacuum. The naive (pe) model of the neutron contradicted the facts. Indeed, it was known already that the pe bound state is the hydrogen atom. Neutron is much smaller than the atom. Therefore, it would be unusually tight binding, and perhaps with something elese involved that keeps the size small. By the way, this \something elese" could also save the energy conservation law. In 1930, Wolfgang Pauli suggested that in addition to the electron, the ﬂ decay involves another particle, ”, that is emitted along with the electron and carries away part of the energy. For example, 234 90Th ¡! 234 91Pa + e¡ + „” : (19.6) This additional particle was called neutrino (in Italian the word \neutrino" means small neutron). The neutrino is electrically neutral, has extremely small mass (maybe even zero, which is still a question in 2004) and very weakly interacts with matter. This is why it was not detected experimentally till 1956. The \bar" over ” in Eq. (19.6) means that in this reaction actually the anti-neutrino is emitted (see the discussion on anti-particles further down in Sec. 19.9.2). 19.9.2 Particle physics In an attempt to explain the ﬂ decay and to understand internal structure of the neutron a new branch of physics was born, the particle physics. The only way to explore the structure of subatomic particles is to strike them with other particles in order to knock out their \constituent" parts. The simple logic says: The more powerful the impact, the smaller parts can be knocked 368 out. At the beginning the only source of energetic particles to strike other particles were the cosmic rays. Earth is constantly bombarded by all sort of particles coming from the outer space. Atmosphere protects us from most of them, but many still reach the ground. Antiparticles In 1932, studying the cosmic rays with a bubble chamber, Carl Anderson made a photograph of two symmetrical tracks of charged particles. The measurements of the track curvatures showed that one track belonged to an electron and the other was made by a particle having the same mass and equal but positive charge. These particles were created when a cosmic quantum of a high energy collided with a nucleus. The discovered particle was called positron and denoted as e+ to distinguish it from the electron, which sometimes is denoted as e¡. It was the ﬂrst antiparticle discovered. Later, it was found that every particle has its \mirror reection", the antiparticle. To denote an antiparticle, it is used \bar" over a particle symbol. For example, „p is the anti-proton, which has the same mass as an ordinary proton but a negative charge. When a particle collides with its \mirror reection", they annihilate, i.e. they burn out completely. In this collision, all their mass is transformed into electromagnetic energy in the form of quanta. For example, if an electron collides with a positron, the following reaction may take place e¡ + e+ + ; ¡! (19.7) where two photons are needed to conserve the total momentum of the system. In principle, stable antimatter can exist. For example, the pair of „p and e+ can form an atom of anti-hydrogen with exactly the same energy states as the ordinary hydrogen. Experimentally, atoms of anti-helium were obtained. The problem with them is that, surrounded by ordinary matter, they cannot live long. Colliding with ordinary atoms, they annihilate very fast. There are speculations that our universe should be symmetric with respect to particles and antiparticles. Indeed, why should preference be given to matter and not to anti-matter? This implies that somewhere very far, there must be equal amount of anti-matter, i.e. anti-universe. Can you imagine what happens if they meet? Muon, mesons, and the others In yet another cosmic-ray experiment a particle having the same properties as the electron but 207 times heavier, was discovered in 1935. It was given the name muon and the symbol „. For » a long time it remained \unnecessary" particle in the picture of the world. Only the modern theories harmonically included the muon as a constituent part of matter (see Sec 19.9.3). The same inexhaustible cosmic rays revealed the … and K mesons in 1947. The … mesons (or simply pions) were theoretically predicted twelve years before by Yukawa, as the mediators of the strong forces between nucleons. The K mesons, however, were unexpected. Furthermore, they showed very strange behaviour. They were easily created only in pairs. The probability 369 of the inverse process (i.e. their decay) was 1013 times lower than the probability of their creation. It was suggested that these particles possess a new type of charge, the strangeness, which is conserving in the strong interactions. When a pair of such particles is created, one of them has strangeness +1 and the other 1, so the total strangeness remains zero. When decaying, they act individually and therefore the strangeness is not conserving. According to the suggestion, this is only possible through the weak interactions that are much weaker than the strong interactions (see Sec. 19.9.4) and thus the decay probability is much lower. ¡ The golden age of particle physics began in 1950-s with the advent of particle accelerators, the machines that produced beams of electrons or protons with high kinetic energy. Having such beams available, experimentalists can plan the experiment and repeat it, while with the cosmic rays they were at the mercy of chance. When the accelerators became the main tool of exploration, the particle physics acquired its second name, the high energy physics. During the last hal
f a century, experimentalists discovered so many new particles (few of them are listed in Table 19.3) that it became obvious that they cannot all be elementary. When colliding with each other, they produce some other particles. Mutual transformations of the particles is their main property. family photon leptons hadrons Lifetime T1=2 (s) stable stable 2:2 10¡6 £ 10¡13 stable stable stable 2:6 0:8 1:2 0:9 5:2 10¡8 10¡16 10¡8 10¡10 10¡8 £ £ £ £ £ 10¡18 stable 900 particle photon electron muon tau electron neutrino muon neutrino tau neutrino pion pion kaon kaon kaon eta meson proton neutron lambda sigma sigma sigma omega symbol e¡, e+ „¡, „+ ¿ ¡, ¿ + ”e ”„ ”¿ …+, …¡ …0 K +, K ¡ K 0 S K 0 L ·0 p n ⁄0 §+ §0 §¡ ›¡, ›+ mass (MeV) 0 0.511 105.7 1777 0 » 0 » 0 » 139.6 135.0 493.7 497.7 497.7 548.8 938.3 939.6 1116 1189 1192 1197 1672 £ £ £ £ £ Table 19.3: Few representatives of diﬁerent particle families. 2:6 0:8 6 1:5 0:8 10¡10 10¡10 10¡20 10¡10 10¡10 Physicists faced the problem of particle classiﬂcation similar to the problems of classiﬂcation of animals, plants, and chemical elements. The ﬂrst approach was very simple. The particles were leptons (light particles, like electron), mesons divided in four groups according to their mass: (intermediate mass, like pion), baryons (heavy particles, like proton or neutron), and hyperons 370 (very heavy particles). Then it was realized that it would be more logical to divide the particles in three families according to their ability to interact via weak, electromagnetic, and strong forces (in addition to that, all particles experience gravitational attraction towards each other). Except for the gravitational interaction, the photon ( quantum) participates only in electromagnetic interactions, the leptons take part in both weak and electromagnetic interactions, and hadrons are able to interact via all forces of nature (see Sec. 19.9.4). In addition to conservation of the strangeness, several other conservation laws were discovered. For example, number of leptons is conserving. This is why in the reaction (19.6) we have an electron (lepton number +1) and anti-neutrino (lepton number 1) in the ﬂnal state. Similarly, the number of baryons is conserving in all reactions. ¡ The quest for the constituent parts of the neutron has led us to something unexpected. We found that there are several hundreds of diﬁerent particles that can be \knocked out" of the neutron but none of them are its parts. Actually, the neutron itself can be knocked out of some of them! What a mess! Further eﬁorts of experimentalists could not ﬂnd an order, which was ﬂnally discovered by theoreticians who introduced the notion of quarks. 19.9.3 Quarks and leptons While experimentalists seemed to be lost in the maze, the theoreticians groped for the way out. Using an extremely complicated mathematical technique, they managed to group the hadrons in such families which implied that all known (and yet unknown) hadrons are build of only six types of particles with fractional charges. The main credit for this (in the form of Nobel Prize) was given to M. Gell-Mann and G. Zweig. At ﬂrst, they considered a subset of the hadrons and developed a theory with only three types of such truly elementary particles. When Murray Gell-Mann thought of the name for them, he came across the book "Finnegan’s Wake" by James Joyce. The line "Three quarks for Mister Mark..." appeared in that fanciful book (in German, the word \quark" means cottage cheese). He needed a name for three particles and this was the answer. Thus the term quark was coined. Later, the theory was generalized to include all known particles, which required six types of quarks. Modern theories require also that the number of diﬁerent leptons should be the same as the number of diﬁerent quark types. According to these theories, the quarks and leptons are truly elementary, i.e. they do not have any internal structure and therefore are of a zero size (pointlike). Thus, the world is constructed of just twelve types of elementary building blocks that are given in Table 19.4. Amazingly enough, the electron that was discovered before all other particles, more than a century ago, turned out to be one of them! After Gell-Mann, who used a funny name (quark) for an elementary particle, the fundamental physics was ooded with such names. For example, the six quark types are called avors (for cottage cheese, this is appropriate indeed), the three diﬁerent states in which each quark can be, are called colors (red, green, blue), etc. Modern physics is so complicated and mathematical, that people working in it, need such kind of jokes to \spice unsavoury dish with avors". The funny names should not confuse anybody. Elementary particles do not have any smell, taste, or colour. These terms simply denote certain properties (similar to electric charge) that do not 371 family elementary particle symbol charge leptons quarks electron muon tau electron neutrino muon neutrino tau neutrino up down strange charmed top (truth) bottom (beauty) e¡ „¡ ¿ ¡ ”e ”„ ”¿ 2=3 1=3 ¡ 1=3 ¡ +2=3 +2=3 1=3 ¡ lepton number 1 baryon number /3 1/3 1/3 1/3 1/3 1/3 mass (MeV) 0.511 105.7 1777 » » 0 0 0 » 360 360 1500 540 174000 5000 Table 19.4: Elementary building blocks of the universe. exist in human world. Hadrons There are particles that are able to interact with each other by the so-called strong forces. An10¡15 m), other name for these forces is nuclear forces. They are very strong at short distances ( and very quickly vanish when the distance between the particles increases. All these particles are called hadrons. The protons and neutrons are examples of hadrons. » As you remember, we learned about the existence of huge variety of particles when trying to look inside a nucleon, more particularly, the neutron. So, what the neutron is made of? Can we get the answer at last, after learning about the quarks? Yes, we can. According to modern theories, all hadrons are composed of quarks. The quarks can be combined in groups of two or three. The bound states of two quarks are called mesons, and the bound complexes of three quarks are called baryons. No other numbers of quarks can form observable particles1. Nucleons are baryons and therefore consist of three quarks while the pion is a meson containing only two quarks, as schematically shown in Fig. 19.7. Comparing this ﬂgure with Table 19.4, you can see why quarks have fractional charges. Counting the total charge of a hadron, you should not forget that anti-quarks have the opposite charges. The baryon number for an anti-quark also has the opposite sign (negative). This is why mesons actually consist of a quark and anti-quark in order to have total baryon number zero. 1Recently, experimentalists and theoreticians started to actively discuss the possibility of the existence of pentaquarks, exotic particles that are bound complexes of ﬂve quarks. 372 ”•u ”•u „‚ „‚ ”•d „‚ proton ”•d „‚ ”•d „‚ ”•u ”•u ”•„d „‚ „‚ „‚ neutron …+ meson Figure 19.7: Quark content of the proton, neutron, and …+-meson. Particle reactions At the early stages of the particle physics development, in order to ﬂnd the constituent parts of various particles, experimentalists simply collided them and watched the \fragments". However, this straightforward approach led to confusion. For example, the reaction between the … ¡ meson and proton, (19.8) would suggest (if naively interpreted) that either K 0 or ⁄0 is a constituent part of the nucleon while the pion is incorporated into the other \fragment". On the other hand, the same collision can knock out diﬁerent \fragments" from the same proton. For example, ¡! …¡ + p K 0 + ⁄0 ; which leads to an absurd suggestion that neutron is a constituent part of proton. …¡ + p ¡! …0 + n ; (19.9) The quark model explains all such \puzzles" nicely and logically. Similarly to chemical reactions that are just rearrangements of atoms, the particle reactions of the type (19.8) and (19.9) are just rearrangements of the quarks. The only diﬁerence is that, in contrast to chemistry where the number of atoms is not changing, the number of quarks before the collision is not necessarily equal to their number after the collision. This is because a quark from one colliding particle can annihilate with the corresponding antiquark from another particle. Moreover, if the collision is su–ciently powerful, the quark-antiquark pairs can be created from vacuum. It is convenient to depict the particle transformations in the form of the so-called quark ow diagrams. On such diagrams, the quarks are represented by lines that may be visualized as the trajectories showing their movement from the left to the right. For example, the diagram given in Fig. (19.8), shows the quark rearrangement for the reaction (19.8). As you can see, when the pion collides with proton, its „u quark annihilates with the u quark from the proton. At the same time, the s„s pair is created from the vacuum. Then, the „s quark binds with the d quark to form the strange meson K 0, while the s quark goes together with the ud pair as the strange baryon ⁄0. The charge-exchange reaction (19.9) is a more simple rearrangement process shown in Fig. 19.9. You may wonder why the quark and antiquark of the same avor in the …0 meson do not annihilate. Yes they do, but not immediately. And due to this annihilation, the lifetime of … 0 is 100 million times shorter than the lifetime of …§ (see Table 19.3). 373 …¡ ‰ p ( d „u u u d d „s s u d K 0 ⁄0 ) Figure 19.8: Quark-ow diagram for the reaction …¡ + p K 0 + ⁄0 . ¡! …¡ ‰ p ( „u d u u d „u u d u d …0 n ) Figure 19.9: Quark-ow diagram for the reaction …¡ + p …0 + n . ¡! Despite its simplicity, the quark-ow diagram technique is very powerful method not only for explaining the observed reactions but also for predicting new reactions that have not yet been seen in experiments. Knowing the quark content of particles (which is available in modern Physics Handbooks), you can draw plenty of
such diagrams that will describe possible particle transformations. The only rule is to keep the lines continuous. They can disappear or emerge only for a quark-antiquark pair of the same avor. However, the continuity of the quark lines is valid only for the processes caused by the strong interaction. Indeed, the ﬂ-decay of a free neutron (caused by the weak forces), as well as the ﬂ-decay of the nuclei, indicate that quarks can change avor. In particular, the ﬂ-decay (19.10) or (19.6) happens because the d quark transformes into the u quark, n ¡! p + e¡ + „”e ; (19.10) d ¡! u + e¡ + „”e ; (19.11) due to the weak interaction, as shown in Fig. 19.10 Quark conﬂnement At this point, it is very logical to ask if anybody observed an isolated quark. The answer is \no". Why? And how can one be so conﬂdent of the quark model when no one has ever seen an 374 „”e e Figure 19.10: Quark-ow diagram for the ﬂ decay of neutron. isolated quark? Basically, you can’t see an isolated quark because the quark-quark attractive force does not let them go. In contrast to all other systems, the attraction between quarks grows with the distance separating them. It is like a rubber cord connecting two balls. When the balls are close to each other, the cord is not stretched and the balls do not feel any force. If, however, you try to separate the balls, the cord pulls them back. The more you stretch the cord, the stronger the force becomes (according to the Hook’s law of elasticity). Of course, a real rubber cord would eventually break. This does not happen with the quark-quark force. It can grow to inﬂnity. This phenomenon is called the conﬂnement of quarks. Nonetheless, we are sure that the nucleon consists of three quarks having fractional charges. A hundred years ago Rutherford, by observing the scattering of charged particles from an atom, proved that its positive charge is concentrated in a small nucleus. Nowadays, similar experiments prove the existence of fractional point-like charges inside the nucleon. The quark model actually is much more complicated than the quark-ow diagrams. It is a consistent mathematical theory that explains a vast variety of experimental data. This is why nobody doubts that it reects the reality. 19.9.4 Forces of nature If asked how many types of forces exist, many people start counting on their ﬂngers, and when the count exceeds ten, they answer \plenty of". Indeed, there are gravitational forces , electrical, magnetic, elastic, frictional forces, and also forces of wind, of expanding steam, of contracting muscles, etc. If, however, we analyze the root causes of all these forces, we can reduce their number to just a few fundamental forces (or fundamental interactions, as physicists say). For example, the elastic force of a stretched rubber cord is due to the attraction between the molecules that the rubber is made of. Looking deeper, we ﬂnd that the molecules attract each other because of the electromagnetic attraction between the electrons of one molecule and nuclei of the other. Similarly, if we depress a piece of rubber, it resists because the molecules refuse to approach each other too close due to the electric repulsion of the nuclei. Therefore the elasticity 375 of rubber has the electromagnetic origin. Any other force in the human world can be analyzed in the same manner. After doing this, we will ﬂnd that all forces that we see around us (in the macroworld), are either of gravitational or electromagnetic nature. As we also know, in the microworld there are two other types of forces: The strong (nuclear) forces that act between all hadrons, and the weak forces that are responsible for changing the quark avors. Therefore, all interactions in the Universe are governed by only four fundamental forces: Strong, electromagnetic, weak and gravitational. These forces are very diﬁerent in strength and range. Their relative strengths are given in Table 19.5. The most strong is the nuclear interaction. The strength of the electromagnetic forces is one hundred times lower. The weak forces are nine orders of magnitude weaker than the nuclear forces, and the gravity is 38 orders of magnitude weaker! It is amazing that this subtle interaction governs the cosmic processes. The reason is that the gravitational forces are of long range and always attractive. There is no such thing as negative mass that would screen the gravitational ﬂeld, like negative electrons screen the ﬂeld of positive nuclei. Force Relative Strength Range Strong 1 Electromagnetic 0.0073 Short Long Weak Gravitational 10¡9 10¡38 Very Short Long Table 19.5: Four fundamental forces and their relative strengths. Towards the uniﬂed force Physicists always try to simplify things. Since there are only four fundamental forces, it is tempting to ask "If only four, then why not only one?". Can it be that all interactions are just diﬁerent faces of one master force? The ﬂrst who started the quest for uniﬂcation of forces was Einstein. After completing his general theory of relativity, he spent 30 years in unsuccessful attempts to unify the electromagnetic and gravity forces. At that time, it seemed logical because both of them were inﬂnite in range and obeyed the same inverse square law. Einstein failed because the uniﬂcation should be done on the basis of quantum laws, but he tried to do it using the classical concepts. Electro-weak uniﬂcation Now it is known that despite the similarities in form of the gravity and electromagnetic forces, the gravity will be the last to yield to uniﬂcation. The more implausible uniﬂcation of the electro- 376 magnetic and weak forces turned out to be the ﬂrst successful step towards the uniﬂed interaction. In 1979, the Nobel prize was awarded to Weinberg, Salam, and Glashow, who developed a uniﬂed theory of electromagnetic and weak interactions. According to that theory, the electromagnetic and weak forces converge to one electro-weak interaction at very high collision energies. The theory also predicted the existence of heavy particles, the W and Z, with masses around 80000 MeV and 90000 MeV, respectively. These particles were discovered in 1983, which brought experimental veriﬂcation to the new theory. Grand uniﬂcation The next step was to try to combine the electro-weak theory with the theory of the strong interactions (i.e. quark theory) in a single theory. This work was called the grand uniﬂcation. Currently, physicists discuss versions of such theory that predicts the convergence of the three forces at aw1017 MeV. The quarks and leptons in this theory, are the uniﬂed leptoquarks. fully high energies » The grand uniﬂcation is not that successful as the electro-weak theory. It has the problem of mathematical consistency and contradicts to at least one experiment. The matter is that it predicts the proton decay, that does not conserve both the baryon and lepton numbers, with the lifetime of The measurements show, however, that the lifetime of the proton is at least 1032 years. » 1029 years. p ¡! e+ + …0 ; Theory of everything Some people believe that the grand uniﬂcation has an inherent principal aw. According to them, one cannot unify the forces step by step (leaving the gravity out), and the correct way is to combine all four forces in the so-called theory of everything. There are few diﬁerent approaches to unifying everything. One of them suggests that all fundamental particles (quarks and leptons) are just vibrating modes of string loops in multidimensional space. The electron is a string vibrating one way, the up-quark is a string vibrating another way, and so on. The other approach introduces a new level of fundamental particles, the preons, that could be constituent parts of quarks and leptons. The quest goes on. Everyone agrees that constructing the theory of everything would in no way mean that biology, geology, chemistry, or even physics had been solved. The universe is so rich and complex that the discovery of the fundamental theory would not mean the end of science. The ultimate theory of everything would provide an unshakable pillar of coherence forever assuring us that the universe is a comprehensible place. 19.10 Origin of the universe Looking deep inside microscopic particles, physicists need to collide them with high kinetic energies. The smaller parts of matter they want to observe, the higher energy they need. This is why they build more and more powerful accelerators. However, the accelerators have natural limitations. Indeed, an accelerator cannot be bigger than the size of our planet. And even if we manage to build a circular accelerator around the whole earth (along the equator, for example), 377 it would not be able to reach the energy of mental interactions takes place. » 1017 MeV at which the grand uniﬂcation of funda- So, what are we to do? How can we test the theory of everything? Is it possible at all? 1017 MeV, should be looked for in the cosmos, Yes, it is! The astronomically high values, like » of course. Our journey towards extremely small objects eventually leads us to extremely large objects, like whole universe. Equations of Einstein’s theory of relativity can describe the evolution of the universe. Physicists solved these equations back in time and found that the universe had its beginning. Approximately 15 billion years ago, it started from a zero size point that exploded and rapidly expanded to the present tremendous scale. At the ﬂrst instants after the explosion, the matter was at such incredibly high density and temperature that all particles had kinetic energies even higher than 1017 MeV. This means that at the very beginning there was only one the uniﬂcation energy sigle force and no diﬁerence among fundamental particles. Everything was uniﬂed and \simple". » You may ask \So what? How can so distant past help us?". In many ways! The development of the universe was governed by the fundamental forces. If our theories about them are correct, we should be able to reproduce (wi
th calculations) how that development proceeded step by step. During the expansion, all the nuclei and atoms in the cosmos were created. The amounts of diﬁerent nuclei are not the same. Why? Their relative abundances were determined by the processes in the ﬂrst moments after the explosion. Thus, comparing what follows from the theories with the observed abundances of chemical elements, we can judge validity of our theories. Nowadays, the most popular theory, describing the history of the universe, is the so{called Big-Bang model. The diagram given in Fig. 19.11, shows the sequence of events which led to the creation of matter in its present form. Nobody knows what was before the Big Bang and why it happened, but it is assumed that just after this enigmatic cataclysm, the universe was so dense and hot that all four forces of nature (strong, electromagnetic, weak, and gravitational) were indistinguishable and therefore gravity was governed by quantum laws, like the other three types of interactions. A complete theory of quantum gravity has not been constructed yet, and this very ﬂrst \epoch" of our history remains as enigmatic as the Big Bang itself. The ideal \democracy" (equality) among the forces lasted only a small fraction of a second. 1032 K and the gravity separated. By the time t The other three forces, however, remained uniﬂed into one universal interaction mediated by an extremely heavy particle, the so-called X boson, which could transform leptons into quarks and vice versa. 10¡43 sec the universe cooled down to » » » When at t 10¡35 sec most of the X bosons decayed, the quarks combined in trios and pairs 10¡10 sec, to form nucleons, mesons, and other hadrons. The only symmetry which lasted up to was between the electromagnetic and weak forces mediated by the Z and W particles. From 10¡10 sec) until the universe was about one the moment when this last symmetry was broken ( second old, neutrinos played the most signiﬂcant role by mediating the neutron-proton transmutations and therefore ﬂxing their balance (neutron to proton ratio). » » Already in a few seconds after the Big Bang nuclear reactions started to occur. The protons 378 ? ? ? ? ? ? ? BIG BANG ? ? ? ? single uniﬂed force gravitational force separated strong force separated weak force separated n+” ! p+e¡ , p+ „” ! n+e+ p+n ! 2H+, 2H+2H ! 4He+ pp{chain today 1032 K 1028 K 1015 K 1010 K 109 K 107 K 2:9 K » ? temperature 0 10 10 10 ¡43sec ¡35sec ¡10sec 1 sec 10 sec 500 sec » ? 15 £ 109years time Figure 19.11: Schematic \history" of the universe. and neutrons combined very rapidly to form deuterium and then helium. During the very ﬂrst seconds there were too many very energetic photons around which destroyed these nuclei immediately after their formation. Very soon, however, the continuing expansion of the universe changed the conditions in favour of these newly born nuclei. The density decreased and the photons could not destroy them that fast anymore. During a short period of cosmic history, between about 10 and 500 seconds, the entire universe behaved as a giant nuclear fusion reactor burning hydrogen. This burning took place via a chain of nuclear reactions, which is called the pp-chain because the ﬂrst reaction in this sequence is the proton-proton collision leading to the formation of a deuteron. Nowadays, the same pp-chain is the main source of energy in our sun and other stars. But how do we know that the scenario was like this? In other words, how can we check the Big{Bang theory? Is it possible to prove something which happened 15 billion years ago and in such a short time? Yes, it is! The pp-chain fusion, pp-chain: p + p e¡+p + p p + 2H 3He + 3He 3He + 4He 2H + e+ + ”e 2H + ”e 3He + 4He + p + p 7Be + ! ! ! ! ! 379 . & p + 7Be 8B 8Be⁄ 7Li + ”e 8Be + 4He + 4He e¡+7Be p + 7Li 8Be ! ! ! 8B + 8Be⁄ + e+ +”e 4He + 4He ! ! ! is the key for such a proof. ‰=‰p 6 1 ¡2 ¡4 ¡6 ¡8 10 10 10 10 ¡10 ¡12 10 10 helium deuterium - 10 102 103 104 t (sec) Figure 19.12: Mass fractions ‰ (relative to hydrogen ‰p) of primordial deuterium and 4He versus the time elapsed since the Big Bang. As soon as the nucleosynthesis started, the amount of deuterons, helium isotopes, and other light nuclei started to increase. This is shown in Fig. 19.12 for 2H and 4He. The temperature and the density, however, continued to decrease. After a few minutes the temperature dropped to such a level that the fusion practically stopped because the kinetic energy of the nuclei was not su–cient to overcome the electric repulsion between nuclei anymore. Therefore the abundances of light elements in the cosmos were ﬂxed (we call them the primordial abundances). Since then, they practically remain unchanged, like a photograph of the past events, and astronomers can measure them. Comparing the measurements with the predictions of the theory, we can check whether our assumptions about the ﬂrst seconds of the universe are correct or not. Astronomy and the physics of microworld come to the same point from diﬁerent directions. The Big Bang theory is only one example of their common interest. Another example is related to the mass of neutrino. When Pauli suggested this tiny particle to explain the nuclear ﬂ-decay, it was considered as massless, like the photon. However, the experiments conducted recently, indicate that neutrinos may have small non-zero masses of just a few eV. In the world of elementary particles, this is extremely small mass, but it makes a huge difference in the cosmos. The universe continues to expand despite the fact that the gravitational forces pull everything back to each other. The estimates show, that the visible mass of all galaxies is not su–cient to stop and reverse the expansion. The universe is ﬂlled with a tremendous number of neutrinos. Even with few eV per neutrino, this amounts to a huge total mass of them, which is invisible but could reverse the expansion. Thus, the cooperation of astronomers and particle physicists has led to signiﬂcant advances in our understanding of the universe and its evolution. The quest goes on. A famous German 380 philosopher Friedrich Nietzsche once said that \The most incomprehensible thing about this Universe is that it is comprehensible." 381 Appendix A GNU Free Documentation License 2000,2001,2002 Free Software Foundation, Inc. Version 1.2, November 2002 Copyright c 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document \free" in the sense of freedom: to assure everyone the eﬁective freedom to copy and redistribute it, with or without modifying it, either commercially or non-commercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modiﬂcations made by others. 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meet ; with this point of intersection as a centre and with any line as a radius, describe a circle ; let L be the length of the arc of this circle intercepted between the two lines and let R be the length of the radius of the circle ; then by definition the ratio — is the numerical value of the angle. Thus, in the figure, AB and CD are the two lines lying in a plane ; 0 is their point of intersection; OP is the radius of the circle, i.e. R; PQ is the intercepted arc, i.e. L ; hence the PQ angle has the value — ^- This definition of the value for an A p , angle is adopted because it is known — R— — i from geometry that the ratio of the FIG. i.- Numerical^ value of the intercepted arc to the radius of its i8^ circle is the same for all" values of the radius ; and therefore it is not necessary to specify the latter. If Nis the numerical value of the angle, this relation can be expressed N= =\|, or L = RN; in words, the length of the intercepted arc equals the product of the length of the corresponding radius and the value of the angle. A unit angle is called a " radian " ; it is an angle such that the lengths of the intercepted arc and the radius are equal. In ordinary language angles are expressed in "degrees," "minutes," and "seconds" ; there being 60 seconds in a min- ute, 60 minutes in a degree, and 90 degrees in a right an<rl<«. INTRODUCTION a , e s u c f i R It is easy to find the relation between a radian and a degree ; e b a the arc intercepted by two lines making a right angle, where TT= 8.141»l approximately ; and therefore — , or 1.5708, is the — value of a right angle in terms of radians. Hence, l.")708 radians = 90°, or 1 radian = 57°.2958 = 57° 17' 45". ? / An angle has a sign as well as a numerical value. Thus, if OA be chosen as a fixed direction, there is a difference between the angles (AOB) and (y40(7), although they are numeric- ally equal. The latter corresponds °' to a rotation like that of the hands of a watch ; the former to a contrary rotation. One angle, it is immaterial which, is called plus ( + ); the other, minus (— ). When an angle becomes very small, the ratio of the value of its sine to its own value approaches unity. For, referring to the cut, the value of the angle between OA and OB is the ratio of the arc AB to the radius; the value of the sine of this angle is defined to be the ratio of CB to the radius; therefore, the ratio of the an^le t»» hs sine is the ratio of the an- , 1 />' t.> t '/>'. Fio 8 -In the limit when '^lis (M!ll;ils tn<> rati° °f tne arc /; ' /; to its ch'ord BCD; and, as the angle is made smaller and smaller, this ratio /t)-(AOB). . approaches unity, because in tin- limit an arc and its chord eqaaL 3. Vectors and vector quantities. — A vector is a limited port i«»ll nf a MlMl-ht illir ill a definite diivrtinn. TllUS tin- INTRODUCTION straight lines AB and CD are vectors ; their lengths are the distances between A and B and between C and D ; and their directions are indicated by the arrows. Three ideas are involved: the direction of the line, the sense of this direction (i.e. a distinction is made between a line drawn to the right and one drawn to the left, etc.), and the length of the line. The position of the line is immaterial; so two vectors of the same length and in the same direction, wherever placed, are equal. A vector, then, is a straight line traced by a point moving from one position to another, as is indicated by the use of an "arrow" in the line. FIG. 4. — Two vectors. The process of "addition of vectors" is defined as follows: move one vector parallel to itself until one of its ends meets that end of the other which causes the arrows to indicate continuous ad- vance from the free end of one vector to that of the other, and then join the former free end to the latter by a straight line. The "sum" is therefore a vector. Thus AB and CD may be added in two ways : (1) move CD parallel to itself until C coincides with B, — the arrows now indicate continu- ous advance from A to D, — and join these points by a PIG. 5. — Three methods for tho_additton of the vectors AB and CD. tffTRODUi TION 27 straight line, tlius forming the vector AD: (2) move CD parallel to itself until I) coincides with A, — the arrows now indicate continuous advance from C to 1?, — and join these points by a straight line, thus forming the vector CB. It is evident from geometry that these two vectors are identical, having the same length and the same direction and sense. (If a parallelogram is formed, having the two vectors as adjacent sides, both starting from the same point. the diagonal is their sum.) This process is called " geometrical addition v ; and it can obviously be extended to three and more vectors. The simplest case is evidently that when the two vectors are in the same straight line : if they are in the same sense, the numerical value of the sum is the ordinary arithmetical sum; while, if they are in opposite senses, it is their arithmetical difference. If, then, two vectors are in the same line and in the same sense, both may be called posi- A > B tive ; but if they have opposite senses, D < c we should call one positive ( + ) and the other negative ( — ) ; and their A D B,C geometrical sum equals in numerical FIO. B.-A.I • i«ni- value the algebraic sum in both cases, and has the direction of the two vectors. Its sense of direc- tion in the former case is that of both vectors; in the latter, that of the greater. Looking at this process of geometrical addition in a con- verse manner, it may be said that the vector j&? is the geo- metrical sum of AH ii\\<\ ( 7>, where .1 // and f'/>aiv any two vectors such that, when added, their initial and final points are A and D: the vector AD is said to be "resolved into components." The case when the two components are at right angles is the most important. Let* AD be any vector and (JT any straight line; drop 28 INTRODUCTION perpendiculars AA' and DD' upon OP\ A'D' is called the "projection of AD upon OP." Draw through A a line par- allel to OP; it intersects DD' in B. Then the vector AD B,C FIG. 7. — Resolution of the vector AD into components. A' D' FIG. 8. — Projection of the vector AD upon the line OP. equals the geometrical sum of the vectors AB and BD. Let the lengths of AB, BD, and AD be 5, v, and A; then, by geometry, A2 = b2 -f v2 ; and, if JVis the angle (BAD), by the definitions of trigonometry : ^ = sine N, - = cosine N, - = tangent N, h n b or, as ordinarily written, v = h sin TV, b = h cos N, v = b tan N. So the projection of AD on OP equals the product of AD and cosine JV.* The vector ^..B is called " the component in the direction OP of the vector AD" (If AB and BD were not perpen- dicular, i.e. if (ABD) were not a right angle, the latter vector might be so resolved as to have a component in the direction OP ; and in that case the former would not be the only component of the vector^/) in this direction.) But, as just shown, AB = AD cos N. The general rule, then, for * In a similar manner if perpendicular lines are dropped upon a plane from the points forming the contour of any limited surface, the area inclosed by the feet of these lines is called the projection on this plane of the limited surface. If this surface is plane and has the area A, if the projected area is Aij and if the angle between the two planes (i.e. between lines perpendicular to them) is N, it is seen that A\ = A cos N. INTRODUCTION 29 obtaining the numerical value of the component in a particu- lar direction of a given vector is to multiply the numerical vulue of the vector by the cosine of the angle between it and the specified direr' It is obvious from geometry that the component of any or AC along any line OP, i.e. Afi^ is the algebraic sum of the components along this same line of any two vec- tors whose geometrical sum is AC, e.g. AB and BC; viz., The geometrical addition of the two vectors AB and BC to produce the vector AC may be expressed in words as follows: In order to ob- tain the vector AC by joining a vector to AB, it is necessary to add the vector BC. Therefore, Itc is i In' "geometrical difference" between the _ I vector .1 < ' and the vector 5 AT" BT~ cT P AB. Thus, the nile for ^IQ „._ ^ projection' of the vector ^T upon Sllht 1 -acting One Vector or e<lual8 the_sum ofjhe projections of the two from another is evident; thr difference is such a vector that, if added to the former, it L,rivcs thr latter. component* AB and BC. There are many physical quantities which require for their description a numerical value and a direction, e.g. velocity, force; they can be represented graphically by a vector, and are called "vector quantities." L Average or mean values. — If there is a set of similar quantities, ar ay •••,«„, their "arithmetical mean " is defined tobe-1 — ^— — • 1 1 of ten happens, however, th // to the physical or mathemati. -al < -(.n. lit ions, the various quan- tities at. DO! «.t thr same importance. For instance, suppose thr mean age of a class of students is desired: let the age 30 INTRODUCTION of 2 members be 16, of 5 be 17, of 10 be 18, of 1 be 19 ; the ages are then 16, 17, 18, and 19, but the importance of 17 in the average is 5 times that of 19. In this case evidently the proper mode of finding the mean value is to multiply 16 by 2, 17 by 5, etc. ; add these products and divide by the num- ber in the class, i.e. 2 + 5 4- 10 -f- 1. So in the general case, if m1 is the importance of ar m2 that bf «2, etc., the mean value is w?, + ra2 + ••• + mn If a quantity assumes different values in succession, but in a continuous manner, e.g. the speed of a falling body, it is often necessary to find its mean value for a definite interval of time. This can be done by a graphical method, using the theory of limits. Let «j be the value of the quantity at the instant of time T-± ; a2 its value at T2 ; etc. Choose the intervals of time /77 fj? frt fji pfp «7 «C L Ta T3 T4 T6 To T7 T8 Ta FIG. 10. — Graphical method for determini
ng the mean value of a series of quantities. equal. The importance in the average of a given value, e.g. am, is the length of time it lasts. The actual quantity dur- ing the interval T2 — T^ is changing from a1 to «2 ; during the interval T8— T2 it is changing from a2 to #3 ; etc. As an approximation, let us assume that the quantity keeps its value a1 during the inter- val T2— Tv then suddenly assumes the value av which it keeps during the interval T3— Tv etc. By the above defi- nition, the mean value of a is, therefore, (T2 - T\)a\ + C7^ - ^s)qa + 1" (Tn - !Tn_i)an_i (T2 - Tj) 4 (T8 - TJ + ... + (Tn - Tn-i) The value of the denominator is evidently Tn— Tv or the total interval of time. The numerator can be represented 1NTROJ)1< T1OJ9 graphic-ally. On a horizontal line mark points Tr T Tn at equal intervals apart, and at these points erect vn: lines of length av av • ••, an. Construct rectangles having 'i{ and (^2~ ^i)* av all(1 ( ^8~^i)» etc' * sides, as shown I >y the unshaded rectangles in the cut. The numerator of the above fraction is the value of the sum of the areas of these parallelograms. This sum divided by the total inter- val of time is the approximate mean of ar av etc. Another approximate mean may be obtained by assuming that Up- value of the quantity during the interval T<i—Tl is aa; dur- ing 23— T2 is a8; etc. The mean is then (T, - T,) + (T, - Tj + ... + (Tn - I',-!) The denominator is, as before, the total interval of time : and the numeral i»r is the sum of the areas of the rectangles whose sides are (T2— 2\) and a2, (T3 — T2) and d.r etc., as shown in the cut by the rectangles formed by adding the shaded portions to the former ones. These two numeratnix are evidently not equal ; but, as the intervals <»f time T^ — Tr T9— T.r etc., are taken smaller and smaller, the two sets «»f rectangles appp>a< -h the same limit: vi/., tin- area included bet \\een the base line, the two vertieal lines of lengths al and aB, and the line (curved or .straight ) that passes through the ends of the vertical lines erected at the points Tv Tr etc., as these in the limit bee consecutive. So the true mean value of the quantity a is the ,i of the curve " as just described, divided 1»\ the value of the total interval of time. Amplest case is when the qnant it :IILT uniformly : i.e. ~ ever T^ Tv etc., and T^an- : for the line described above as the locus of the ends of the lines ah Oy • 1 a., is evident 1\ a straight line. Its " area " is kn-\\n from geometry to be INTRODUCTION the iiu'aii value of the quantity is this divided by (Tn — Tj), the total interval of time; i.e. % (an + a^) or the arithmetical mean of the initial and final values. FIG. 11. — Special case, when the quantity whose mean Is desired is varying at a uniform rate. Another mode of taking means has its origin as follows: Let ar a2, ay etc., form a "geometric series"; i.e. «2 = rav aa = ra2 = rzav etc. It is seen that a2 = Vd^ag. So, in gen- eral, the " geometric mean " of two similar quantities a and b is defined to be MECHANICS AND PROPERTIES OF MATTER INTRODUCTION \\'E have recognized three so-called fundamental proper- «»f matter: inertia, weight, and tin- one which includes the varied characteristics of size and shape. Kach of these will now l>e considered in !_;Tealer or less detail. As has been said before, the science of Mechanics is that branch of Physics which deals witli the inertia of matter. It is often divided into two parts. ••Kinematics" and •• Kinetic^": the former is the science of motion considered apart from matter; that is, it treats of possible motion; the latter is -trictly the science of the inertia of matter. If there is no change of any kind in the motion, what we call "rest " beiiiLT ecial case of this, the science is called "Statics"; while if the motion is chan^in^, the science is called "Dynamics." These two sciences are branches, then, of Kinetics. Statics can. however, be considered as a special limiting case of Dynamics; and this plan is adopted in the present book: 8O Mechanics will be treated under the two divisions. Kine- matics and Dynamics. In the former, the (jnestion as to the different possible kinds of motion will be discussed; in the latter, the physical conditions under which these types of motion occur. Kinematics is a geometrical science; dynamics, a physical one. \\YMii .\ill l»c discussed under the more general head of .Station. This will be f«ill..\\,-d by several chapters on the properties of solids, liquids, and gases. AMES'S PHYSIC W CHAPTER I KINEMATICS General Description. — Kinematics has been defined as the science of motion apart from matter ; that is, it is concerned with the study of the possible motions of the geometrical quantities : a point, a plane figure, and a solid figure. For the sake of illustration, many material bodies will be referred to ; but all the statements and theorems are meant to apply to figures, not to bodies, unless the contrary is expressly noted. If the motion of any actual body is observed (for instance, a stick thrown at random in the air, a moving baseball, the wheel of a moving wagon), it is seen that there are two types of motion involved : the object moves as a whole, and it also turns. These motions are independent of each other; one may occur without the other. In the up and down motion of an elevator, in the motion of a railway car on a straight track, etc., there is no turning ; in the motion of the fly- wheel of a stationary engine, in the opening or closing of a door, etc., we may say that the motion is one of turning only. The name " translation " is given to that kind of motion during which all lines in the figure remain parallel to their original positions ; further, all points of the figure move through paths that are geometrically identical. Thus, to describe completely any case of translation, all that is neces- sary is to describe the motion of any one point of the moving figure. The name " rotation " is given to that kind of motion dur- ing which each point of the figure moves in a circle ; e.g. a door as it opens or closes. The planes of these circles are 34 KIM-MAUL'S 35 parallel, and their centres all lie on a straight line which is called the --axis." If a plane section, perpendicular to the axi>, is taken through the rotating figure, all the lines of the figure in this plane have identically the same angular mo- tions: otherwise the figure would break up into parts. To • leseribe, therefore, motion of rotation at any instant, we must know two things: the position and motion of the axis and the angular motion of any line fixed in the figure with reference to any line fixed in space, provided the ^ lines lie in the same plane per- pendicular to the axis. Thus, con- sider the rotation of a figure like that of a grindstone ; its axis is the FIO. i2.-Rouuon of figure ..inn,, of the axle. In the cut, ^^^^^ which represents a cross section by fixed in »pace, and />e, » line fixed a plane perpendicular to the axis, let PQ be a line fixed in the moving figure and AB be a line fixed in space; the position and motion of the figure at any instant are given by a knowledge of tin anurle l>etwern these lines and of its changes in value. It should l»e noted that this is a special case of rotation, because tin- axis does imt ci. position, as it does in general. To describe the most complicated motion, tin -i must consider it resolved into two parts, a translation and a rotation, and must discuss each separately. Translation In translation, as lias been already explained, it is neces- tn describe tin* motion of a point only. The simplest case of this i> motion almi<r a straight line; but the \| general case, that of motion along a curved line, is not ditli- eult. To describe this nmtion the first thing that it is neces- to know is the position of the point at .my inMant with re nee to soi i figure. 36 MECHANICS Linear Displacement. — Let the path of the point with reference to some fixed figure be represented by the curve in the cut. Let 0 be its position at any instant, and P that at a later time. The vector OP is called the "linear displacement" of the point with reference to the fixed figure during this interval of time. This same vector might be the displacement for any motion that passed through 0 and P ; or, in other words, a point may pass from 0 to P by various paths. The displacement is then a vector quantity and may be resolved into components in as many ways as we choose ; con- FIG. is. -The vector OP versely, two or more displacements may is the linear displacement of \>Q compounded by geometrical addition. P with reference to O. ,„, . The importance of mentioning the "fixed figure of reference " may be seen from an illustration : if a stone is dropped from the top of the mast of a moving steamer, it will fall at its foot ; the displacement with refer- ence to the steamer is a vertical line, while with reference to the earth it is an oblique one, being the geometrical sum of the vertical line and the displacement of the steamer. Linear Velocity. — If the interval of time taken for the displacement is extremely small, P is very close to 0; and, in the limit, the displacement OP coincides with the actual path along the curve, and has, in fact, the direction of the tangent to the curve at the point 0. If, as the displace- ment becomes very small, its length is represented by Aa?, and the corresponding interval of time by A£, the ratio — in the limit is called the " linear velocity " at the point 0 with reference to the fixed figure ; that is, it is the " rate of change" of the displacement. It is evidently a vector quantity for it is defined by its numerical value, which is that of — in the limit, and by the direction and " sense " of KINEMATICS 37 the displticement in the limit : viz., its direction is that of tht? tangent at 0 drawn from 0 to P, when P is close to 0. The numerical value of the linear velocity is called the k- linear speed"; so t
hat the velocity at any point is char- acterized by the value of the speed and by the direction and "sense" of the tangent to the path at that point. If the motion is uniform along a straight line, that is, if the velocity is constant (both in amount and in direction), the speed is equal numerically to the distance traversed in a unit of time; and if the motion is not uniform, the speed at any instant is the distance which the point would travel during the next unit of time if the motion were to remain uniform. The unit of linear speed on the C. G. S. system is the speed of "1 cm. in 1 sec."; and the unit of linear velocity in a definite direction is the unit of speed in that direction. (This C. G. S. unit speed has not received a name; in fact, the only system of units in which there is a unit speed which has received a name is that based on the nautical mile — 6080 ft. — as the unit length, and the hour as the unit time : the speed " one nautical mile in one hour " is called a "Knot." The expression "16 knots per hour" is therefore incorrect; for a knot is a speed, not a length.) Since the linear velocity is a vector quantity, it can be resolved into components; and, conversely, two or more linear velocities may be compounded by geometrical addition. These statements are illustrated by many familiar facts: if a man \\alks across a moving railway carriage, his velocity with reference to the ground is com- pounded «>f that of the train and of that which he would have 38 MECHANICS if the train were at rest ; if a boat is rowed across a river, the actual velocity with reference to the earth is the geometrical sum of that of the water of the river and of that due to the oars ; the velocity of a raindrop with reference to the window pane of a moving carriage as it strikes it is the geometrical sum of a velocity equal but opposite to that of the carriage and of its own downward velocity at that instant ; if a man walks in a northeast direction with a speed of 8 cm. per second, his velocity may be represented by a vector PQ whose length is proportional to 8 and whose direction is northeast; and his velocity in a northern direction is given by the component, PR, in the direction north and south, whose numerical value is 8 cos 45°, or, more properly, s cos j- Similarly, one velocity may be subtracted from another, the difference being also a velocity. We will consider two illustra- tions: a body falling freely toward the earth and an extremely small particle mov- ing in a circle with a constant speed. In the first case, the velocity at any instant is represented by a vertical vector AB and FIG. is. - Rectilinear at some later instant by another vertical motion: AB and CD are -~=r f . , the velocities at different vector CD ot greater length, because as BD l8 th6lr the body falls' its 8Peed increases. Call the length of AB sr and of CD «2. The change in velocity is the difference between these vectors ; that is, it is a vertical vector BD of length equal to 2 ~ r In the second case in which the particle is moving in a circle let the constant speed be «; and let the direction of motion be that indicated by the arrows. When the particle is at the point A, its velocity has the direction of the tangent and the numerical value «; it can therefore be represented KIXEMATirs 30 by the vector PQ which is parallel to the tangent at A, and has a length proportional to «. Similarly, when the part irk- is at the point B, its velocity can be represented by the vector PS which is parallel to the tangent at B and whose length is equal to that of 7Vtv ( since the speed does not alter). The change in the velocity in the time taken for the particle to move from A to B is the difference between the vectors PS and PQ; that is, it is the vector s v . Linear Acceleration. - - To Fio.J_6. — Uniform motion In » circle. PQ and PS are the velocities of the point at A and B. return to the original problem, that of describing the general case of the motion of a point in a curved path, ire have defined the displacement and the velocity at any point, the latter being the rate of change of the former. The velocity may change, however, both in direction and in speed ; and its rate of change at any instant is called the " linear acceleration " at that instant ; that is, if At; is the change in the velocity dm MIL: the time At, the limiting value of — - is the acceleration. Moreover, since the change in the velocity, A>\ is a vector quantity, so is also the accel- eration. Its numerical value is that of -- in the limit ; and At its direction is that of A'- in the limit. We may consider separately two cases; in one of which the din-rtion remains constant hut the speed changes, -while in the other th« speed remain^ constant but the direction changes. As an illustration "f the former we may take the mot; : falling l»odv: and of the latter, the uniform 40 MECHANICS motion of a particle in a circle. These two cases have already been partially discussed. In the former motion let the change in speed from s1 to s2 take place in the interval of time T2 — T^\ then the accel- when the interval eration has the numerical value T2 — Tl is taken infinitely small, and its direction is ver- tically down. If j:he acceleration is constant, it is, therefore, the change in the speed in a unit of time. In the latter case, that of uniform motion in a circle, let the interval of time during which the particle moves from A to B, and the velocity accordingly changes from PQ to PS, be taken extremely small, so that the length of the arc AB becomes minute also ; then, if this interval of time is called A£, the acceleration at the point A is the limiting value of fvector Q8\ Call the lengths of the various straight lines in the diagram by the letters marking their terminal points : by geometry the triangles (SPQ)_,in\_(BOA) are similar, hence, -^ = ^-r ; but PQ equals s, the value of the constant speed ; OA equals the radius, r, of the circle; and in the limit, when A£ is infinitely small, the length of the chord AB equals the length of the arc AB, the value of which is s • A, because s is the distance tra- versed in one second, and hence the distance 8 - A£ is traversed in A£ seconds. It follows, then, that the numerical value of the acceleration at " x AB\ s : x OA J A* xr FIG. 17. — Uniform motion In a circle: the acceleration is the limit- QS ing value of ~. the point A, viz. lim = KINEMATICS 41 'II a- Direction of the acceleration is that of the vi in tin- limit, when B approaches A, and therefore S approaches Q. Since PQ and PS are of equal length, in the liin 3 perpendicular to_P@. But PQ is paral- lel to tin- tangent at A\ and QS is therefore parallel to tin- radius OA of the circle in which the point is moN and is directed toward the centre. In conclusion, tin -n. when a point is moving in a circle with constant speed, its acceleration at any point is along the radius drawn from ~2 the point toward the centre and has the numerical value — , where « is the value of the constant speed and r is the length of the radius. (It is thus seen that, calling At> the in- finitesimal change in the velocity in the infinitesimal interval of time A£, Av is a vector at right angles to the one repre- senting the velocity at any point; and that their ^eomet: sum is a vector, lying in their plane, whose numerical value in the limit is the same as that of the original velocity hut whose direction is different; in fact, its direction is th, the tangent at a point <>f the circle infinitely near the original point. In words, l>y continually adding to the velocity an infinitesimal velocity perpendicular to it, the speed does not change, while the direction does.) Since an acceleration is a vector quantity, we may resolve on.- in any manner; or, conversely, we may add two <>r re accelerations geometrically. Illustrations aiv -i.>n menoe. Kxperiments show, as will 'ained ! that a body falling frerly near the earth acquires an accel- 1OH whose numerical value i> ahoiit '.»s«l mi the C. ( • i.e. in one second its speed increases by the amount 980cm. per Second); thil it ordinarih written : I the B for all hodics, whatever th- -haj»c, or mass, pro- d they fall i mm. hut differs xli-htlyat diffc latitudes on the earth's surface. If, then. .1 l.,.dy is DOl Qan "inclined plane " filial i>. a 10 lined t<> the 4:2 MECIIAM' .^ horizontal plane) which is so smooth that friction can be neglected, the acceleration down the plane is the component parallel to the plane of the vertical acceleration g, if we assume that accelerations in nature can be "resolved." Let the latter be given by the vector AB\ its component parallel to the plane is the vector AC, where BC is perpendicular to the plane. If the angle between the plane and the earth is called N, it is evident that the angle (ABC) = N-, and the length of the vector AC equals g sin N. (As N is made small, this component acceleration be- inclined plane of a vertical vector AB is COniCS Small; and it may become so small that it can be measured FIG. 18. — The component along' an easily, whereas g itself is so large that any direct measure- ment of it is extremely difficult. This experiment was -first performed by Galileo.) Again, when a small piece of iron is brought near a magnet, each moves toward the other, if it is free to do so, and acquires an acceleration; and if the piece of iron is allowed to fall freely past the end of the magnet, its acceleration at any instant is the geometrical sum of that which it would have if the magnet were not there, viz. #, and of that which it would have if there were no gravity. (The fact that in nature the accelerations produced by forces can be added geometrically involves the definite assumption that forces act independently of one another.) The acceleration of a moving point is not necessarily con- stant either in direction or amount ; but no name has been given the " rate of change of linear acceleration," and, in f
act, it is a quantity of no importance physically, as we shall see later. Special Cases. — There are several special cases of trans- lation which deserve detailed description owing to their groat importance in problems of Physics. KINEMATICS 43 1 . A point moves in a straight Hm- with a constant accelera- Since the acceleration is constant, its numeri- cal value equals that of the change in speed in one second; and therefore, if the value of the speed at any point Pl and instant of time 2\ is sr that at the point P2 which is reached at the instant T2 is given by the following equation, in which a is the value of the accel- Fl°- l9- ~ e ration : MT motion. sa-Sl = a(T^- T,), or s, = Sl + a(Ts- 7\). (1) (It should be noted that, if the sense of the direction of a is that of the velocity, the speed increases in value with the time, e.g. a falling body ; whereas, if the sense of the two diivrtions are opposite, a and * have different signs, and the formula iH'coim-s s^ = s1 — a(T% — 2j) if a is here the numerical value of the acceleration ; therefore the speed decreases, e.g. a body thrown vertically upward -or a body thrown along a slnTt of ice.) 1 1 tin- speed were constant, the distance traversed in t sec- onds would be the product of this by the value of the speed; but the latter is varying from instant to instant. Since the speed is, however, by assumption changing at a uniform rate, its mean value (see page 32) between the instants 7, and 5P, is the average of «2 and sr i.e. ""; and the distance passed over iu the interval of time (!Ta — 71,) is &^il x (T, - r,) = s^T, - T,) +J a(r, - r,)'. If anv point 0 in the line of motion is chosen as a] reference or "origin," and if the distances OI\ ami OPt are • •all. --I A, and /.,. ili.- displacement in the interval of time 7!,- T, /., - Lv That is, 44 MECHANICS L2- Ll = 9l(Tt - 7\) + \| a(T2 - r,)2, or ^•=^i + «i(r,-r1)+Ja(r,-rl)^. (2) (If the acceleration is an opposite sense to the speed, a must be given a negative value.) Thus, if the acceleration is known, and if the position and speed at any one instant are given, they can be predicted for any future instant. Con- versely, if any motion is found to obey either of these laws (for one is a consequence of the other), it is known that the acceleration is a constant. These two formulae assume their simplest form when we agree to measure time and distance from the instant and position in which the moving point is at rest. For instance, let the point be at rest at Pv i.e. s1 = 0 ; then we will choose 0 to coincide with Pr i.e. L^ = 0 ; and also choose this in- stant as the one from which to measure time. Hence the formulse become These are due to Galileo, and it was by showing that when a body moved down an inclined plane the displacement varied as the square of the time taken, that he convinced himself of the constancy of the acceleration parallel to the plane. Since this is constant, so is that for a body falling freely. In the general formula (2) it is seen that the displacement, L2 — Lv is made up of two parts : s^T2 — T^) is the distance the point would have gone if there had been no acceleration ; %a(T2— j^)2 is therefore the additional displacement owing to the acceleration. Another general formula may be obtained from equations (1) and (2) by eliminating (T^ — T^) from them : substitute in (2) the value of (T^- TJ obtained from (1,) viz. This gives a(L2-L1)=KV-V> (3) KINEMATICS Comparing equations (1) and (3), it is seen that the former defines a as a function of the interval of time, viz., , while the latter expresses it as a function of the 1 displacement, viz., a == - 2 -. A i>"lid moves with a constant acceleration in one direction with a constant velocity in a direction at r'ujht anjlt's to this. Let the constant accelera- tion have the numerical value a; and let distances in its direction, measured from the point at which the speed in this direction is zero, be called y. Then, by count in^ time from the instant when the point is in the position for which y=0, we have the relation y = J at2 for later time t. any Similarly, let the constant .'0. — Parabolic motion : composition Velocity have the numerical a uniCm-m horizontal motion with • uniformly value i; and Let distances in -^<«<«» ™ ''' its direction, measured from the point at which y = 0, be (ailed jr. Then, after the interval t. jr = xt. The resultant displacement is the <jeomet rical sum of r and y. Tin- path of the mo\in^ point may he found by actual netrical construction, giving to t different values; or, more simply, by the elimination n the two equations. jm<1- i( \|M'S lvlilt>"11 between * and v This gives y = n~Z is plotted, it will give the path of the point. The curve is evidently a parabola, having as its axis the direction of acceleration. 2 8 * We need not, however. »n as beginning at ' \\ In -re y = 0. Let two lines be chosen at right angles to each other, DC 46 MECHANICS in the direction of the constant acceleration, OB perpendicular to it. Let a point be projected obliquely to OB with the velocity V and making the angle N with it. This velocity can be resolved into two, V cos N along OB, V sin N along OD, but in the opposite sense. Therefore, assuming that the two motions at right angles to each other are inde- pendent, the point will maintain through its motion a constant ve- locity V cos N parallel to OB ; and, if the constant acceleration along OD is a, the point will move in the direction opposite to OD for an in- terval of time t, where la = V sin N. From this point on, the condi- tions are exactly as in the pre- vious problem ; so the point describes a parabola. But since the path is now a parabola, by symmetry, it must have been so before this point was reached. In the diagram, this point which marks the instant when the motion along OD in the opposite sense ceases and that along OD begins, is shown by C. The time taken for the moving point FIG. 21. — Motion of a projectile: OV is the direction of the initial velocity ; OB is the horizontal trajectory. to reach it has just been shown to be t = — — ; and the displacement in this time along OB is the product of t and V cos N, or — a This is the distance OA on the diagram. The line A C has the length V sin N 2 at2 or g 2 ^he moving point in its further motion will cross the line OB at a point B, where OB = 2OAorOB = - — , F2sin 2 N which may be written — — For given values of V and a, this distance OB has its greatest value when sin 2 N = 1 ; that is, when 2 N = £, or when N = j or 45°. In y h T the path of a projectile or ball thrown obliquely upward. The distance OB is in this case the horizontal trajectory. Of course, the resistance of the air seriously influences these results. It is evident, too, that the hori- zontal trajectory is greater if the point of projection 0 is above the earth's surface, and the moving body is free to fall down to the surface.) This problem of a projectile was first discussed by Galileo, 1638 KI\I-:UATICS 47 .1 finint is moviiKj in <t /•//•<•/,• //•/>// rnnstant speed. — It has heen shown already that in this ease tin- acceleration is toward the centre along the radius drawn to the moving point at any instant, and has the numerical value — ; where « is the value of the constant speed and r is the length of the radius. This result can be expressed differently. Since r is the radius of the circle, the length of the circumference is 2 T is the time taken to complete one revolution, the speed s = — =-• Hence the acceleration may be written • Or, if ^Vis the number of complete revolutions in a unit of time, N = . and the acceleration may be written Again, since one complete revolution corresponds to an an^le whose value is 2 TT, the angle turned through in a unit of time by the radius drawn to the moving point is 2?r^V; so, writing this quantity A, N= — , and the acceleration becomes /•//-. h is called the "angular speed": it evidently equals •ecause 8 is the arc descriln-d in a unit of time by the radius whose length is r, and th.-refore - is the value of the aiiL,rh' described in this time. (•rulilem may be solved in a ilitT-P-iit way whirh is «lu.- to Newton. ircle of motion be that shown /• '•tit. and since at any position. /'. its \. is along the tangent, a velocity be added alonu ..» toward the ! lf r ' ' J i. Draw a diameter PR 48 MECHANICS through P. Call the numerical value of this acceleration a, and that of the constant speed .v. In an infinitesimal interval of time, t, tin- distance the point moves in the direction of the acceleration is £ at- ; but the actual displacement is the chord of the arc whose length is nt. Let this chord be given by PQ, in the cut; then its projection on the diameter through P, PS = \ a/2 ; and in the limit ~PQ = «/, since the arc and chord coincide. But by geometry P(f = PS x ~PH ; or, calling the radius of the circle r, sW = J at2 x 2 r. That is, a = - . r 4. A point moves to and fro along a straight line, being the projection on any diameter of a point moving in a circle with constant speed. {Simple Harmonic Motion.) In the cut let P be the point moving in a circle with constant speed ; let AB be any diameter ; then Q is the projection of P on this ; and, starting from any instant, as P describes a com- plete circumference, Q moves to and fro along the diameter, returning to its original position and condition; that is, at the end of this time, which is called a "period," the point Q is not alone in •a circle, Q is its projection in the same direction with the same speed and acceleration that it had at the beginning of the period. The motion of Q is called "simple harmonic." The acceleration of Q is evidently the component of that of P resolved parallel to AB ; because Q has identically the same motion as that of P resolved in this direction. The acceleration of P is toward 0, and has the numerical value rA2, if r is the radius of the circle and h is the angular speed ; its component parallel to AB is
therefore rh2 cos (POQ). If the displacement of Q with reference to 0 is called rr, and if it is defined to be positive when drawn to the right, cos (P0$) =-; and hence the numerical value of the K1XKMATICS 49 acceleration of Q is /r./\ its direction hein^ toward 0. which i lied the, "origin." If we cull all vectors positive when drawn toward the right, the acceleration is given, both in amount and in direction, by — lrj\ the minus sign indicat- ing that when Q is on the right of 0, i.e. when x has a positive value, the acceleration is negative, and therefore is to\\urd the left", and when Q is on the left of 0, i.e. when a; has a negative value, the acceleration is positive and there- fore is toward the right. Since h is the constant angular speed of P, the period, or the tinu- taken for a complete revolution, is — — • One half 27T h the extent of the swing of the vibrating point Q^ i.e. the maximum value of the displacement 3, is called the "ampli- the radius. tude" of the vibration. It therefore equals the length of It there are two points moving around the same circle with the same speed, there will be two projected points in harmonic motion with the same period and the same ampli- tude; yet their motion is not the same at any instant, for their displacements, etc., are different, — one lags behind the other: they are said to differ in "phase." Thus, if one point is passing through the origin toward the ri^ht when the »»t her i> at the ri^ht-hand end of its swini^, the difference in phase is a quarter of a period, from the standpoint of time : Or, since One period em-responds to an an^le '1 TT. a .piarterof a period corresponds to an angle ^, and so we call it a "dif- iee ,,f phase of £•" Again, if one point is moving m through the origin toward the ri-ht when the nth. :ic_r through tl ...... -i-ri,, toward the left, the difference in pi. 'iie half a period, or TT. lar in every respect to that of the vilu-utini: extremely common: any point of a stretched :•_: \ il»rat iii'_r in a Dimple manner, the end of a tu: AMK"' -—4 50 MECHANICS fork, a body hanging from a spiral spring and set vibrating, etc. In all these cases the acceleration at any instant if found to obey the relation —6%, where x is the displace- ment at that instant of the vibrating point from its position when at rest, and c2 is a constant quantity depending upon the nature of the vibrating system (c2 is written for the constant quantity instead of c, for purposes of simplicity, as will appear immediately. Of course, C or any other symbol would serve equally as well, so far as its meaning is concerned). We may, then, discuss the motion of a point in a straight line whose acceleration at any instant is — <?2#, where x is the displacement of the point from its origin and c2 is a constant quantity. Such motion is " simple harmonic." Obviously, by comparison with the motion of the projected point Q in the problem just described, this harmonic motion is periodic, with the period — . The amplitude is the c maximum value of the displacement. Two harmonic motions of the same period may differ in phase. The motion of the projected point Q in the original problem may be described in a different way : the value of its displacement may be stated instead of that of its accel- eration. As before, let P be a point moving in a circle of radius OP with a constant angular speed A, and let the motion of FIG. 24._— Harmonic mo- jts proi'ection Q on any diameter be con- tion: if OQ-x, OA = R, r J \ / — angular speed of p=h, sidered. Call the displacement OQ, x\ «-«>•<- jr>. the iength of the radius, R\ let time be counted from the instant at which the moving point P was at the point $, and call the angle (SOA), N. Then by trigo- nometry x _ R cog ^A op^ = R cog ^sop _ SQA ^ = R cos (SOP- N). h'IM-:UATICS 51 lint h is the angular speed, and time is counted from the in- stant when P \\ as at .V: therefore the angle (SOP) equals At, it t is the interval of time taken for the point to move from § P. Hence , = a «» (ju - A). Iii this expression, # is the displacement; the amplitude is 72; the period is r-E; mid (ht - N) is called "the phase." So two similar motions having the same periods, i.e. the same value of A, may differ in phase (they may have different values of N) ; and the "difference in phase" is (ht — Nt) — (ht — -ZVj) = ^V2 — N'1. In words, the position in the cir- cle of the moving points when one begins to count time is different in the two cases. In general, therefore, motion of a point in a straight line whose displacement is given by the formula x =a A cos (ct — N) is harmonic; and in this r\pi vssimi the amplitude equals -A, • •riod equals - , and the phase equals (ct — N). 2 7T c Harmonic motions may be compounded by geometrical addition ; and several sp. i ial cases of importance physically will be discussed lat< i. Tin- velocity of the point Q equals the component of the '•ity of P msohrd parallel to BA. Tin- linear velocity of P is along the tan^-nt and has tin- vain.- A'//: its com- ponent alon- />M is -/{/, sin < .lO/») or - Rh»in (ht - JV), it' \elocitiestoward the ri^ht an- calli 'ive. So, in the velocity of the harmonic motion x = A COS (ct — N) • > .,.. is — Ac sin (ct — IT) ; or, calling the period T, .since T =* — , this equals — A -£- sin (ct — JV). Similarly, the acceleration of P /'". ihcn that of Q is - R&c<M(AOP) or B ' - N). This, therefore, equals - 52 MW11AXIC8 Rotation As already stated, the name "rotation" is given the motion of a geometrical figure when, at any instant, each point of the figure moves in a circle ; the planes of these circles are all parallel, and their centres lie on a line called the "axis." It was shown further how to describe the angular motion around a fixed axis by means of two lines, one fixed in space, the other in the moving figure, but both lying in a plane perpendicular to the axis. Angular Displacement. — In the cut, as before, let AB be a line fixed in space and PQ a line fixed in the rotating figure ; let 0 be the point where the axis cuts the plane. As the figure turns around the axis, the angle between AB and PQ varies: It is called the "an- gular displacement" at any instant of PQ with refer- ence to AB. To describe --B this displacement more fully, however, we must know the position of the axis and the FIG. 25. — Angular displacement: 0 is trace ,, . r • u i~ i_ J of axis, AB is a line fixed in space, PQ is a line " S6nS6 in Which the body fixed in the figure. js turning; that is, whether, looking at the lines in the cut, PQ is rotating like the hands of a watch or in the opposite sense. These three ideas — the numerical value of the angle, the position of the axis, and the sense of the rotation — can all be represented by a vector placed so as to coincide with the axis; for its posi- tion indicates that of the axis, its length can represent the value of the angle through which the figure turns, and its direction can be made by some agreement to indicate the sense of the rotation. The connection between the direction of the vector and the kind of rotation which is usually adopted is as follows : if an ordinary right-handed screw partially in a board is placed so as to coincide with the K I \EMATICS 53 \ector and is turned in the sense of the rotation, the direc- tion which it moves lengthwise into or out of the board is that L,ri\ -en the vector. Thus, in the cut, if AB is the vec- tor, and if an observer looks at a rotating figure in the direction from A toward B* the rotation is like that of the hands of a clock. If tin- arrow were in the opposite direction, the vector would indicate rotation in the opposite direction. This connection between the directions of the vector and the rotation is called the "ri^ht-handed screw relation/' Fin. 26. — Rlffat- A Vector located 111 a definite position 111 this handed screw relation manner is called a "rotor." It may be proved J^^J,** without dillicnlty that one can add two rotors geometrically : (1) if they lie in the same line, when the addition is therefore algebraic, because rotation in one sense is posit ive ( -f- ), and in the opposite is negative ( — ), and tin- two rotors are lines either in the same sense or in opposite senses; (2) if their axes lie in the same plane, i.e. if they meet at a point, and if the angular displacements to which they are proportional are infinitely small. < this pn.of reference maybe made to any treatise on Media such as that of Xiwet or Williamson.) Angular Velocity. - I he rate of change of the angular displace, in-lit around a given axis is called the "angular velocity around that u numerical value of this velocity is called the " angular speed " ; 80 that in order to describe an B :ill.-ular vdodty three things um>t be Specified : the position <>( tin t\ i^. the sense of the rota- *7.-AmruUr Speed. til,n ,]„. jmLrnl,lP gpeed. If the ilar motion is uniform, the \alue of the angular speed is that of the angular displacement in a unit of time. In 64 MECHANICS the cut let, as before, AB and PQ be the two fixed lines of reference, one in space and one in the figure; but in this case let them be drawn through the axis at 0. Q is then any fixed point in the figure; call its distance from the axis, i.e. OQ, r. If the axis is fixed in space, the length of the arc described by Q is evidently equal, by the definition of the value of an angle, to the product of r and the value of the angular displacement. It follows, then, that the linear speed of Q at any instant, i.e. the rate at which Q passes along the arc of its circle, equals the product of r by the value of the angular speed of that instant, i.e. the rate at which the radius describes the angle. To return to the idea of angular velocity, it is evident that it can be represented by a rotor ; and since an angular velocity is the rate of change of an angle, i.e. is the limit of the ratio of the value of a small angle to that of a cor- responding small interval of time, i
t is proportional to an infinitely small angle, and therefore two rotors representing angular velocities may be added geometrically, if the two axes lie in the same plane. Illustrations of angular velocities are common ; a few may be described as follows : that of a flywheel is given by a line having a definite sense and length, coinciding with the central line of the axle ; that of a door or gate by a limited portion of a vertical line drawn so as to coincide with the central line of the hinges ; that of a cylindrical barrel rolling down an inclined plane by a line coinciding with the line of contact between the cylinder and the plane — in this case the axis is moving parallel to itself down the plane. As a hoop rolls along a floor, the rotor giving its angular velocity is a horizontal line perpendicular to its plane. If the hoop were at rest in an upright position, a sidewise push at the top would give it a rotation around a horizontal axis in its own plane ; therefore, if a sidewise push is given a rolling hoop at its top, the rotor of the resulting motion is the geometrical sum of the two separate rotors, and is in a horizontal plane but in a different direction from either — this explains why pushing a rolling hoop sidewise at the top changes the direction of its path. (The independent action of two or more forces is assumed again.) KIM-:.MATIC8 55 Angular Acceleration. — The rate of change of the angular velocity is called the angular acceleration. There are two special cases : in one, the direction of the axis remains fixed in space, while tin- angular speed varies, e.g. a door when it is opened <»r closed, a grindstone when it is set in motion or radually stopped, etc.; in the other, the angular speed remains constant and the direction of the axis changes, e.g. a spinning top whose axis is not vertical, a rolling hoop turn- ing a corner, etc. In the general case, of course, both the angular speed and the direction of the axis change. In tin- case of rotation around a fixed axis there is evidently a simple connection between the angular acceleration and the linear acceleration of any point of the figure. If A is the angular acceleration and a the linear acceleration of a point at a distance r from the axis, it follows at once from the defini- tion of the value of an angle that a = rA. (See page 47, where a similar formula for the velocities is proved.) There is also harmonic motion of rotation around a fixed . analogous to that of translation; it is illustrated by vibrations of the balance wheel of a watch, by those of a .) ^ clock's pendulum, etc. It is defined as follows: if the value of the angular displacement in a particular sense is called N^ harmonic motion of rotation is such that the angular accel- '.on equals — f?y, where c3 is a constant, depending upon the vibrating system. The amplitude is the maximum value of N; the period may be proved to be - id two harmonic c vibrations may differ in phase. It may be shown, fun that this definition is equivalent to saying that the angular displacement, equals A cos (ct — M ), where A is the ampli- tude: ~ — . the period : and (ct — AT), I lie phase. C General Remarks. - The displacement ,,f a point in any direction is independent of a displacement ll u at ri'^ht angles to ihix, ,-.;/. a man I 'liuard does not 56 MECHANICS move to the east or to the west ; and, since it is possible to draw from any point three lines that are mutually perpen- dicular, like the three lines meeting at the corner of a room, a point, and therefore a solid figure, may have three indepen- dent directions of translation. Similarly, rotation around any axis is independent of rotation around an axis perpendicular to it; and therefore a solid figure has three independent modes of rotation. So, in general, a solid figure may be displaced in any one of six independent ways — three of translation and three of rotation; it is said to have six " degrees of freedom." Freedom of motion may of course be hampered by various constraints ; thus a figure in a straight groove has only one degree of freedom of translation, a figure turning on a fixed axis has only one degree of freedom of rotation. It is always possible to produce a given displacement of a figure in several ways. Thus, if a figure like a wire, AB in the cut, is displaced by a rotation around 0 into the position A^v this same final position may be obtained by a translation from AB to A2B2 and by a rotation around 01 from A%B2 to A^BY It is often a matter of importance to select the simplest mode of displacement to meet the requirements of a given problem. FIG^ 28.— The rotation of AB about an axis_at O into the position A1Bl is equivalent to a translation from AB to A%Bt and a rotation about an axis at There are several simple theorems which may be stated without proof, although such proofs are not difficult. The displacement of a plane figure in a plane into any other posi- tion can always be produced by a single rotation around an axis perpendicular to the plane ; any change of position of a solid figure, one of whose points is fixed, may be produced by a single rotation around an axis through the fixed point; any change of position of a solid figure may be produced by KINEMATICS 57 a translation in a definite direction and a rotation around this direction as an axis, i.e. l»y a U80ren motion." Analogy between Translation and Rotation. — It may be ul to arrange in parallel columns the properties of translation and rotation that correspond to one anoth Translation Rotation A point moving n a _rht line. a. line of motion b. displacement linear speed if. 1 incur acceleration e. harmonic motion A figure turning on a fixed axis. xis of rotation b. aiiLMil.tr <lisj>lacement c. angular sj>eed rf. angular acccl«T<it ion e. harmonic motion acceleration = — c^ angular acceleration = — cW, - • M x = A cos (ct - C), or angular displacement AT = .lcos(< period = ~7r- \ point moving in acuru d /.»•. the direction of its motion is rhaii'_rinur. Lin. ;u spi-i'd is altered by a<l<liiiLT a velocity in tin- sunn- direction. I)iivction of motion is al- period = A liLMir-'. the tliiv. \\hosr a\i> of rotation is changing. Angular speed is altered 1 < 1 i i ig an angular velocity around the same a Tin- diivrti-.n ,,f il i. 1 1)\ adding a relocitj in altered 1»\ ad.lii: ;ular a (litTt-rrnt velocity around a dill- . tlir two IvillLT ill tll«' CHAPTER II DYNAMICS Introduction. — In describing our fundamental ideas of nature, emphasis was laid upon the conditions under which we feel the force sensation. Among the most important of these are the following : when we support a body free from the earth, when we change the size or shape of a body, when we alter the motion of a body. These conditions may be brought about by the action of material bodies, instead of by our muscles ; and, when this is the case, we say " a force is acting," one body " is exerting a force on another," etc. We do not, however, mean to imply the existence of a thing, but of a condition. We must now devise some method of measuring forces ; and we shall begin by discussing certain illustrations. Consider a vertical wire whose upper end is clamped to some support and at whose lower end hangs a heavy body. The wire exerts an upward force on the body, and in the process it is stretched, its molecules are displaced from their ordinary position ; the wire also exerts a downward force on its support ; this, in turn, exerts an upward force on the wire, and in doing so it is bent and its molecules are slightly displaced; the support must rest upon the earth in some manner, and here again enter two forces. If the wire is broken, the heavy body falls with an acceleration toward the earth, thus showing that there is a downward force on the heavy body — even when it is hanging from the wire — due to the earth. In the case of the stretched wire, the bent support, etc., we see that the force is associated with the deformation of a body, that is, with the displacement of its 58 /M \AMICS 59 molecules; iii tin- ca.xe of tin- heavy body and the earth i is — to our eyes — no connecting median ism and no deforma- tion of matter ; in both cases the presence and "action" of a material body is essential for the production of the force. Similarly, if two moving bodies strike each other, the motion of each is changed ; so each exerts a force on the other. We cannot directly investigate the motion of molecules, nor can we understand or even describe the action between the falling body and the earth, partly owing to the great size of the latter. For these reasons we shall first discuss, as an illustration of forces, the accelerations of material bodies of ordinary size, when these are produced by the interaction of the bodies themselves. If two billiard balls meet, the velocities of both change, i.e. they are accelerated ; if a magnet and a small piece of iron are suspended by strings at the same level, each moves toward the other with an accel- eration ; if a man stands on a box which rests on a smooth floor, and jumps off sidewise, the box moves in a direction opposite to that iii which the man jumps; etc. It is a familiar fact, too, that if one of the moving bodies is much heavier than the other, its acceleration is much less; and, as the bodies are varied, there is apparently a connection between some property of the body and its acceleration. Fundamental Principles 1. Principle of Inertia. — In none of the cases described above is there an acceh-ration of one of th- without Hi-' re being at the same time an acceleration of the other. ve make, as the fundamental assumption in regard to iiat the acceleration of a body depends upon itt potition reference to neighboring bodies and upon their velocities. ii asure these velocities and accelerations, some suitable geometrical Hum re «,f reference must be selected.) There is no way of proving this assumption or the following ones; but all observation- .uv in iOO
Ord with them. 60 MECHANICS 2. Principle of Independence of Action of Forces. — Again, we shall assume that, when a body is tunh'r the influence of several forces, the action of each one is independent of the actions of the others. 3. Principle of Action and Reaction. Definition of " Mass." — Then, if we have an isolated system of two bodies, each will have a" linear acceleration ; and, in order to speak defi- nitely, we shall consider the bodies as being so small that they occupy points. Such bodies are called "particles." We assume that their accelerations are in the straight line joining the particles, but in opposite directions, and that the ratio of these accelerations is a constant quantity. We can, therefore, assign a number to each body such that, if m^ and w2 are these numbers, and al and #2 are the accelerations, m^ = —m2a2. Similarly, if we have a third particle, we can assign a number to it by allowing it to "act" upon the first particle, using ml as its number, or upon the second one, using m2 for it. Experiments prove that the numbers thus obtained for the third particle are the same. Therefore, if we adopt an arbi- trary number for any one particle, the numbers obtained for all other particles are definite. These numbers are the values of what is called the " mass " of a body. (The system of masses in ordinary use will be described presently.) 4. Definition of " Force." — When a particle of mass m has an acceleration a, the product ma is a vector with a definite value and direction ; and it is defined to be the value of what we have called the force. Thus, in the case of the two par- ticles, there are two equal and opposite forces ; and we say that the force of a particle A upon a particle B is equal numerically but opposite in direction to that of B upon A, or that "action and reaction are equal and opposite." Since, then, F = ma, calling the value of force F, a= — , or m when a given force acts upon a particle, the acceleration is in the direction of the force and its numerical value varies •pi I>Y .;i inversely as the mass of the particle. That i>, if m is large, -mall : and conversely; so m measures the inertia of [Kirtirle with reference to translation. 5. Centre of Mass. — In practice we cannot obtain particles, for all material 1 todies occupy finite volumes ; and so we can- not apply the definition of a force directly. Further, under the action of a force a body, as a rule, has both linear and angular acceleration ; for instance, if a rod lying on a smooth table is struck at some point near the end by a ball rolling on the table the rod will move in the direction of the blow, and it will also rotate. Hut we can prove, as will be shown • •ntly, that, if we assume tJmf <i / 1>ody may be «w- red as built up of particles, there is a geometrical point e-iiineeted with tin- body that has the same linear accelera- tion when the body is acted on by a force, as if this a- on a particle whose mass equaled that of the body. This point is (ailed the "centre of mass" of the body. A few illustrations of this fact may be interesting. If a particle were to fall freely from a projectin. itli would l»e vertically down ;• \vniiM have a constant acceleration. Similarly. when a plank or a falls without striking an obstacle, there is a jt«»ii tliat mOVBfl vertically down with a con-taut acceleration, however much the l»ody turns. If aparticlewereBtruckabloNN.it \s»uM move in ion of thy force. Similarly, if a rod or a chair is five to move 11 struck a hlo\\. tin-re is al\\ay> a point connected with it that I in a straight line in the direction of the blow, although rotation may be produced also. Consequently, whenever we give illustrations of forces, etc., and refer to the linr.n- acceleration -of the body." the linear acceleration of it- cenl re nl - meant. Measurement of Mais The Theory and Practice of the Measurement - [.article ha- IM-.-M defined in ten '* «»f one p.irtiele on aiiothrr: i he definition "f a force, we may say that, if two parti '.• ve accelr \\\i\g to 62 MECUA N1CS the action of the same force, their masses vary inversely as their accelerations. For, if m1 and m2 are the masses and ax and a2 the accelerations, the value of the force F must satisfy the two equations : F=mlal> F = w2a2. Hence m^ = m^a2J <i «2 or ml:m2 = ~:-. In order to give a number, then, to the mass of any body, some piece of matter must be arbitrarily selected to which the number 1 is given ; and the mass of the body must be compared with this " unit mass " by the experiment just described, assuming, what will be proved presently, the prop- erty of the centre of mass as just stated. One manner in which this comparison can be imagined done — although it is not practicable — is to place the two bodies on a smooth, horizontal table, so that gravity has no action and that the motion is not affected by friction ; and, attaching to each in turn a spiral spring, to drag it in a hori- zontal direction in such a manner and at such a rate that the spring is elongated by the same amount. The spring will therefore exert the same force on the two bodies in turn ; and, if flj and a2 are the measured accelerations, TWI«I = m2a2. (We assume in this that when a spring is stretched a definite amount it exerts a definite force, regardless of the time or place.) Another method is to make use of a fact which will be dis- cussed later, viz., the acceleration with which a body falls toward the earth — due allowance being made for the effect of the atmosphere — is the same at any one point on the earth's surface for all bodies. A body, then, is acted on by a force due to the earth, which we call " weight," whose value, in accordance with the definition of force, equals the product of the values of the mass of the body and the acceleration which DYNAMICS 88 it would have if falling freely. Call the mass ro and the leiation^; then the weight of the bod \ is ///</. It follows, then, that if the body is supported from the end of a spiral spring, — an ordinary "spring balance" such as is shown in the (lit, — the spring will be elongated and therefore exerts an upward force on the body. Since the body is at rest, — relatively to the earth and the spring, — this upward force must equal mg. If two bodies when suspended in turn from the same spiral spring produce the same elongation, their weights must be tin- same, proper allowance being made for the buoyancy of the air 6 page 165); i.e. ea li- the masses m1 and KTl hence 7H1==wia, or their masses are the same. n dard body is chosen whose mass is called 1, another body can be taken which elongates a spring slightly more than does the former when suspended by it : and then, by chipping or filing off minute quantities, it maybe so altered as to produce the sam. • . • • i tion as does the Standard body. Thus \\e obtain a Ml body of mass 1. Similarly, we can obtain a lx»d\ of mass 2 by first suspending together the two bodies of unit mats, noting the elongation, and then deterrninun: a third 1 which produces this same elongation ; etc. To obtain a 1 Fto. ». — Sprint BaUnc «M^\| fur rln,: 64 MECHANICS whose mass is -\|, we must make two bodies of equal mass — as shown by producing the same elongation — which when suspended together will produce the same elongation as does the body of unit mass. Proceeding in this manner, we may obtain a u set " of bodies all of whose masses are known in terms of the standard. Then to obtain the mass of any body, it may be suspended from the spring and that com- bination of bodies from this set determined which will produce the same elongation. (Other and more accurate methods are used in practice, as will be shown later in speaking of the "chemical balance.") In using a spiral spring for ordinary purposes a different method is followed from that just described. Experiments show that the elongation of a spiral spring is proportional to the stretching force, most approximately ; and a divided scale may be attached to the frame carrying the spring, the readings on which are proportional to the elongation of the spring. Then if a body whose mass is ml produces an elongation hv and one whose mass is unknown but which may be called m produces an elongation A, h^ : h = m^g : mg, since these last are the forces. Therefore, h^ : h = ml : m ; or m = ^-^ As a rule the instrument maker divides and marks the scale so that it gives the values of the masses directly on some known system ; that is, when a body of mass 1 is suspended, the pointer which marks the elongation stands at division 1 on the scale, etc. Mass and Weight. — It should be carefully noted that this method of comparing the masses of two bodies is in reality one which compares their weights; but, since g at any one locality is the same for all kinds and quantities of matter, the weight of a body is proportional to its mass. In other words, two bodies that have the same weight at any one point on the earth's surface also have equal masses. This 'l\ DYNAMICS i;:, fact is not self-evident ; for the weight of a body is de- pendent upon tlie presence and proximity of tin- earth; its mass, upon the acceleration it would receive from a deli force anywhere in the univew. 'Hie former is a variable quantity, as we shall see. because it is different for the same body at different latitudes on the earth and at different heights above sea level ; while the latter is, to the be> our knowledge, an unvarying constant quantity for a given body. There is no more a jn-t'>ri reason tor believing that two 1 mdies which have equal masso have equal weights, than for believing that they have equal volumes, which is ob- viously not true in general. It is a question to be settled by experiment, and depends upon proving the constancy of i all kinds and amounts of matter at any one point on the earth's surface. This was first shown by (ialileo in 1590, who allowed two bodies of different weights to fall from the top of the Leaning Tower of l'i>a. an
d observed that • lied the ground at the same instant, thus proving that they had the same acceleration. (Galileo, however, had no conception of the property of mass, and performed hi* ex- periment with a different object.) Newton was Un- to real i/e the fact that a material body had a pi < iiieh we have called -mass"; and he devised a most ingen experiment for the purpose of learning \\hether .-/ was the same for all bodies. His method depends upon the use of and will be described later. It ha- been used Bessel ; and all experiments pm\e ih.it we may .7 as a constant at any one point. Conservation of Matter. l-Vw quantitiea can he measured with the exactness of the weight, and 161106 mass, of tic use of a "chemical balance" 80 railed. One ,.(" the questions ti: loans of it was as to the < stancy of mass of a body, o: when combined . . I rom \\hat has been said in the -ion. evident ti imh'rgo van AMI> 6 66 MECHANICS changes ; it may have its shape, its size, its temperature, etc., altered ; it may explode into fragments ; it may be " electrified " or " magnetized " ; it may be melted if it is a solid, or evaporated if it is a liquid, or vice versa. Simi- larly, if two pieces of matter are brought together, they may stick to each other like putty and glass ; or they may unite to produce new substances, like a piece of coal burning, a process in which the oxygen in the air unites with the car- bon in coal to form a new gas, called "carbon dioxide." In all these changes, however, there is, so far as we know, absolutely no alteration in the total mass of the body or bodies concerned. This fact is sometimes called the "Principle of the Conservation of Matter " ; or, more properly, the " con- servation of mass." The Unit of Mass. — The standard body whose mass forms the basis of the accepted system of physical units is a piece of platinum which is kept in Paris, and which is called the Kilogramme des Archives. It was officially adopted in 1799, at the same time as the metre bar. Its mass is called a kilogram (Kg.) ; and a body whose mass is one thousandth of this is said to have a mass of 1 "gram" (g.)' A. mass of one tenth of a gram is a " decigram " ; one of one thou- sandth of a gram is a " milligram," etc. When the kilogram was originally made it was designed to have a mass equal to that of 1000 cu. cm. of pure water at a temperature of 4° C., because under those conditions the water is more dense than at any other temperature. (The temperature must be specified, because, as it changes, the volume of a given quantity of matter varies.) More exact experiments have, however, shown that this relation is not quite exact. In fact, the mass of 1000 cu. cm. of pure water at 4° C. is about 999.96 g. In England and the United States the commercial Unit Mass is that of a piece of platinum kept at the Standards Office at Westminster, marked " P. S. 1844 1 lb.," and called the " Imperial Avoirdupois Pound." It has been determined by experiment that the number of grams in one pound is 453.5924277. This unit is subdivided in such a way that 16 ounces equal one pound. /»'.v.i M 67 C.G.S. System. — In all scientific work the units in terms of which lengths, intervals of time, and masses are expressed are the centimetre, the gram, and the mean solar second. This is called the "C.G.S. system." Force F= ma The Unit of Force. — The unit of force on this C. G. S. em is that which corresponds to the product ma being unity ; that is, a force which produces in a body whose mass is 1 g. an acceleration of 1 cm. per second each second (or an acceleration 2 in a body where mass is J, etc.). This force died a "dyne." A force of one million dynes, i.e. 106 dynes, is called a "megadyne." As estimated by our muscles, a dyne is extremely small ; for, as we can find by experiment, the value of the acceleration of a falling body is not far from 980 on the C. G. S. system, so the weight of a body whose mass is a milli'jrnm i* the product of 0.001 and 980 or 0.980. Conse- quently, we feel approximately the force of 1 dyne when we hold a milli- gram "weight" in our hands. On the pound-foot-second system the unit force is one which gives in 1 sec. an acceleration of 1 ft. per second to a particle whose mass is 1 Ib. (This unit therefore equals 13,825 • approximately.) Other unit forces often used are the "weight of a body whose mass is 1 g.," or of one whose mass is 1 Ib. : these units have the great disadvantage of being variable, owing to variation in " g." Force Effects. — In defining the numerical value of a force, we made use of the idea of a particle and of the "fo effect " acceleration. But the production of acceleration is, as we have seen, not the only effect of a force. This is evi- dent frnin ili,' formula of definition itself, which has the form /= ma, if / ivjuvsi-nts the t e, m the mass of the par- ticle, and a the acceleration. I r \\«- li.tvo assumed that M aet independently; that is, they are vector quantities. 11 the fiM-mul;,. / may be the geometries 1 sum of several s. In \|,,irti. -nlsr, if a = 0, /= 0 ; hut this does not mean necessarily that th«-re U n<» foroe; it may mean that there are two equal and opposite forces acting on the particle. 68 MECHANICS Thus, if a particle is suspended " at rest " at the end of a vertical wire, it has no acceleration, owing to the fact that the two forces acting on it, its weight and the tension in the wire, are equal and opposite. It may also happen that when a particle is under the action of two opposing forces, its acceleration is not zero ; in this case one force is greater than the other. So, in general, we may say that a second effect of a force is to overcome or neutralize, more or less completely, another force. (If we speak of matter when being accelerated as offering an opposing " force of inertia " equal to ma, we may say that the effect of a force is always to overcome another force.) This second effect of a force is illustrated in many ways : it requires a force to stretch a spring, to bend a stick, to twist a wire, to push a body over a rough table, to drive in a nail, etc. As the properties of matter are gradually better understood, we hope to explain all these effects in terms of the acceleration of particles of matter. We can do this in certain cases already, as we shall see later. (See Kinetic Theory of Matter, Chapter IV.) Similarly, as we have shown before, forces may be pro- duced in various ways. If a stretched spring or wire or cord is fastened to a body, it will be accelerated unless there is an opposing action ; if a moving body strikes another, each exerts a force on the other, etc. Measurement of a Force. — We make use of some one of these force effects in order to measure a force in practice. We know that a body which is supported at rest free from the earth must be acted upon by an upward force whose numerical value is mg, where m is its mass and g is the accel- eration of a body falling freely in a vacuum under the influ- ence of gravity. We have shown how to measure m, and methods will be described shortly for obtaining the value of g. Thus, we can observe how much a spring elongates under the stretching action of different bodies ; and, assuming that 69 the spring does not change during the operations, we may thus obtain readings on a scale attached to the spring, which correspond to known forces. This process is called ** calibra- tion " of the spring. Then, to measure any force, we can observe how much it elongates the spring. (It -h to exert a known force in a definite direction upon a bodv, we can attach one end of a cord to the body and the other to a spiral spring, and then pull the farther end of the spring in the desired direction until it elongates the proper amount. ) < )ther elastic bodies maybe calibrated in a similar manner and used for the measurement of forces. Linear Momentum. — The general formula for the value of a force, F = ma, may be expressed in a different manner. Since the acceleration is the rate of change of the velo. the value of the force is the product of the mass by this rate of change ; and, since the mass is a constant for a given particle, the force equals the rate of change of the product mv, where v is the velocity. This product is called the •• linear momentum" of the particle ; so the force equals the rate of change of the linear momentum. If the acceleration is constant, i.e. if the speed is changing at a uniform rate but the direction of motion is unchanged, writing i'o and ?'j as the velocities at the end and the be- ^inniiiLC of an interval of time t. the rate of elian^e, of the velocity is -*— — -• Hence, we may write F Ft = m(v9-vl). The product Ft is called the "impulse" If a sudden blow is struck the {.article, its momentum in the direction of the iWe.o will 1)0 changed; ami the amount of this change measures the impulse of the blow. If a particle is moving with a constant velocity, there is restdtanl force acting: but to produce e in the ;orce is required. A useful ill nila F=ma is furnished by suspending a heavy body 70 MECHANICS by a long cord and attaching a thread to it also, so that it may be drawn side- wise. If it is pulled slowly, a small force is required ; if it is jerked sud- denly, the force required may be so great that the thread breaks. Illustrations of Forces. — There are two types of acceleration, one when the speed changes and the direction does not, the other when the direction changes and the speed does not ; and correspond- ing to each of these is a definite type of force. In the one the force produces a change in speed ; in the other, a change in direction of motion. a. Rectilinear force. — Thus, if we ob- serve that a particle is moving in a straight line with varying speed, we know that there is at any instant a force acting in the direction of the line, whose numerical value is ma, where m and a are the values of the mass of the particle and its accelera- tion at that instant. Il- lustratio
ns of this type of force are afforded by fall- ing bodies, F= mg ; by an elevator when rising or falling at a varying rate ; a railway train "getting g and mtff are Up " speed, etc. When an elevator is rising at a uni- Tis the tension in the . FIG. 30. — Atwood'f the forces due to gravity, string. DYNAMICS 71 form rate, the upward force exactly balances the downward force of weight (and friction) ; but, if it is accelerated, an additional force is required. A similar statement is true of the train. If a particle is given a velocity obliquely upward, it will have the path of a parabola as described on page 46, for it will retain a constant horizontal velocity, if we neglect the action of the air; and it will be under the action of a constant downward force. This is an illustration of the independence of two motions, one of uniform velocity, the other of uniform acceleration. ..tin, let two particles whose masses are m, and m, hang from the •nds of a perfectly flexible inextensible cord which passes over a piiilt-y ; let us suppose that there is no friction, and let us neglect for the time being the mass of the cord and the inertia of rotation of the pi, e are two forces acting on each particle: on the one whose mass is ml there is a force downward owing to gravity and equal to m^, and an upward one due to the tension in the string whose value may be «-n T, hence the total downward force is m,<7 — T\ on the other partirl.-. the total downward force is, similarly, m^g — T, because the due to the tension in the string must be the same at both ends if is no friction and it is inextensihle. The rfinrnu-nrd acceleration on one particle must, however, r<\|iial th«* H/m-anl >i the ;, since the string is inextonsible; the downward acceleration of first particle is, however, m&~ T-\ and the upward one of the other T '"' -^2- Calling this a,,!, -ration a, we have, tl W T— —T mt m, or, eliminating 7, a / ... . if ,„, >!«, " i- poritire, and th« -5H.- will descend v if "», <wr a is negative, and the first particle will ,'iM-rml. The tension in the string is given at once from the formula. T = mtf + m^n. T = mtf - ni shows that in tin- case of an ascending particle, rfcu, the one whose IIIRH> lie tension is greater than the weight by an 72 MECHANICS amount MgO, while in the case of the descending particle it is less than its weight. The same formula for the acceleration may be obtained in a different way ; the total force acting on the system in such a direction as to give the first particle an acceleration downward is itiig — m^g, the total mass having the acceleration is mi + m2, hence the acceleration equals -f in 2 nil + (This apparatus is called Atwood's machine.) An illustration of the independent action of forces is afforded by the slipping of a body down a rough inclined plane board. Let, at any instant, the body whose mass is m be at P. There are three forces: gravity, acting vertically down, whose value is mg] friction, acting parallel to the plane in such a direc- tion as to oppose the motion, whose value may be called F, and the force of reaction due to the board, opposing the force with which the body presses on it, which is perpendicular to the plane of the board and upward, and whose value may be written R. If N is the angle the inclined plane makes with the level surface of the earth, the component of mg parallel to the plane is mg sin N and its direction is down the plane ; F is parallel to the plane, but directed upward ; R has no com- ponent parallel to the plane, because it is perpendicular to it. The total FIG. 81.— Motion down a rough inclined plane. . , . . , , . , , force down the plane is, then, and so the acceleration is this divided by m ; i.e. mg sin N — F, mg sin N — F b. Centrifugal force. — In order to change the direction of motion of a particle a force must be applied at right tingles to this ; and conversely, if a particle is moving in a curved path under the action of a certain force, and the force is removed, the particle will continue to move with a constant velocity in ths direction of the tangent to the curve at the point where it Avas at the instant the force ceased. If a particle is moving with constant speed in a curved path which 73 lies in a plane, the acceleration at any instant is toward the centre of curvature of the path and has for its numerical value — , where « is the speed and r is the radius of curvature. ( Whatever the path is, a circle can always be drawn which will coincide with the curved path at any point, and its centre and radius are called the "centre and radius of curvature" at that point.) There must then be a force whose direction is perpendicular to the line of motion and whose numerical value is - s . in order to make the particle whose mass is m and speed change its direction of motion and move in a circle of radius r. This force, F= — , is called " centrifugal force." The less the force, the less the change in direction, i.e. the greater the value of r\ or, if the speed is increased, tin- radius must be also, unless the force is increased; and \vhen a j »article is revolving in a circle, if the force toward the centre is diminished, or, if the speed is increased, tin- particle moves farther away from the centre, if such motion is possible, so as to have a larger radius of motion. This fact i.s SM],I, -times described by saying "a particle re- volving around a centre tends to move as far a\\ay from it as possible." Thus, a wet mop may be freed from the water by revolving it rapidly; clothes may be dried by inclosing them in a perforated Cylinder which is made to rotate rapidly : etc. The force that is required to hold a particle in a circle \ aries dhvctly as its ma>s ; and. if the force applied i than " ~. the particle will move toward the centre. \\ hile, if it is less than this, the particle will move away from the centre. Therefore, if an emulsion of two liquids, one imnv dense than the othi M in milk, is put into a raj. idly rotat- ing cylinder, the heavier of the two — milk — will go to the "inside wall, while the lighter — cream — will come closer to the ,: 74 MECHANICS c. Pendulum. — If both the speed and the direction of motion are changing, the force must be oblique to the latter, i.e. it must have a component in the direction of motion and one at right angles to it. One particular case of this kind of force is furnished by the action of the earth on a " simple pendulum," i.e. on a particle of matter suspended by a mass- less cord from a fixed support so that it is 'free to move in a vertical plane. When at rest, it hangs in a vertical direc- tion ; but, if disturbed slightly, it will make vibrations. (Of course a simple pendulum cannot be made, but we can ap- proximate to one by using a small but heavy bob and a very fine wire to support it.) Let OM be a vertical line through the point of support 0 ; let OP be the position of the pendulum at any instant (not necessarily at the end of the swing) ; let a circle be drawn with radius OP, thus indicating the path of the particle; draw through P two lines: one vertically down, to indicate the direction of the force of gravity ; and one tangent to the circle. Call the mass of the particle w, and the length of the pendulum I. There are two forces acting on the particle ; its weight mg vertically down, and the tension in the supporting cord acting along the line P 0. The latter has, however, no effect on the speed, serving merely to change the direction of motion ; similarly, only that component of mg which is along the tangent has any influence on the speed. This component has the value mg sin N, where N is the angle (MOP). Let us suppose the arc of vibration is made so small that its chord coincides with it; in this FIG. 82. — Simple pendulum. DYNAMICS 75 limit the sine of an angle equals the angle itself (see j and so the force at any instant along the tangent to the curve — which in the limit is nearly horizontal — has the value mgN. If the displacement of the particle along the arc measured from its lowest point is #, N= - ; and there- fore the force has the value mgj. Its direction is opposite to the displacement ; and therefore its value must be written /y. The acceleration is, then, — &x. It follows that t d the motion of a particle along its infinitesimal arc is harmonic ; and its period is 2 w\— (See page 50.) 9 A method is thus offered for the measurement of g at any point, and therefore for the investigation of the question as to whether it is the same for all kinds of matter and for all amounts. That portion of the above formula which states that the period of a pendulum varies as the square root of its length was deduced by Galileo as early as 1638. Parallelogram and Triangle of Forces. — Another method of stating that tones act independently is to say that they can be added geometrically like vectors ; and accordingly this theorem is sometimes red to as the "parallelogram of forces." For if tho two forces acting on a particle art; repre- " & rented liv /'O -iml K";' •-" . " P»r»lWo,rr»m of Forces ": UJ V ' C»l sum »f r<J and PR U PS. I'll ill tin- rut, tli.-ir ft. "Trianfle of Forew": th« fwoMtrlttl tarn of FQ U'coinrtrical sum is ^ "d §6r° < '.sen ted by PS, the diagonal of the parallelogram which has PR and PQ for two adjacent sides. It isrvidnit that it tlnj particle is acted on by three forces, 76 MECHANICS PR, PQ, and one equal but opposite to PS, their geometri- cal sum is zero ; for, adding these lines geometrically, a closed triangle is formed. Conversely, if a particle under the action of three forces has no acceleration, they will, if added geo- metrically, form a closed triangle. This theorem is called the "triangle of forces." This principle was first stated, for a special case, however, by Stevin, as early as 1G05. JFio.84. — Composition of Forcesj_ OA + OB= OC\_ Or, ~OA= OD+l>A ;_OB= OE +EB^_DA = - EB and OD=EC\ there- fore OA+OB=OC. The composition of forces is illus
trated by the motion of a boat that is being rowed across a river, there being two forces, one due to the oars, the other to the current ; and by many other similar motions. If a cord carrying at its two ends particles whose masses are ml and m2 is supported by two pulleys, and if a third particle of mass m3 is attached to it at some point between the pulleys, as shown in the cut, the system will come to rest under the action of gravity in some definite position. There are now three forces acting at the point O : m^g, vertically down; mlgl in the direction OA, because all that the pulley does is to change the direction of the force of the earth on the first particle ; w2gr, in the direction OB. If OA is a length proportional to the product m^, 77 and OB is a length proportional to m^, their geometrical sum OC will be a vertical line proportional to m^g. (Another mode of considering the geometrical sum of two vectors OA and OB is to see that, if OA is resolved into two vectors OD and ZM, where OD is along the diagonal OC and DA is perpendicular to it, and if OB is resolved in a similar manner into OE and EB, DA = - EB and OD + OE = OU.) Action and Reaction. — In the case of the interaction of two particles, the force exerted by one on the other is equal and opposite to that exerted by the latter on the former, and is in the straight line joining them. That is, if m1 and ro2 are the masses of the particles, and al and a., their accelera- tions owing to their mutual action, these accelerations are in tin- line joining the particles, and their numerical values are such that mla1 = — w2a2. This law may be expressed in terms of the linear momenta. It may be written m^ + m^ = 0 ; and this is equivalent to saying that the rate of change of the sum (^m1vl -f w2v2) is zero, where v1 and v2 are the com- ponents at any instant of the velocities of the two particles in the same direction along the line joining than. If the rate of change of a quantity is zero, the value of the quantity itself must remain unchanged. So we may write the formula in the form : m1v1 + 7?i2t»2 = a constant, so long as one particle is influenced only l»y tin- other. This product of tin- mass of a particle by its velocity in a definite direction is, as we have said, called its -lineai momentum " in that direction; and so the above formula states that, when two particles are left to their mutual inter- the sum of their linear momenta along the line join- them does not change, however much the momentum of • •it her one changes. Thus, the two particles may approach, ma\ CM. Hide, may recede, etc. :iw \\ ill I* given a simple geometric! interpretation when we speak of tli-- \ <>f a system <>f particles. In tin- simplest case, -.\lien tli- pa: tides are moving along the same line, it N illustrated by many familiar f:i< N. When a rifle is fired, it recoils; and, if it is suspended l»y cords, it will move in a direction nppoxit.- to that of tin* 78 MECHANICS bullet. If wij is the mass of the rifle and w2 that of the bullet, and vl and v2 are the corresponding velocities, the sum (m^ + w2t>2) remains unchanged after the explosion of the powder. But before the explosion both vl and v2 are zero ; and therefore the sum (m^j + w2v2) must be zero after the explosion also. So m^i\ = — m.2r2 or i\ = -$• Similar illus- FIG. 85. — Impact apparatus. trations of recoil or reaction are given when a man jumps sidewise off a chair, when water is allowed to escape from an opening in the side of a can which is suspended free to move, etc. A simple form of laboratory apparatus for studying the impact or collision of two bodies is shown in the cut. Directions for its use are given in all laboratory manuals. Conservation of Linear Momentum. — The linear momen- tum of a particle is a vector quantity ; and so the quantity m1v1 + m2v2 is a vector along the line joining the two parti- cles at any instant. As there is no force acting on the par- ticles except that due to their mutual action, their momenta in any other direction do not change. The particles are not 7 DYNAMICS 79 in general moving in the line joining them; so, if MJ is the actual velocity of the first particle at any instant, v1 is its component along the line joining the particles, and m^^ is its total linear momentum. Simi- larly, if t/2 is the actual velocity of the second particle, its total linear momentum is rw2w2- ^^ie above statements are, then, equiv- alent to saying that the geometri- cal sum of m^ and w2w2 remains unchanged provided there are no ' , f mi . e Fio. 36. — Conservation of linear mo- external forces. The sum is, of mentum: ^ geoinetrical 6um of ^ COUrse, a Vector. This more gen- llnear moment doe« not change on 1m- i- nil law may be illustrated by the ' impact of two moving billiard balls. Let the total momen- tum of the first particle be represented by the vector BO, and that of the second by A0\ their geometrical sum is CO. i impact, if OE and OD are the two momenta, their geometrical sum OF must be in the same direction as CO and equal to it. 'I' ln-.se statements in regard to the mutual actions of two part id cs may be extended at once to a system of several parti- cles. In this case there are forces acting on any one particle o\vin'_r to tin- action of all the others, but these are equal and opposite to the reaction of this particle on the others. So, if we add together the components of all these internal forces resolved in any fixed direction, the sum is zero. This fact !»«• expressed in terms of momenta. Since the algebraic sum nf the components in any one direction of the linear momenta of any two particles due to their interaction re- mains constant, so must the sum for all the particles; and, il sum of the total momenta of all the particles must ;. -main constant so long as there is no external act inn. This is called the " Principle of the Conservation of Linear Momentum." 80 MECHANICS Centre of Mass. — This principle has a simple geometrical interpretation. Lei the perpendicular distances of the par- ticles from a fixed plane at any instant be called xr x^ x3, etc. If we assign to each of these distances as a measure of its u importance " the value of the mass of the particle at that distance, i.e. m1 to xv m% to x2, etc., the mean distance of the system of particles from the plane is mlXl + m^ + ... (gee 3Q } ml + m2 + -. This distance may be called x ; and, writing for the total mass of the system, m1 -+- mz + •««, M, we have or MX = m1x1 + But the particles are moving, and each one has a velocity whose component perpendicular to this plane may be called u with a proper suffix. Thus uv this component of the velocity of the first particle, is the rate of change of x1 ; etc. Owing to these motions x does not, in general, remain con- stant ; and calling u its rate of change, we have from the above definition of #, taking the rate of change of both terms of the equation, M- = m^ + m^ + ... But the principle of the conservation of momentum states that this sum, m^ + m2u2 + •••, remains constant so long as there are no external actions ; therefore, M u is a constant or u is constant. Further, the total momentum of the system away from the fixed plane is Mu. In order to describe definitely the position of a particle, its distances from three fixed planes at right angles to each other must be given ; for instance, the position of the corner of a table in a room can be described by stating its height above the floor and its distances from two of the walls that make a corner. Thus, to describe the positions of the particles that are being considered, two other planes at right angles DYNAMICS 81 ich oilier and to the first must be chosen; and we have throe distances for each particle, which we may call x. each with its proper suffix. We can then define a point x, y, z, for which y and z have values similar to that of x\ Mz = mft If v and w are the rates of change of y and z at any instant, they are also constant, provided there are no external forces. Therefore the geometrical sum of w, v, w* which is the actual linear velocity of the point a;, ?/, z", remains unchanged. (The product of this velocity by M* the total mass, is evidently equal to the total momentum of the system, the momenta of the separate particles being added geometrically.) The point defined by the distances ir, y, z is called the " centre of mass " of the system of particles ; and the state- i units just proved in regard to its velocity are seen to be equivalent to stating that " the centre of mass of a system of particles free from all external actions moves with a constant velocity." This is another expression of the principle of the nervation of linear momentum. Tin- acceleration of the centre of mass is. in a perfectly similar manner, seen to be given by three equations, Ma = m,aj + m^a, + •••, two others, in which at is the rate of change of tj, etc. l.ut ///,//j is the total force on the particle \\li«^e mas> i^ ///,. and if there are no external forces, this is due to the action of the .it her pai tides, resolved parallel to the direction o! etc. And, since all the internal forces neutralize each other. this sum. j/yij -^m^-\ ---- .must e<\|iial /ero. 1 a is -. as was shown in the preceding paragraph. External Forces. — L«'t us now suppose that the systen acted upon 1>\ some externa . such as Lrra\it\. 1 dole will then he under the action «.f two sets <>t AMES'S piirnirs — 0. 82 MECHANICS internal and external ; so that if a is the acceleration in some particular direction of a particle whose mass is w, it is due to the sum of the components in this direction of the forces acting on this particle due to the two sets. If, then, we form the sum m^ 4- wz2«2 + "'» ^ s ^ne sum °f tne compo- nents in this direction of all the forces, internal and external ; but the sum of the components of the internal forces is zero, and so only the sum of the components of the external forces need be considered. Call this sum X.
Then m^a^m^a^ — equals X. We have just shown, however, that this sum equals Jfefa, where a is the component of the acceleration of the centre of mass in the chosen direction. That is, M a = X or a = — . In words, the acceleration in any direc- tion of the centre of mass of a set of particles equals the sum of the components of the external forces in that direction divided by the total mass of the system. We thus see the exact agreement between the properties of a single particle with those of a set of particles, the centre of mass of the set playing the part of the point occupied by the single particle. j£- The General Problem of Dynamics. — So far we have con- sidered only the applications of forces to particles; but in nature, of course, material bodies are never in this form. The actual cases of forces are always those concerned with extended bodies; and it is evident that the effect of a force on an ordinary material body depends upon three things: its numerical value, its direction, and its point of application. Tims, as has been explained, if a blow is struck a body, the effect depends upon the point where the blow is applied as well as upon its numerical value and direction; and as a rule both translation and rotation are produced. We shall inves- tigate these two questions separately; that is, we shall deduce first the effect of a force in producing linear acceleration and then its effect in producing angular acceleration. DY.\AMICS 83 Translation Translation of an Extended Body. — We can at once apply all our deduetions for a set of particles to a material body that lias a finite volume or to a system of such bodies, if we assume that we ean regard one as made up of particles. The centre of mass of such a body is defined by the same equations as for a set of particles; and its position can be calculated in many simple cases, e.g. a sphere, a uniform rod, etc. The linear momentum of such a body is the sum of the momenta of its parts, and therefore it equals the product of its total mass by the linear velocity of its centre of mass. The accel- eration of its centre of mass in any direction is equal to the sum of the components in this direction of all the external forces divided by the total mass ; and, therefore, the actual acceleration of the centre of mass is in the direction of the geometrical sum of the external forces and is numrric -ally tMjual t<> tin- value of this resultant force divided by that of the total mass. This is the great physical property of the centre of mass. In general language we may say that no matter what the shape or size of the body or where the exter- nal forces are applied, no matter how it turns or spins in its motion, the centre of mass follows the same path with the same velocity and acceleration that would be observed if these force* w. re transferred parallel to themselves and acted on a particle whose mass was that of the body. 1 '•!« or two illustrations maybe given. If a parti' 1- i- tlr oMiijiit-ly in the air, it will describe the path of a parahola; .similarly. \\ li.-n a chair i- thrown in tin- air. or wln-n an acrobat jump* otV a >\| i. turning •'» >'>m«-rsault, tli.-ir ••••ntrrs of maw describe paral I liMinKxhrll «'xpl«Mli's in mid air, the centre of mass • fragments describes a parabola as they gradually fall «« .\\anl the earth. • .in.l. .m Mow is struck a body, its centre of mass moves off in tin- tion <>f the blow. Translation of a System of Bodies. — It > ve a system of bodies, it may l>o considered, 90 far n is con- 84 MECHANICS cemed, as a system of particles coinciding with the centres of mass of the bodies and having their masses. This follows at once from what has just been said in regard to the properties of a single body. It is this fact that justifies us in using the illustrations we have given in discussing the laws concerning the motion of particles. It also justifies the language used in speaking of mass and its measurement. The properties of matter in rotation will be considered in the next Section. Illustrations of the calculation of the position of the centre of mass. It is easy to calculate the position of the centre of mass of any regular solid provided the matter is distributed uniformly throughout it, e.g. a cylindrical wire, a cube, a sphere, etc., and also of a system of bodies whose masses and the positions of whose centres of mass are known. 1. Uniform rod. — The centre of mass of a uniform rod is its middle point. For, consider the rod as made mnninnniniiufininiiiH 2 up of equal separate particles ; and _ x _ ^ let m1 and w2 be two which are at the ends. Take as the plane of reference FIG. ST. -centre of mass of a one perpendicular to the rod, and let uniform rod: m, and »it are mi- x^ and #2 be the distances of ml and 7W2 from the plane. Then, by defini- tion, their centre of mass is given by the equation : 2 But m1 = ra2 ; hence x — X^ _ x%, i.e. the centre of mass of these two particles is halfway between them. A similar statement is true for the other masses which make up the rod, always combining those which are equidistant from the two ends ; and therefore the centre of mass of the rod is this same point. Q.E.D. DYNAMICS 85 The centre of mass of a uniform sphere (or spherical shell ) Jso its eentre of ligure. 2. Uniform triangular board. — Draw the three medial lines Aa, Bb, Co, connecting the vertices with the middle points of the opposite sides. They meet in a point 0. Since the straight line Bb divides the tri- angle into two equal halves, the centre of mass must lie on it; for the triangle may be considered built up of a great number of strips parallel to the side AC, and as the centre of mass of each of W lies on the medial line Bb, the centre of mass of the entire triangle must lie on it also. Similarly, it must lie on Aa and Cc ; that is, it must be the point 0, their common point Fio. 88.— Centre of mass of a uniform triangular board. of intersection. 3. A uniform rod, mass 7»8 = 25, carrying two symmetrical bobs whose masses are ^=15, #^=20; the dimen- sions and distances being as indicated ill the CUt. The centre of mass of the rod itself is its middle point ; which is at a dis- tance 15 cm. from the ends. Take as Fi«,. .T.». Ontn- .if rn weighted bar. at its left of» the plane from which to measure « tances one perpemlieuhir to the rod Then m, = 15, r, = 5; mf = 20, rf = 20 ; 1113 = 25, ar, = 15; and tl - _ mlxl + m, + m, + m, 75 + 400 + 875 80 14.17. centre of mass must, then, lie HI .nee of 14.17 cm. tin plan, at the end of the rod; and since the bobs are 86 MECHANICS symmetrical, it must lie in the axis of the rod at that distance t'nun the end. masses m 4. A rigid framework lying in a plane ; two bodies, whose 20, 7W2 = 10, are connected by massless wires to a uniform rod whose mass 7H8 = 10; the dimensions being as shown in the cut. Take as the two planes of reference one perpendicular to the rod at its lower end, the other through the rod perpendicular to the two wires. Hence nij = 20, xl = 0, yl = 10, m2 = 10, x2 = 20, y2 = 5, FIG. 40. — Centre of mass of a rigid framework. So i = \V = 7.5; y = W=6.25. That is, the centre of mass is a point at a distance 7.5 cm. from the plane perpendicular to the rod at its lower end; and a distance 6.25 cm. from the rod itself in a direction parallel to the wires ; therefore it is at the point 0 as shown. Rotation Introduction. — Let us now consider the rotation of a material body when it is acted on by a force. A simple case is that of a body pivoted on an axle, e.g. a door. If a push is given it, in general an angular acceleration will be produced ; but if the push is so directed that its line of action passes through the axis, it has no effect on the rotation. It is a fact easily observed that the effect increases as the direction of the line of action of the push is removed farther and farther from the axis. Moment of a Force. — To be definite, let the cut represent a cross section of the body so taken as to be perpendicular to the axis and to include the point of application of the DYNAMICS 87 force. Resolve the force into two, one parallel to the a the other in this plane section. Only the latter has any effect on the rotation. Let P be the intersection of the plane l>y tin- axis, and F the position and direction of this component of the force. Prolong the line of action of the force, if necessary, and let fall upon it a perpen- dicular from P ; this is called the " lever arm " of the force F with reference to the axis through P. If its length is J, experi- ence shows that the change in rotation produced by the force F varies directly Pro. 41.— Moment of force: plane section of body perpendicular to axis at P; F is component of force in this plane. M /' ami as l\ if either is increased, the angular acceleration is increased, and vice versa. To determine the exact effect that the two quantities F und I have in producing angular acceleration, the simplest method is to apply a second force to the pivoted body in such a direction as to neutrali/.e the rotating effect of the first. Ex- periments lead us to believe that if the relations between the two forces and their t\vo lever arms is such that FJ^FJy one force balances the other so far as rotation is concerned. The } product of a force by its lever arm, _ with reference to any axis, is < ailed the ••moment of the force with reference Fro. 42. — Rigid body pivoted by an axis through P, and In to thlSBXlS J and this is evidently ti^fo^r U^r-%TUOn °f the P1"!"1 measure of the rotating effect of a foroe. The moment hears the same relation to rotation that foroe does to translation. A moment is detim-d 1»\ tin- /»•///»// < if its axis, its amount, and the BeUM ol its rotating action. ( \ moment in tin- sense of tin- hand- 'i\en an al-vl.r.iir M-H, different from one in the opposite sense.) A moment is. thru, a rotor. 88 MECHANICS Moment of Inertia. — We must next determine the rela- tion between the moment of a force and
the angular acceler- ation produced owing to it. The simplest case of a rotat- ing body that we can imagine is that of a particle attached to a pivot by a massless rod. Let the particle of mass m be at A ; let the pivot be at P ; and let there be a force F in the direction AB. Draw the perpendicular PQ upon the line of action of the force ; let its length be I ; and let the angle it makes with the massless rod AP be called N. Let the length of this rod be called r. The lever arm of the force F is ?, and it equals rcos^"; hence the moment of the force F around the axis at P is Fr cos N. There is also a force on the particle owing to the rod; but this force has a lever arm zero, since its line of action passes through P. As FIG. 48. -A particle of mass a result of the force F the particle m is constrained to move in a .-,-, . , •• , . circle of radius r about P, and Wl11 receive an angular acceleration, is under the action of a force, F, WhoS6 Value may be found as follows: at any instant. _ . resolve F into two components, one along AP, the other perpendicular to it along AC-, the value of the latter is .FcosJV, because the angles (BAG) and (APQ) are equal ; and this is the only component that pro- duces any angular motion of the particle. Therefore, the linear acceleration of the particle in the direction of this component is — — But it was proved on page 55 that m the angular acceleration of the moving point equals its linear acceleration divided by r. So, calling the angular accel- eration A, A = — — ; and its connection with the mo- F cos N mr ment of the force Fr cos N is evident; viz., Fl = mr2A. Writing a single symbol, L, for the moment, this equation becomes L = mr^A or A = — -• * r m 1)Y XAMICS 89 Since "moment of force, v L, plays the same part in rotation that force does in translation, and since angular acceleration in one corresponds to linear in the other, it is seen that, in the rotation of a particle round an axis in a circular path of radius r. ni/~ takes the place of mass in translation; that is, it measures the inertia of the matter for rotation; if it is large, A is small, and conversely. Similarly, if a material body, considered made up of particles, is rotating around a fixed axis, the total inertia of the body for rotation is m^rf -f rw2r32 -f . . ., where rx is the radius of the path in which the particle who>e mavs is m^ is moving, etc. This sum is, of course, an arithmetical one. It is called the "Moment of Inertia" of the body with reference to the axis of rotation, and is ordinarily written /. Moments of inertia may be calculated for the regular solids provided the matter is uniformly distributed in them, and in many other special cases. Processes of the infinitesimal calculus are, however, required. A few illustrations may be given. Fora sphere of radius H and mass M^ the axis being a diameter, I— \MJ&-, for a cylindrical cylinder of radius //and mass .)/. the axis being the axis of symmetry. I=^MR. (If, however, the axis is a generating line of the surface, /= % ^f/22.) If the length of the cylinder U //. and if the axis p;i>s«-s ilir«»ngli its middle point perpendicular t o its length, /= M -f 7 Equation of Motion for Rotation of a Rigid Body around a Fixed Axis. — \Ve must next discuss the rotation of H body of definite si/.e and shape round a tixed axis under the action ot an ,-xtenial moment. Such a b<.dy is called in dynamics auri_urid" one. AS before, describe a cross section through the body perpendicular to the axis and including the point the force; and let the external force be re- solved int.. t\\,, components, one parallel to the axis, the other in the plane. L,-t the latter i»e represented in the cut '.M) MECHANICS by F ; let P be the trace of the axis ; and I the lever arm. The moment of the force is Fl. It should be expressly noted that this quantity has the same value wherever the point of application of the force is, provided only that the force keeps its line of action, i.e. provided I does not change. (Thus the force may be applied at Av or Av or Ay etc., in the line of action.) In discussing translation it was shown that the effect of a force on the motion of the centre of mass did not depend upon the position of the line of action nor of the point of application of the force, but. FIG. 44. -A rigid body . under the action of a force simply, on its amount and direction; con- F. A^ A* A, are points sequently, the total effect of a force upon in the line of action of F. 1 J a rigid body depends upon its amount, its direction, and the position of its line of action with reference to the body, not upon the position in the line of action of its point of application. (If the body is made up of particles so connected as to form a figure of variable size and shape, e.g. an elastic body, I and I would change.) There are also internal forces between the particles; in any actual case there is friction between the material pivot on which the body turns and the body itself ; and, further, the pivot in general exerts a force on the body. The moment of this last force is zero, because its lever arm is zero ; and we shall assume for our present purposes that there is no friction. The moments of the internal forces neutralize each other, moreover, because the forces between any two particles have been assumed to be equal and opposite and in the line joining them, so the lever arms are equal, and there- fore the two moments are equal, but in opposite senses of rotation. The total moment around the axis is, then, that of the external force F\ and calling, as before, Fl=L, we have DYNAMICS 91 the fundamental equation for the rotation of a rigid body around a tixed axis, ^ where A is the angular acceleration, L is the moment of the external force, and I is the moment of inertia of the body, the last two quantities being referred to the fixed axis. Illustration. — A simple but important illustration of this be a vertical formula is afforded by a pendulum. Let line through the point of support, and OP the position of the pendulum at any instant, so that the angular displacement is the angle (M OP), or N. If m is the mass of the bob of the pendulum, and r the length OP, the moment of inertia of the pendulum around an axis at 0 perpen- dieular to the plane of motion is mr2. Tl it- re are two forces acting on the pendu- lum hob: its weight, mg^ whose moment around the axis through 0 is mgr sin N, F,O. «.- simple Pendu- and the tension of the string, whose moment is zero. But this moment due to gravity is in an opposite direction to the displacement; hence, in the mr*. Therefore, formula A = — , L = — m</r sin N and /= marsinff a . ,_ ^ = 15 — = — -smiV. If the amplitude of vibration is made infinitesimal, we may replace sin ^V by JV; and A = — -N. Therefore, the motion is harmonic, and the Period is 27r\/-, as was found before. v Composition of Moments. — It is evident, since forces act iii'leprmlentlv, that if t here are several external forces acting, A ifl the sum of tin- moments of the separate forces, proper attention brinir irivrn the ul^ehraie signs. 92 MECHANICS It is easily proved by geometry that if a rigid body is acted on by two forces that lie in a plane, the sum of ilic moments about any axis perpendicular to this plane equals the moment of their geometrical sum, provided its line of action passes through the point of intersection of the two forces. For, let the two forces have their lines of action in the lines OA and OB, and let them be represented in direc- tion and amount by the vectors OA and OB. Their geo- metrical sum is then 00, and its line of action is here represented as passing through 0, the intersection of OA and OB. Let P be the trace in this plane of the perpen- dicular axis ; draw the lines PO, PA, PB, and PC. The numerical value of the moment of the force OA about the axis through P is the product of the length of OA by the perpendicular dis- tance from P to this line ; that is, it equals twice the numerical value of the area of the triangle (POA). Similarly, the moment of OB is FIG. 46,-composition of the moments given by twice the area of the tri- of OB and OA about axis at P. -\ /-T-»/^r>x j 1-1 c /T7T i angle (POB), and that of OO by twice the area of the triangle (P0(7). But the area of the latter triangle equals the sum of the areas of the three tri- angles (POB), (PBC), and (6>£<7); and the combined areas of these last two equals the area of the triangle (POA), because all three have bases of the same length, viz.; OA or BC, and the combined altitude of the first two equals that of the third. Therefore the area (P<9<7) equals the sum of the areas (POB) and (POA); and it follows at once that the moment of 00 equals the sum of the moments of OA and OB. (In the cut, these last two moments are in the same direction, viz., that corresponding to rotation counter- clockwise. If P is placed elsewhere, the two moments might be in opposite directions ; in which case they would DYXAMH S 0,3 be given opposite signs, and their algebraic sum must be taken. ) Conservation of Angular Momentum. — Returning to the general formula, it is seen that A = 0, if L= 0 ; that is, the angular velocity of a body turning on a fixed axis remains constant if either the moment of the external force is zero or if the sum of the moments of the external forces is zero. This is perfectly analogous to the case of translation when ^=0, and is illustrated by the rotation of a wheel whose frietion with its axle can be neglected. If the value of the angular velocity at any instant is written h. the product Ih is called the "angular momentum" of the ri-_rid body around the given axis; and the general law may rated by saying that the moment of the force around the equals the rate of change of the angular momentum about the same axis. So, if the total moment is zero, the angular momentum remains constant. If the rotating body is not rigid, the angular momentum is the sum where mr rr /^
apply to one particle of the body ; etc. The statement that this sum is constant when the external moment is zero is still true, however. Several illustrations are worth noting, if the angular velocity of all the particles is the same, the angular moment um may be expressed (wjfj2 -f nyf + —)h\ and now, if o\viii«jf to any internal cause the values etc., become smaller, the value of h must increase. This \\;is the case with the planets in their early history and is s<> \\ith the sun at present. There are forces acting <>n these bodies, but their moments about the axes of rotation : and the formula may be applied. As time goes on, the planets and <\\\\ have contracted owing to internal gravi- s ; and therefore, as proved above, their angular velocities have increased. A^ain, as an acrobat turns a :lt in the air. while at the same time he jumps over 94 MECHANICS an obstacle, his centre of mass describes the path of a parab- ola ; but he can increase his angular velocity by drawing in his arms and legs, thus diminishing his moment of inertia, because there is no moment due to the force of gravity. Illustrations of Rotation. — If a rigid body is turning on a fixed axis, a moment round the same axis will change the angular speed, either increasing or decreasing it, as is illus- trated by setting in motion a grindstone by means of a crank handle or in stopping one by means of a brake. If, however, the moment is around an axis at right angles to that of the existing angular velocity, the direction of this axis will be changed ; this is illustrated by the motion of a rolling hoop whose upper edge is pushed sidewise, as explained on page 54, or by the motion of a spinning top whose axis is inclined to the vertical. Principal Axes. — When a material body is rotating on a fixed axle there are in general certain forces and moments which the body exerts on the axle and which are borne by the bearings that hold the axle. If the axis does not pass through the centre of mass, there is a pull on the axle toward this point as it moves in a circle around the axis, due to the reaction of the centrifugal force. Its amount is Mrh?, if r is the radius of this circle, M the mass of the body, and h its angular speed. At any point of the moving body there are three directions, called "the principal axes at that point," such that if the axis of rotation does not coincide with one of them there is a twist on the axle tending to make it turn. This push and twist must of course be withstood by the axle or its bearings. So, if the body is to turn freely, producing no forces or moments on the axle, the axis of rotation must pass through the centre of mass and must be one of the prin- cipal axes at this point. In other words, to make a body maintain its axis of rotation in a definite position and direc- tion other than one which is a principal axis at its centre of mass, a force or moment is required ; and, if no such force DYNAMICS 95 or moment is applied, the position or the direction of the axis of rotation will change. But, if a body is set spinning about a principal axis at its centre of mass, it will maintain its rotation unchanged in every respect, if no moment acts upon it. This last statement is illustrated by the throwing of a quoit, whose axis remains parallel to itself if it is set spinning in the proper way ; by the motion of the earth on its axis, which moves in space parallel to itself (omitting small perturbations and the effect due to the protuberances at the equator); by the motion of projectiles shot out by "rifled" guns; etc. Translation and Rotation It is interesting to arrange in parallel columns correspond- ing properties of translation and of rotation around a fixed axis. Translation of a Particle Rotation of a Rigid Body a. mass a. moment of inertia l>. force A. moment of force c. linear momentum c. angular momentum F = ma L = I A Forces act independently. Moments act independently. If F= 0, the linear momen- If L= 0, the angular mo- tuin remains constant. mentum remains constant. [f the direction of the force If the axis of the moment is perpendicular to that of the is perpendicular to that of the motion, th«- direction of the. motion, the direction of the latter is changed. latter is changed. Motion in General of a Material Body General Description. A material ln.d\ will, in general, receive both linear and angular acceleration when acted upon by external forces; but these are independent <>f each other. The i . utre of DIMS of the body will re.ri \ c a linear arc. 96 MECHANICS tion; and, as this point moves in space, the rotation will take place about it exactly as if it were a fixed point in the figure. Several illustrations have been given already ; viz., the motion of an acrobat, that of a chair thrown in the air, etc. If a rigid body is struck a blow at random, its centre of mass will move in the direction of the blow, and the body will rotate, in general ; but, if the line of action of the blow passes through the centre of mass, there is no rotation. Consequently, if two lines of action are found such that blows along them do not produce rotation, they must intersect at the centre of mass. Therefore, to discuss completely the most general problem in dynamics, all that is necessary is to know the laws of motion of translation and those of rotation about an axis passing through a fixed point. Resultant. — There are certain cases in which the resulting accelerations of a body under the action of several forces- might have been produced under the action of a single force; if such is the case, this force is called the " resultant " of the others. The action of a single force on a body is to accel- erate the centre of mass in its direction and to cause angular acceleration around an axis through the centre of mass at right angles to the plane including it and the line of action of the force. If the various forces acting in a body all lie in a plane, or if they are all parallel, it may be shown that they have a resultant, with the exception of one case, which will receive due attention. Non-parallel Forces. — Let the body be acted upon by two coplanar non-parallel forces Fl and F2. Their geomet- rical sum R may be found as usual ; and its effect in accel- erating the centre of mass equals the combined effects of Fl and Fz. But if R is to be the resultant, a position for its line of action must be found such that its moment shall equal the combined moments of F-^ and F^. It has been shown on It Y \AM1CS 97 '.':! that this is the case if the line of action of R passes through the intersection of the lines of action of .Fj and Fv Therefore the obvious geometrical method of determining the resultant of two coplanar non-parallel forces is to take a plane section through the body so as to include the lines of action of the two forces, prolong these lines until they meet at a point, 0; lay <>ff from 0 along the lines of the forces, distances OA and OB propor- tional to their numerical values; and complete the parallelogram. The di- agonal OC represents in direction, <»pi«nar forces /\ »n,i amount, and position the resultant /2; its point of application may be anywhere along the line OC, e.g. at Q. , ... ,, ,. „ Their resultant Is R. In a perfectly similar manner, by continuing the process, the resultant of any number of coplanar non-parallel forces may be determined. It M is the mass of the body; /, its moment of inertia about an axis through tin* centre of inertia perpendicular to the plane of the forces; and /. the perpendicular distance from the centre of inertia to the line of action of the re- sultant It \ the linear acceleration of the centre of mass is /' // — ; and the angular acceleration of the body is -— . Equilibrium. — It is evident that if there are three forces :•_!• MM the body, Fl% Fy and one e<pial and »>pp<> aloiiLT the same line of arti.m. they will neiitrali/e eaeh uthei- completely: there will Ix? no accelerat ion of .my kind. Such a state is called one of juilibriuin." The general law max then be stated that, if a body is in equilibrium under the n of three non-parallel forces, they must lie in a plane, their lin- [on must meet in a point, and any one must \MI \|\| i-in -i. H — 7 98 MECHANICS be equal and opposite to the geometrical sum of the other two, — or, what is the same thing, if the three are added geometrically, they will form a closed triangle. (It was by considering the special case of equilibrium of a body on an inclined plane that Stevin was led to the statement of the parallelogram of forces. See page 76.) Parallel Forces. — If the two forces acting on the body are parallel, their resultant, so far as translation is concerned, must be a force parallel to them, whose numerical value is their algebraic sum. That is, if Fl and F2 are their numerical values, that of the resultant is Fl + F^ if the forces are in the same direc- n tion; but, if they are opposite directions, and i is the greater, the resultant is in the direction of this force, and has the value F% — Fr Further, in order to satisfy the requirements in regard to rotation, this resultant must have such a position thajb its moment around any axis equals the algebraic sum of the moments of Fl and F2 around the same axis. Describe a plane section through the body, including the parallel lines of action of Fl and FY The line of action of the resultant must also lie in this plane ; other- wise the resultant would have a moment about any axis Fm. 48. — Rijrid body under the action of two parallel forces 1\ and Ft in the same direction. Their resultant is Ft + Ft. lying in it, which Fl and F2 do not. We shall consider first the case when the two forces are in the same direction. Imagine an axis perpendicular to their plane, and let its trace on the plane be 0. From 0 draw a line OA 0 perpendicular to the lines of the two forces ; if the DYNAMICS parallel force (^Fl -f P., ) is to be the resultant, it must be so placed that its moment a
round the axis through 0 equals the si 1111 of the moments of Fl and Fv Let its position be indi cated as shown in the cut, its intersection with the line OAC being at B. The condition that B must satisfy is that (F, + F2) OK = F}OA + FtOC, or FlAB = F3BC,i.e.j£ = %. This may also be expressed as follows: i.e. or These relations are independent of the position of 0, and therefore hold true for any axis. They determine uniquely the line of action of the resultant. If the forces are parallel but in opposite senses of direction, and if JPj is the greater, the resultant is F^— Fl and is in the direction of FV and it is so placed that its moment around any axis is equal to the difference in the moments of I'\ and Fr The same formula. as above apply, giving Fl in these a negative sign. As in the previous case, the acceleration of the centre of mass is in the direction of the resultant and has the value - ^ ; and the angular i celeration has the F 4- F Fto. tt.-Rlfftd body am). of two fttnUM fbTOM /', and value ' l T * , where I is the perpendicular distan the centre of mass to the line of Action of the resultant, 100 and / is the moment of inertia about an axis through the centre of mass and perpendicular to the plane of the forces. This process may be continued so as to determine the resul- tant of any number of parallel forces. F.-F, " Couples. " — An ambiguity arises when the two parallel forces are equal but in opposite directions, i.e. when JF1= — Fr In this case there is no resultant. (On substituting these values in the previous solution, it is seen that, if there were a resultant, its value would be zero, and its line of application would be at an infinite distance.) Such a combination of two equal but opposite parallel forces is called of a "couple." Fio. 50. — Rigid body under the action a "couple." Their sum is zero, therefore the linear acceleration of the centre of mass is zero, i.e. it has a con- stant linear velocity ; but there is an angular acceleration around an axis through the centre of mass perpendicular to the plane of the couple. Describe a plane section through the body so as to include the two parallel forces, and consider any axis perpendicular to this plane. Let its trace be 0; and from it drop a per- pendicular OAB upon the lines of action of the forces. The sum of their moments, taken contrary to the direction of rotation of the hands of a watch, is F1OB—F1OA = F1AB. This product is called the "strength of the couple," and is evidently independent of the situation of the axis. There- fore, while the centre of mass of the body retains a constant velocity, the angular acceleration around an axis through the centre of mass and perpendicular to the plane of the couple equals its "strength" divided by the moment of inertia of the body about this axis. I>Y \AMICS 101 Equilibrium. — If a body is under the action of three parallel forces that lie in a plane whose algebraic sum is zero, and the algebraic sum of whose moments around any axis is zero, there is neither linear nor angular acceleration: tin- bndv is in equilibrium. Conversely, if a body is in equilibrium under the action of three, or of any number o/, parallel forces, their algebraic sum is zero, and that of their moments around any axis is zero. A couple can evidently be balanced by another couple that is in the same plane, but that is not necessarily parallel to the first, of equal strength but of opposite sign. Centre of Gravity. — The most important illustration of parallel forces is furnished by gravity. A body, considered made up of particles, is acted upon by as many forces as there are particles; and these forces are parallel because their lines of action join the separate particles to the centre of the earth. Their resultant is their arithmetical sum, and is called the "weight of the body " ; its numerical value is the product of the total mass of the body by g, the acceleration of a falling body; and the position of its line of application mav he found as follows: Let m* and >//., l>e the masses of any the action of gravity. Their tun {.articles of the body at an in van- able distance apart, AS. The forces of gravity are vertical and their numerical values are mtf and wi^Mlirir resultant centre of gravity Is C. is Q^+rttg).?, so placed that (ml + m^gQR = (m^PR, wh6TC / ontd line, intersecting the lines of the three forces in the points P, R, Q. That is. (m^m^QR = mlP/f. This line of action «.f the resultant meets the line joining the particles in a point (7, such that, by similar triangles, AB : CB = PR : QR- 102 MECHANICS But PR : QR, as we have just seen, equals — J -- 2- There- OR ml m1 + ra2 fore, 41 = or G^S = _i_ . 45. It is seen, then, that the point on the line joining the two particles through which their resultant weight passes, depends upon the masses of the particles and upon their distance apart, but is inde- pendent of the position or direction with reference to the earth of the line joining them ; it is therefore a fixed point ori this line. Similarly, in the general case, there is a point fixed, with reference to the particles, through which the line of action of the resultant weight passes, however the body is situated with reference to the earth. This is called the " centre of gravity " of the body ; and it is seen from the above equation that it coincides with the " centre of mass." N.B. — The above proof of the existence of a centre of gravity and of its coincidence with the centre of mass depends upon the fact that " g " is a constant for all amounts and all kinds of matter. Equilibrium The state of equilibrium of a body has already been defined as that in which there is no acceleration, either linear or angular; and the obvious conditions are that both the sum of the components of the forces in any direction and the sum of the moments around any axis should be zero. (We may speak in the same way of the equilibrium of a system of bodies.) If the body is at rest with reference to any standard figure, — e.g. a book lying on a table is at rest with reference to the table, — the equilibrium is called "statical"; while, if the body is in motion, — unaccelerated, of course, — the equilibrium is called " kinetic," e.g. a sphere rolling on a smooth horizontal table. There are several kinds of equi- librium, however, depending upon what changes in the motion of the body (or system of bodies) take place when a slight impulse or blow is given it. 103 Stable. — If the equilibrium of the body is such that, as a result of the impulse, it does not continue to move away from its former position, but makes oscillations about it, it is said to be "stable." This is illustrated by practically all bodies in nature that are in equilibrium. An ordinary pendulum when at rest, a block when it rests on a table, a body hang- ing at rest from a spiral spring, etc., are illustrations of statical stable equilibrium. If an ellipsoidal body is set spinning around its longest or shortest axis, the motion is stable. Unstable. — But if the equilibrium is such that as a result of the impulse the body departs farther and farther from its former position, it is said to be "unstable." This is illus- trated by a Wrd balanced in a vertical position on one corner, by a conical body balanced on its point, etc., or by an ellipsoidal body spinning around its intermediate axis. It is obvious that when a body is in unstable equilibrium Fio. M. - G la the centre of gravity ; If the vertical line fall* inside the base £<?, there Is equilibrium. w for impulses in some din •< 'tions, it may be stable for others; and again a body may be stable for an extremely small impulse and unstable f..r a larger one, so that there are ••degrees of Stability." Thus a block shaped as shown in the rut and resting on a hori/.ontal support is in stable equi- librium, because the force of gravity acting vertically do\\n through the centre of Lrnmty is balanei-d b\ an upward force due to the table. P.iit this last fOTOC CM pO88 through the 104 MECHANICS centre of gravity and therefore neutralize gravity completely only so long as the line of action of the force of gravity falls inside the area of contact between the block and the table. If the shape of the block is so changed that this line of action approaches the edge of this area, the stability becomes less and less — for an impulse in the proper direction. When the line of action of the weight falls outside the edge, this down- ward force forms a couple with the upward force due to the table, and the block will turn over. Neutral. — Another state of equilibrium is also recognized; namely, that in which, when an impulse is given the body, the change in motion produced remains permanent. This kind of equilibrium is called " neutral," and is illustrated by a sphere or a cylinder lying on a smooth horizontal table, by a body pivoted on an axis passing through its centre of gravity, etc. It is evident that when the condition of a body in unstable equilibrium is disturbed, it passes over into either a stable or a neutral condition; and, as disturbances are always occur- ring in nature, the condition of unstable equilibrium can exist for only infinitesimal intervals of time. Principle of Stable Equilibrium. — Any disturbance of stability must produce a reaction which tends to restore the body or system to its previous condition ; and this principle can be applied to any stable condition, whether it is a purely mechanical one or not. Consider some illustrations of stability. (1) A body hanging suspended by a spiral spring is in stable equilibrium. If a blow downward is given it, the initial velocity will be decreased owing to the increased tension of the spring. Hence, if the tension of a stretched spiral spring is increased by any means, it will raise the suspended body. (2) An iron bar surrounded by some medium, e.g. water, at a constant temperature is in stable equilibrium ; for if its temperature is suddenly increased in
any way, the tendency will be for it to return to the tern- DYNAMICS 105 perature of the surrounding medium. Now, when the tem- perature of an iron bar is increased, its length is increased ; but this act of increasing in length produces a tendency for tin- bar to return to its former temperature. That is, if an iron bar is stretched by mechanical means, its temperature will fall. (•>) Just the opposite effect happens with a piece of rubber cord from which a weight is hanging. When its temperature is lowered, it elongates; consequently stretch- ing a rubber cord raises its temperature. .»{. Illustration of tlirvr kiml.s of «-<\|iillit>riiiiii. Work and Energy Measurement of the Effect of a Force. — In the previous sections of Dynamics we have; considered, generally spcak- •nly one property of a force, viz., the fact that it pro- duces a change in momentum. It was shown, however, on page 45 that when a particle is m«>\in<r in a straight line with a constant ROOeforaf i»u. this .juaiitity could be expressed in two ways : where x, is the speed ..\|' the particle at th-- instant 7\ when 106 MECHANICS it has reached a point at a distance x1 from a fixed point of reference in the line of motion ; and «2 is the speed at the instant T2 and position xv Multiplying each of these values - f_ of a by M, the mass of the moving particle, we have the force. Thus, X2 - X 1 •> J ~ The former is the ordinary expression for the value of a constant force, stating that it equals the change in the linear momentum in a unit of time. The product/ (T2 — ^i) ^s called the impulse of the force ; so this formula expresses the fact that the impulse of the force equals the change in linear momentum of the particle. If the force varies, we must understand by f (T2 — T^ the sum of a series of terms each of which is a force multiplied by its time of action. Definition of Work and Kinetic Energy. — The latter for- mula, however, is a new expression. Suitable names have been given its terms : \ ms2 is called the " kinetic energy " of the particle whose mass is m when its speed is s ; / (z2— x^) is called the " work done by the force " / in the distance x2 — xv provided the speed is increasing, or the " work done against the force " / if the speed is decreasing — thus this equation reads : either, " the work done by the force in the distance x2 — xl equals the increase of the kinetic energy of the moving particle in that space " ; or, " the work done against the force/ in the distance xz — xl equals the decrease of the kinetic energy of the moving particle in that space." Several things should be noticed : 1. The distance #2 — x1 is measured in the line of action of the force ; if the line of motion makes an angle N with the line of the force, the work is the product of F cos N and O2-i)- 2. Tho idea of work involves both force and motion in the direction of the force / no work is done unless there is motion ; 107 and this motion must be in tin- direction of the force. Tims, a pillar supporting a building does no work, neither does a horizontal table on which a ball rolls. 3. In the expression for the kinetic energy # is the speed, not the velocity ; in other words, kinetic energy does not depend upon the direction of motion. This is evident, be- cause to produce a change of direction (and no change of speed) the force must be at right angles to the direction of the motion, and therefore, by what has just been said, no work is done. 4. The same relation between work and change in kinetic energy holds true even if the force is not constant ; because we can in that case consider the force as constant for a short distance, during which the formula holds, thru assume another constant value for a short distance, etc. The total work done — that is, the sum of the amounts done by the separate forces — will then equal the total change in the kinetic energy. Illustrations. — Let us consider several cases of motion from the standpoint of both momentum and kinetic energy. When a ball is thrown, the momentum gained depends upon the impulse of the force, i.e. upon the time during which it acts; while the kinetic energy gained depends upon the work -lone by the force, i.e. upon the distance through which it aets. If a bullet is set in motion 1>\ a compressed spring, {is in a toy gun, the spring in relaxing axerta an impulse ami also does \\ork. producing momentum and kirn-tic energy. As a bullet fired from a rille enters a wooden target, the distance it penetrates depends upon its kinetic energy ; the interval of time required to bring it to rest depends upon its momentum. The destm. ti\e power of a moving l»»>'ly is due to its kinetic energy; its power . to its moment uin. Conservative Forces. — For the time being we shall limit ourselves to the consideration of what we m.i\ call purely 108 Ml < IIANICS mechanical forces, e.g. those produced by gravity, by elastic bodies which are deformed, such as a bent bow, compressed spring, etc. In all these cases we make the assumption, which is based upon the results of experiments, that the work done by a force during a displacement from one point to another depends upon the initial and final positions, not upon the path followed ; such forces are called " conservative " for reasons that will soon appear. This may be illustrated by the force of gravity. Describe two horizontal planes PQ and P1Q1 at a distance h apart. Draw an oblique line AB, making an angle N with the ver- tical line AO\ its length is Let a particle of mass cos^V m move under the action of gravity down BA ; at any point the force in a vertical direction is mg, and therefore the component along the line BA is mg cos N. The work done, then, between B and A is P, * < 1 mg cos N • — or man. cosN Similarly, if the particle in passing from B to A _ _A~ -Q had followed the path EQ FIG. 64. — PQ and I\Ql are two horizontal planes at an(J (JJ^ ^}ie WOrk done would nave been the a distance apart, h. same ; for, in the path BO the motion is perpendicular to the force of gravity, and therefore no work is done ; while in the vertical motion OA the work done is mgh. Since any line curved or broken may be considered made up of straight lines, it is seen that when a particle of mass m falls from any point in one horizontal plane to any point in a lower parallel plane at a vertical distance A, the work done by gravity is mgh, and is independent of the path. This is a consequence of the fact that the force of gravity is vertical and is con- stant in amount at all points near the surface of the earth at any one locality. I)Y \AMICS 109 Potential Energy. — Since, then, in all the cases that we are to consider at present the work done by forces depends upon only the initial and final points, we may write whore Fis a quantity whose value depends simply upon the point considered. Thus, our fundamental formula becomes or This means that during the motion the quantity F-f Jw2 remains constant. Consider an illustration: a ball being set in motion by a compressed spring. The above formula states that as the speed of the ball increases, the value of V decreases; or, vice venta, if the ball is made to strike the spring and compress it. as the speed of the ball decreases, tin- value of V increases. Again, in the case of a falling body, as the speed increases, the value of V decreases; and, if tin- ball is thrown upward, as its speed decreases, the value of V increases. The quantity whose value is V is called "potential energy"; and it is seen by the above illustrations that when a spring is compressed, the potential energy increases; when a particle is raised vertically upward, the potential energy increases ; and. conversely, when the spring relaxes or the particle falls, the potential ener-\ decreases. We say that the 4 com pressed spring has potential energy," and that the "system of the particle and the earth has potential energy"; or, in the latter case, more simply, "the particle ; IMS potential energy"; but these words are only a description of the expei i men ts just mentioned. In the case of the p article a -id t lie earth, the former has not changed its size, its shap IM, or any of its physical properties; it has therefore n,.t l.rrii clian'_:«'d nor has anything been added to or taken from it; but it> relation to the earth has been 110 MM'1IA\ICS altered. The same is true of the particles of a compressed spring; their relative positions are changed. In a similar way, a twisted wire, a bent bow, a clock spring that has been wound up, etc., all have potential energy ; and, in gen- eral, a body or a system of bodies has potential energy if the particles composing it are in such a condition that a force is required to maintain it. The formula F(x% — x^) = F"t — F^ gives a means of cal- culating only the change in the potential energy ; and so what is meant by " the potential energy for a given position or condition " is the work required to bring the system into that condition from some other one which is taken as the standard one. Thus, in dealing with gravity, it is customary to reckon from the surface of the earth ; and the potential energy of a particle of mass m at a vertical height h above the earth's surface is therefore mgh. In compressing a spring, the standard condition is that when the spring is entirely relaxed ; and since experiments show that the force which the spring exerts at any instant when compressed varies directly as the amount of compression, this force may be written ex, where x is the compression and c is a constant to indicate the proportionality ; but, as the spring is com- pressed more and more, the force varies, and therefore during the compression from 0 to #, the mean value of the force is £ ex (the average of 0 and ex — see page 32) ; and the potential energy of the compressed spring is the prod- uct of this mean value of the force by the distance, i.e. it is \cx*. Conservation of Energy. — If a particle has kinetic energy, or if a syst
em has potential energy, it is in a condition such that it can do work. A falling body can compress a spring or bend a board, thus overcoming a force ; or it may strike another body and change its speed, thus doing work also. Similarly, a bent bow may change the speed of an arrow or it may raise a body up from the earth. Two things should DYNAM1' 8 111 l)e noted : (1) There are two ways of doing work corre- sponding to the two types of forces referred to on page 68, namely, producing acceleration in a particle, in which case it gains kinetic energy ; or overcoming some opposing force, e.g. gravity, in which case the system on which the work is done gains potential energy. (2) If a particle or a system does work on another particle or a system, the latter gains energy and the former loses energy. The exact relation between this gain and loss is stated in the general formula V+ \ m& = constant, which is true only for so-called "con- servative" forces. (See page 108.) This says that, if a tern has both kinetic and potential energies, the sum of the two remains constant; if one decreases, the other increases by an equal amount. This is a special case of the principle of the "Conservation of Energy." (See page 115.) Tims, if one part of a system does work on the other, e.g. a com- pressed spring and a ball, a bent bow and arrow, one loses a certain amount of energy, the other gains it. Similarly, in the system made up of the earth and a falling body, the potential energy decreases by an amount equal to that by which the kinetic energy increases. If there are no changes in the potential energy of a system, the total kinetic energy does not change, the loss in one part equals the ^ain in another; an illustration is ^iven by the impact of two billiard balls. (All cases of impact between inelastic bodies are excluded t'r«un considerat i«>n here, because the forces acting durin^'the impact of such bodies do not satisfy our assumption made above in regard to mechanical forces. As will be shown shortly, part of the energy in the case of impact of inelastic bodies disappears from view and is mani- to our senses in the production of heat effects, such as rise of temperature, etc.) Unit of Work and Energy. — The act of transfer of energy t'n.m one particle or one system to another involves what we called -work." Its numerical \alue is. tVom iis deli- 112 MECHANICS nition, the product of the values of the force and the dis- placement in the direction of the force. Work, kinetic energy, and potential energy are, then, all similar quantities and are all measured in terms of the same unit. On the C. G. S. system, this is the work done by a force of one dyne in a displacement of one centimetre, or the energy of a particle whose mass is two grams moving with a speed of one centimetre per second ; it is called an " erg " ; but it is too small for practical purposes, and so 10,000,000, or 107, ergs is the unit in common use ; it is called a " joule," in honor of the great English physicist who did so much to teach correct ideas in regard to energy. Another unit often used is the "foot pound," or the work required to raise in a vertical direction a distance of one foot a body whose mass is one pound. This, then, equals approximately 1.356 joules, assuming g to equal 980. Motion in a Vertical Circle. — One case of transfer of energy deserves special notice ; it is that of a particle sus- pended by a massless cord and making vibrations in a vertical circle under the action of gravity. As it swings through its lowest point, it has its greatest kinetic energy and its least potential ; and, as it gradually rises, the former decreases and the latter increases, until at the end of its swing it has its energy entirely in the potential form. If the arc of vibration is extremely small, this particle is a simple circle under the action pendulum ; but even when the arc is large, we can deduce certain general laws. If the particle moves freely from Pl downward along the circle of radius r, starting from rest, its speed at the bottom A is such that s* = 2 (j QtA, where Ql is the projection of P1 on the vertical diameter. DYNAMICS 113 But, by geometry, we know that \vlit-re Pj-A is the chord, and r is the radius. Hence, s = J\ j y i 2 If tlie particle had moved from P2 down the circle, its 2 r ' speed at the bottom would have been So the speed at the bottom of the path is directly propor- tional to the length of the chord of the arc through whieh it falls. Work and Energy in Motion of Rotation. — When work is done in producing angular acceleration of a rigid body about a fixed axis, somewhat different expressions for the work and the kinetic energy may be deduced, whieh are more useful. Let a plane section be taken through the body, perpendicular to the axis, and pass- ing through the point of application of the force, and let the force be resolved into one parallel to the axis and one parallel to this plane. In the cut let P be the trace of the pivot; F, the com- ponent of the force, whose point of appli- eation is O and whose line of action is OA ; and /M, the lever arm of length /: and call the length of the line PO, r. In the motion neither the component of the force parallel to the udi nor the force exerted by the pivot on of • rljri.l body pei-pen- .llr.ilar to ftxU at /»;/•!• the component of the force in thl. « plane. '•«. — Plane section the body does any work. ( \\'e may suppose that gravity does not act ; or, the axis may be considered as being ver- tical. ) The force /•' may be resolved into two components, nmiot — 8 114 MECHANICS urn- along OP which therefore does no work, the other per- pendicular to OP. Calling the angle (OP A) N, this last component has the value .Fcos.ZV; and, as the body turns, this component remaining always perpendicular to the line OP, the work done equals the product of .FcosJVand the length of the arc described by the point 0. This arc equals the product of the angle turned through and the length of the radius PO. Calling this angle M, the work is then rMFcos N; but r cos N = 7, the lever arm ; and Fl = L, the moment of the force F around the axis through P. Conse- quently the work equals the product of the moment of the force by the angular displacement. The kinetic energy of the rotating rigid body, which we may consider made up of particles, is the sum of the. energy of these taken separately. A particle of mass m1 at the distance rl from the axis has the linear -speed rji, if h is the angular speed of the body, and therefore the kinetic energy 2 mi(ri^)2 or J m^ffi. The sum of the energy of all' the particles is then \(m^r^ + ra2r22 -+- --.)A2 or J/^2, where I is the moment of inertia about the given axis. (Notice the exact correspondence of the values for work and energy in translation and rotation.) Properties of Potential Energy. — The connection between potential energy and force deserves consideration. The formula of definition is F(x2 — x^) = }\ — Vv This is equiva- lent to saying that, if by a displacement from a point 1 to a point 2, the potential energy decreases, i.e. if Vl>Vy F is positive, and is, therefore, in the direction from point 1 to point 2 ; so there is a force acting in the direction of the displacement, whose value is — -. This is illustrated by 22 - X\ the force of gravity, that of a compressed spring, etc. Another interpretation is this : in a system left to itself under the action of its own forces, motions take place, if at all, in such a manner as to produce a decrease in the poten- . " . 1 1 tial energy. If a system is in equilibrium, the total force in any direction is zero, and therefore any slight displacement may be produced without there being any work done ; hence the potential energy remains constant during these displace- ments. This shows, then, that at a position of equilibrium, the potential energy has either a maximum or a minimum value. If the equilibrium is stable, it is not difficult to prove that the value of the potential energy is a minimum; as is illustrated by a pendulum being in stable equilibrium when it is hanging at rest at the bottom of its path. Power. — The rate at which work is done, that is — if this rate is uniform — the work done in a unit time, is called the "activity" or "power." On the C. G. S. system the unit is " one' erg per second " ; but for practical purposes " one joule per second " is taken : this is called a M watt." in honor of the great Scottish engineer who made so many impn>\r- mcnts in the steam engine. Since work equals the product "f force by displacement or of moment of force by angular displacement . power equals the product of force by linear speed or of moment by angular speed. POW.M- is also often measured in terms of a unit called "one horse power," which is defined to he :J:},000 foot pounds per minute. This equals 746 watts approximately. Other Forms of Energy Conservation of Energy. — There are many other forms forces than those which have been considered. S hese correspond to forms of potential energy, such as th< surface tension of liquids, forces due to electric charges ;o magnets, etc. ; others, h«»we\rr. d<> not Am.»n^ tin- latter the force of frietion is the most important. I force .anifest whenever twn pieces of matter in contact with each other move relatively: and in all the cases of motion i-sed the condit «'SSed as to assume the eomph-t.- lion. It is a force 116 MECHANICS that always opposes the motion : and its numerical proper- ties will be discussed later. Let us consider several cases of friction and the immediate results. If two blocks of ice are rubbed together, some of the ice is melted ; if the bearing between a wheel and its axle is not well lubricated so as to avoid friction, there is a "hot box," the bearings become hot and the parts expand; if a paddle is stirred rapidly in water, thus producing friction between different currents of the water, the temperature of the water rises and it will fi
nally boil ; if an inelastic body, like a piece of lead or putty, is deformed, different layers move over one another, there is friction, and the temperature rises. These various changes — melting, boiling, rise in temperature, and expan- sion— are called "heat effects," and will be discussed more fully in the next section of this book ; but what is of funda- mental importance here is to recognize that these effects are all produced when work is done against friction. It will be shown later how we can measure these effects numerically; and experiments show that their amount is proportional to the work thus done against friction. In doing this work, energy is lost by the body or system doing the work ; and so it is natural to assume that the heat effects are manifestations of the addition of energy to those parts of the bodies which are directly affected by the friction, namely, the most minute portions — in certain cases, the molecules. This assumption is completely supported by all experiments and observations. We have seen that in the transfer of energy in purely mechanical rases. Him- is no loss — \\-lui um- liodv loses, an- other gains; so \ve extend this idoa t«> ;ill processes in nature, and state our belief that in no case is there any loss in energy. It may be present as energy of bodies of sensible size, of mole- cules or their parts, or of the ether. This statement is called the "Principle of the Conservation of Energy." Nature of Potential Energy. — A few words more should be said in regard to what is meant by " potential energy." As UYNAMICS 117 have used the expression, it describes a condition of a body with reference to other bodies or of the parts of a b»»dy witli ivtVivnce to each other, which is primarily concerned with the idea of force and its production. We cannot ex- plain it in terms of such simple quantities as intervals of •e or time and mass. We understand, however, its ti formation into kinetic energy; and it is possible that it is the manifestation to our senses of the existence of the kinetic energy of portions of a medium which has inertia and which is intimately connected with ordinary matter, but which does not appeal directly to us. Friction External and Internal Friction. — As has been said in dis- cussing different methods of doing work, there is a force that opposes the relative motions of any two pieces of matter that are in contact; this is called the force of "friction." liscussion does not properly belong to mechanics; but it is convenient to give it here. Distinction must be made between two kinds of friction, internal and external. The latter is illustrated when solid body is made to move in contact with another, or when layer of a fluid flows past another; e.g. a block of wood moving <>ver a tal»le, currents of water produced in a vessel by stirring a paddle in it, currents in the air produced by blowing. In all these cases the relati\. motion is soon stopped unless some force maintains it. Friction between moving layers of fluids is said to be due to " viscosity.' Internal friction is illustrated when a solid body is deformed in an\ \\.iv, for in every case, to a greater or less extent, portions of tl body move over each other. The only case < tiou \\hieh will be considered now is that of one solid moving over another; the discussion of viscosity and of internal friction is defer 118 MECHANICS Sliding Friction. — The most important cases of friction between solid bodies are those when the two surfaces in con- tact are plane and when one body rolls on the plane surface of another. It will be seen that the explanation of the fric- tion in these two cases is quite different. Consider the motion of a rectangular block over a plane. Let AB be the section of the plane by the paper and CDEF be that of the block. Let a force whose value is F produce acceleration of the block parallel to the plane surface ; there is a force of friction opposing this, call its value F^\ then the total force producing the acceleration is (F — F^), and if m TTf TTf is the mass of the moving block, its acceleration is ^. m If there is no acceleration, and therefore the speed remains constant, the applied force exactly balances the friction, F = Fl ; and we have thus an experimental method for determining the force of fric- Fio. 57. — Motion of a block along a rough tion between two given mate- plane. F is the force producing the motion. . , rials, over a definite area of contact, when a definite force presses the two solids together and when the speed has a definite value. Experiments show that the friction is independent of the relative speed of the bodies, if this is small, and of the area of contact, provided the force pressing the two bodies to- gether remains constant ; that it varies directly as this force, and that it is different for different materials and varies with their condition. If P is the force pressing the bodies together and F the force of friction, the above statements may be expressed, F = <?P, where c is a constant for two definite bodies in a definite condition. It is known as the " coefficient of friction." A simple method for the determination of c is as follows : Let a block be placed upon an inclined plane which is gradu- 119 ully made more and more steep until when the block is given .^velocity by a push, it maintains its motion unchanged. AB represent the inclined plane and M the block mov- ing over it. If m is the mass of the block, and N the angle of inclination of the plane, the force witli which the block is pivsst •(! against the plane P is mg cos N, and the force paral- FIO. ss.-Moaon of a block down » lei to the plane due to gravity inclined pi»ne. and friction is >/<// sin N — F. But F = cP = c mg cos N\ and therefore, if the plane is so tipped that there is no acceleration, as above described, the force parallel to the plane must be zero ; that is, N must be such that mg sin N — c mg cos N = 0. Hence c = tan N; and so can be measured. (If the body is at rest on the plane, it must be tipped farther than in the experiment just described before motion will he^in. The coefficient, then, for what may be called "statical " friction is greater than for that which may be called "kinetic.") This coefficient of friction varies greatly with the condition of the surf ace > <>t the body, with the liihrie.ii ion. etc. Rolling Friction. When one solid rolls on another, as for instance a cylinder on a plane, there i> n.. friction, properly speaking, because in a rolling motion there is no sliding of one surface over another; but yet experiments show that a necessary in order to maintain the e\lmder mlliiiL; at a constant speed over a hon/ontal plane. I ;>dueto the f.iei that the latter .surface does not in general remain hori/onlal. bttl i- deformed slightly by the : :ider in such a manner that there is a hump in front of the latter which it staru to .iseeml before it moves on. This so-cnl led line; friction1 i- then due to the fact that a minute f 120 MECHANICS is required to keep the rolling body from sliding back along the hump. In some cases the plane remains horizontal, but FIG. 59. — As a body rolls on a plane surface, both are deformed slightly. the surface of the cylinder is flattened where it is in contact with the plane. The explanation of the "friction" is evi- dent. Rolling friction is always much less than sliding. Dynamometers. — Friction occurs in all actual mechanisms where there are moving parts, but is always diminished in practice as much as possible by all known means. There is one form of instrument, however, that depends upon its presence, the "friction dynamometer," which is devised to measure the power furnished by a revolving shaft. One simple form of this instrument is represented in the cut. The revolving shaft whose cross section is shown at A is clamped between two halves of a block by means of bolts ; and this block is kept from turning by means of a weight attached to a lever as shown. If the adjust- ment is exact, so that the block does not move in either direction, the mo- ment on it due to the FIG. 60. — A Prony Brake. The turning moment on the block, due to the friction as the shaft A re- volves, is balanced by the weight of the hanging weight must exactly bal- ance that due to the fric- tion between it and the revolving shaft. Call this latter moment L, the hanging weight mg, and its lever arm I ; then, when there is a balance, L = mgl. But as the body TO. DYNAMICS shaft revolves it does work against this fractional moment; and, if h is the angular speed of the shaft, the power it fur- nishes is Lh or mylh. Therefore by applying a brake of this kind and balancing it, the power may be measured. There arc many modifications of this instrument, the most important one being the substitution of liquid friction for solid. A set of blades is fastened to the shaft, which is then inclosed in a cylinder containing water and so supported that it may be kept from turning by suitable levers. As the shaft revolves, currents are produced in the water and a moment is required to keep the outer cylinder from turning ; if L is the amount of this moment and h the angular speed of the shaft, the power it furnishes is Lh. (The energy goes into heat ellV< Machines General Principles. — A mechanism rnn>istin^ of rigid or iiiextensihle parts by means of which energy is transferred from one point to another is called a maehine. A force applied at one point overcomes another force applied at a different point. Thus, the lever funning a pump handle is a maehine, hecausc when work is dune by the man pumping at one end, the lever 'does work in raising the water at tin- other end. A pulley over which a cord passes is a machine, because if one end of the eord is attached to a body and the other is held by a man, the latin- l.\ d-.in^ work pulling on tin- c.»rd may raise the body off the ground. In all machines there are parts which move over each other and therefor.-
produce friction. ( '..nsc.jucnt ly, a 'nine never delivers as much energy as it receives; i ic hitter is spent in tion u»d is therefore : " in heat effects. A -a in. if the foffM produces ieceleration of the working parts of the machine, part of the work done is thus spent in producing kinetic energy not in overcomiii ie ; while, if the velocity of the n parta of the machine is decreasing, they t! •* do 122 MECHANICS work in helping to overcome the opposing force. Therefore, in the discussion that follows, the effect of friction will be neg- lected; and we shall assume that there is no acceleration. Con- sequently, the energy furnished the machine equals the work it does. Although this is the case, the force which the machine exerts need not be the same in amount or in direc- tion as that which is exerted on it ; for the distances througli which these forces are displaced need not be the same. In fact, the distinct object of a machine is to obtain in return for a given force a larger one in a suitable direction. The problem, then, is to determine the connection between the force that is doing work on the machine and the force that is being overcome by the machine. The ratio of the latter force to the former is called the mechanical advantage of the machine. There are two general methods for the dis- cussion of this problem. One is to consider the energy rela- tions ; that is, to express by equations the fact that the work done by one force equals that done against the other. The other method is to express the fact that the two forces are keeping the machine in a condition of equilibrium, because we have assumed that there is no acceleration. We shall discuss a few simple machines and deduce the ratio of the two forces. The Lever. — In its general form the lever consists of any rigid body capable of rotation around a fixed axis ; but most levers in actual use consist of straight rods. If a force p Jj is applied to this body at any point, FIO. 6i. - Equilibrium of a it will have a moment around the rigid body which is pivoted at ^^ ^ t p j where j js the lever P, when under the action of two 1 forces^ and F+ Principle of arm ; and, if there is motion, the work done by the force is the product of this moment by the angular displacement. If this moment does work by overcoming another force F% acting on the DYNAMICS body, whose lever arm is lv the work done is FJ2 multiplied by the angular displacement. But since the body is rigid, the angular displacements of all points are the same; and, therefore, since the force due to the reaction of the pivot does no work, the work done by Fl equals that done against Fv i.e. or I ' This relation also expresses the fact that the body is in equilibrium under the action of two forces whose values are I-\ and /-'2, and which are properly directed. Illustrations of the lever are given by a pump handle, a crowbar, a pair of scissors, a pair of tongs, nutcrackers, etc. The formula for a lever was given first by Archimedes in the simple case of a straight bar acted on by two weights. The more general 18 lirst solved by Leonardo da Vinci. The Pulley. — A pulley consists of a circular wheel which has a groove in its edge to hold a cord, and which turns on an axle supported by a framework called the H)lo< k." A pulley is used in two ways: in one its block is fastened to a firm support, and in the other it is kept from falling by being ried in the bi^ht of a cord passing round the whet-1. The former arrangement is called a "fixed pulley"; the latter, a "free" one. In the former case, if a force Fl is applied to one end of the cord and does the work Fio. 09. — A siin- pie form of pulley. ! by producing the displace- ment xv it can overcome a force F% at the other end whose displace- ment is xy provided F^ = / FM.68.-Aflz0dpullry. But if the cord is inextensible, This is the same relation that follows if the pulley is in and IK-IK.- Fl=F 1:24 MECHANICS equilibrium under the two forces Fl and Fv Thus a pulley simply changes the direction of the force. As an illustration of a free pulley, let the arrangement be as shown in the cut : the cord, one end of which is fastened to a fixed support, passes under the pulley, and its two branches make an angle N with the vertical ; to the free end of the cord a force Fl is applied which balances a force F2 applied to the block vertically downward. Since a pulley sim- ply changes the direction of a force, there are two upward forces acting on the block owing to the two branches of the cord, which are equal in amount but inclined at equal angles N to the vertical on opposite sides. If the end of the cord is displaced in the direction of the force a distance x^ the work done is x^F^ ; but the vertical displacement of the block owing to this motion is x2, where x1 = 2 x% cos N. The work done against FIG. 64. — A free pulley. the force F% is F^ ; and therefore Flxl = or 2 F1 cos N= F2. The proof of this relation between xl and x2 is as follows: Let AOB be the original position of the cord ; and A CD its final position after the displacement ; so that A D = A O -f OB. Lay off AC equal to OB, thusjnaking CD = AO\ and drop a perpendicular RE upon AD. Then the vertical displacement of 0, i.e.. r2, is A O cos N\ and the displacement of B parallel to the line OB, i.e. xr is ED. But it is evident from the figure that CD = A O = CB ; and hence ED = CD + CB COS (BCE) = A 0 (1 + COS 2 N) = 2 A O COS2 N. FIG. 64a. — Geometrical treatment of the motion of a free pulley. B Is the end of the cord before displace- ment ; Z>, its position after. II. Mice, [Mica x = 40 cos N. a ; , = L ' This is, again, the condition that the block should be in equilibrium under the three forces. If, as is usually the case, the two branches of the cord are 0, cos-ZV=l; and 2 Fl = Fv i.e. themselves vertical, Pulleys are combined in many ways; but all forms can be easily explained on the above principles. Thus let there be a free pulley and a fixed pulley with two wheels turning inde- pendently on the same axle ; let a cord, one end of which is fastened to the block of the free pulley, be run over them as shown; and let a vertical force act downward on the free pul- ley. Let the pulleys be so mall I'Minpared with their ;>arl that t In- «. — Combination of pullovn: appcr flx«d piilli-v li»« • -; \|..«rr \|>ull various hranrhi's of the cord between the pull dlel. /• . applied in the fn-e end of the »-«.rd and pro- duces a display-men! /-p the free pnll.-v \\ill rise a vertical distance xtt=}-±\ and then ical force F~ can be overcome, wlm-r value is !-\ 8 /' • . . • /' - I, 126 MECHANICS This is evidently the condition that the free pulley should be in equilibrium under the action of Fl and Fv The formulte for pulleys were first deduced by Stevin about 1600. The Inclined Plane. — If a body of mass m1 is moved upward along an inclined plane whose angle of inclination is JV, the force m^g sin^V must be overcome. So, if a body of mass m1 lying on an inclined plane is joined by an inextensible cord, parallel to the plane • and passing over a pul- ley, to a body of mass 7W2 which hangs free, the weight of the latter is just sufficient to balance that of the former provided that m^g = m^g sin JV, or w2 = m^ sin N. FIG. 66. — Principle of the inclined plane. If the first body is displaced up the plane a distance xv the second will sink an equal distance xv Hence the two amounts of work, m^g x sin N and m2gxv are equal. That is, ml sin N= mv as before. The theory of the inclined plane was also given by Stevin. The Screw. — If a piece of paper the shape of a right-angle triangle (ABD) is cut out and wrapped around a circular cylinder, the edge AD being kept perpen- dicular to the axis of the cylinder, the hypot- enuse AB will trace a spiral line on the surface of the cylinder. If a groove is cut fol- i . , i . -i. FIG. 67. — Principle of the screw. lowing this line, we — -2-TT-R- -> have what is called a cylindrical "screw." The portions of the cylinder between the grooves are called the "threads" of the screw ; and the distance from the edge of one thread DY\ 1-27 to tin- corresponding edge i.f the next measured parallel to the axis of tin- cylinder is called the "pitch" of the screw. If a distance equal to the circumference of the cylinder is marked off on the line AD, beginning at ^4, and a perpen- dicular line erected, the length included between the base and the hypotenuse is evidently equal to the pitch of the screw. If R is the radius of the cylinder, tin- length of the circumference is 2irR\ and the vertical distance on the triangular piece of paper which corresponds to this horizontal distance is given by the proportion, vertical distance : 2 irR = BD : AD; or, the vertical distance equals ZirR tan N. This, then, is the pitch. I f a spiral groove Is made on the inner surface of a hollow cylinder, we have what is called a "nut"; and by cutting the grooves of a screw and a nut at the same pit eh an suitable depths, the screw will lit inside the nut. Then, if the nut is held ; -crew can advance through it by a rotation on its axis: or, if the screw is lix.-d. the nut can only advance along it by a rotation round its axis. In either case, for one complete rotation, th<- ad\anee equals the pitch of the screw. If a moment L is applied to the screw . : lifting JM*. turning in a !i\.-d nut, and if by means ! -w .n,i nut "I (his a I'M overcome, which is so applied as to oppose the advance of the screw, the \\ d'.ne by the moment in on.- n.tati.m is the product of the moment by the an-h- L'TT. /.- . L'TT/.. and that done againgt the fon «•/•', / TrR tan N. If the im-mrm A is due to a 7«Tj applied to the circumference «>f th« n such a din «ti.. a as to have the lever ann ///, onsequently r« tan N, 128 or so that Fl = F., tan F, tan Screws are used in letter presses, cotton presses, lifting jacks, etc. It is thus seen that by these various machines a force may be magnified as much as desired, with a cor
responding de- crease in the displacement ; that the direction of a force may be changed ; and that a moment may produce a force. Chemical Balance. — This is not a machine in the ordinary use of this word, but is an instrument involving the principle of the lever, which is used to determine when the weights of two bodies are equal. It consists essentially of a hori- zontal beam, carrying at its ends, by means of knife edges, two " pans," arid sup- ported at its middle point by a knife edge which rests on a fixed support. The bodies whose weights are to be compared are placed one in each pan ; and by adding " weights " from a set, the balance may be brought to a state of equilibrium. The arrangement of the parts of the apparatus is such that the centre of gravity comes below the knife edge. For details of the instrument, reference may be made to Ames and Bliss, Manual of Experiments in Physics^ page 151. Fm. f.O. — Chemical balance. CHAPTER III GRAVITATION Law of Universal Gravitation. -- The property of matter that is railed inertia and forms the basis of dynamics was not recognized until Sir Isaac Newton stated the laws of motion in his great treatise, Philosophies Naturalis Principia Mathematica^ which was published in lb'87; but the prop- erty of weight was familiar to every philosopher. The laws tiling bodies were first stated in 1638 by Galileo, who iiowed that all bodies fall with the same acceleration neglecting the effect of the air. Previous to thK in 1609 and Ml, Kepler had announced the laws that bear his name concerning the motion of the planets around the sun: and in searching for an explanation of these laws and of the in<>ti<»n <>f the moon around the earth. Newton was led to a nerali/.ation concerning matter. He thought that inasmuch as a body falls to the earth, and as there was no • n why such a phenomenon should be limited to the earth, then- might be forces acting between all portions of matter in tin- universe, and that the moon is held in its orbit and the planets in theirs by forces of the nine nature as that which draws an apple as it falls toward the earth. Other philosophers had had this idea 1 •« fore, notably Hooke ; but no .me had expressed it clearly until Newton did M ii. ncipia. The concept ion occurred to him as early as 1666; but for various reasons he did not make it known until 1 He found that the observed facts would l>e explained if he assumed that this force obeyed the foil. \v : Between two particles of matter whose masses are m, and m,and which I-IM-ICH — 9 I-"-' 130 MECHANICS are at a distance r apart, there is a force of attraction propor- tional to 2. This can be expressed by writing /= Gr *2, where Gr is a constant of proportionality and is independent of the material of the particles or their distance apart. This law is in iiccord with all known observations and experiments. Illustrations. — We shall consider a few special cases; and, in discussing them shall make use of the fact first proved by Newton that, if the above law is true, a solid spherical body acts as if all the matter were concentrated into a particle at its centre, provided the body is homogeneous or can be re- garded as made up of spherical shells each of which is homo- geneous. The sun and planets and the various satellites may for present purposes be considered as satisfying these conditions. 1. Fatting bodies. — In the case of a body falling toward the earth, we may let ml be the mass of the falling particle, m2 be the mass of the earth, and r its radius. Then the force between them, acting on each, is as above, /= -- ^— -. Con- sidering the motion of the falling particle, we may write /= m^, where g takes the place of ~-^t and is therefore a constant. This is the ordinary formula for the weight of a body whose mass is m^. If Gr is independent of the kind of matter in the particle and its amount, g should be also, and to test this Newton performed some experiments with pendu- lums ; for, as has been shown in the discussion of the simple pendulum, the period of vibration is 2w'y-, where Z is the length of the pendulum. So, if g is different for different kinds of matter, it would be apparent if the periods of pendu- lums of different materials were determined. This question was investigated by Newton and later by Bessel; and all experiments agreed in showing that g is independent of the y GRAVITATION l-'.l kind of matter. (This fact is also shown by the experiments of Galileo on the two cannon halls falling from the Leaning To\\<-r ai Pisa; for, as they fell in the same time, they had tin- same acceleration.) Variations in "$r."- — Since the earth is not a sphere, but is slightly flattened at the poles, the distance from its centre to the surface decreases as one proceeds from the equator to cither pole, and for this reason g increases. Again, owing to the fact that the earth is spinning rapidly on its axis, a cer- tain force is required to make any particle on its surface move in its circular path; consequently a portion of the force of gravitation is spent in accomplishing this, and the differ- ence between the two produces the acceleration of the falling body. Since the radius of the circle of motion of a particle is greatest at the equator, and therefore the centrifugal force . the weight of a body is least there, or g is least. Therefore, owing to both these causes, .; at diflVn-nt latitudes on the earth, increasing as the latitude is increased. Several formulae have been advanced to connect these two variable quantities: the most satisfactory of which gives as the value of g at any latitude I, g = 978(1 -I- 0.005310 sin8/)- The fact that a pendulum of a constant length had dif- ferent periods ut different points on the earth's surface was recognized as early us 167 Land it was explained 1>\ Iluygens as due to differences in the centrifugal force at these points. -, as we now know, only part «>f the cause.) Of course^ varies also, o\\ in^ t«» 1 ises, such as the rness of a mountain, '^reat inequalities in the constitution of the surface of the earth, etc.; hut these variations are as a rule most minute. Methods of measuring g exactly will be discussed pre^-ntlv. 2. The motion of the moon. — The moon moves around the earth in if ne..rl\ eireul.ir \\ith a p. !j.pni\im.ii.-ly -11 days 8 hr.: and so there must IKS an acceleration toward the centre of its orbit, i.e. the centre of 132 the earth, equal to — , where r is the radius of this orbit and 2 « is the linear speed of the moon; or, substituting for s its 4 rt J_ value in terms of the period T, s = — ?f, this acceleration is . So, since r and T are both known, it can be calculated. If this acceleration toward the earth is due to gravitation and if Newton's law is true, it can also be calculated in terms of the acceleration of a falling body at the surface of the earth, i.e. g. For, using again the general formula, in which ml is the mass of a particle, mz that of the earth, and r the dis- tance from the centre of the earth to the particle, F= m^ • and therefore the acceleration of the particle toward the r earth is —^ . Consequently, calling a the acceleration of the moon, rl the radius of the moon's orbit, and r2 the radius of ri rz the earth, a : g = — ^ : — ~ ; and thus, since #, rv and r2 are known, a can be calculated. Newton showed that the two values, one based upon direct observation, the other upon his law of gravitation, agreed. 3. The motions of the planets. — As a result of a laborious study of numerous observations on the motions of the planets around the sun, and after many futile trials, Kepler suc- ceeded in discovering three laws in regard to these motions, with which all observations are approximately in accord. These are : (1) The areas swept over by the straight line joining a planet to the sun are directly proportional to the time ; i.e. equal areas are described in equal intervals of time. (2) The orbit of a planet is an ellipse, having the sun at one of its foci. (3) The squares of the periods of different planets are proportional to the cubes of the major axes of their orbits. VITAT10N 133 Newton showed that these laws, and many slight variations from tin-in, were direct consequences of his law of gravita- tion. The first law follows because the force of gravitation, acting on a planet, is always directly toward a fixed point, vi/.., the centre of the sun, which in the statement of Kepler's laws is supposed not to move. (Forces like this which are directed toward a fixed point are called -central forces.") The second and third laws follow because gravitation is a central force which varies inversely as the square of the distances between the bodies. There are of course many irregularities in the motions of the moon and of the planets because of the action of other portions of matter than the earth and the sun, of variations in their distances apart, of the departure of the earth from a spherical form, etc. ; but all these irregularities can be fully explained as consequences of this law of gravitation. I is the science of •• <;ra\ national Astronomy." 4. The "Cavendish experiment." — Various experiments have been performed since the days of Newton to see whether the force of gravitation between bodies of ordinary size could be measured. The first of these was carried out by Caven- dish in 1797-8. His method was to place two bodies on the ends of a light rod which was suspended horizontally by a line vertical wire attached to its middle point, then to bring up near these suspended bodies two others so placed as by their force of gravitation to turn th« I thus twist the supporting win-. He observed an effect, and measured the >• exerted. This experiment lias been repeated often and in raiioUfl forms. (In one it was shown that the force d with the masses of the bodies and inversely as the square of the distance.) II .iv ing thus measured the force between two bodies of known mass at a known distance apart, and assumin
g New- s law to be true, one can at once calculate the value of a in the formula. It is 0.000000066570, or 6.6576 x 10-», 134 MECHANICS on the C. G. S. system. This leads to a value for the mass and the average density of the earth. (By the value of the " density " of a homogeneous body is meant the ratio of the value of the mass of a certain portion of it to the value of the volume of this portion. Thus if D is the value of the density, and m and v are those of the mass and volume of any portion, D = — or m = Dv.) We have seen that in accord- ^ r* ance with Newton's law g = ——i where m is the mass and r is the radius of the earth. Thus m=^-'] and all the quan- tities in the second term are known. Further, if the earth can be considered as a sphere, its volume is \| Trr3 ; and there- fore in terms of the average density m — \| TrrD, and accord- 6r ingly D = ^— and can be calculated. Its value is 5.5270 4 TrCrr on the C. G. S. system. As will be shown later, the value of the density of water at ordinary temperature does not differ far from 1 on this same system ; and so the density of the earth is about 5J times as great as that of water. The student should consult Mackenzie, The Laws of Gravi- tation, Scientific Memoir Series, New York, 1900. Centre of Gravity. — A few more things should be said in regard to gravitation as we observe it here on the surface of the earth. It is a force directed toward the centre of the earth approximately ; and therefore the forces acting upon the particles of a body are parallel to each other. Their re- sultant is called the weight of the body, and we have proved that there is a fixed point connected with the body through which this resultant always passes, however the body is turned. This point is called the "centre of gravity." If a rigid body is pivoted so as to be free to turn around a horizontal axis, but is at rest with reference to the earth, a vertical line through the centre of gravity must intersect the axis ; otherwise the weight would have a moment about <;i:.i\-ir.\Tloy LS6 it, and the body would turn. The equilibrium is evidently stable if the axis is above the centre of gravity ; unstable, if it is below ; and neutral, if it passes through this point. The fact that the centre of gravity lies vertically below the of suspension when the equilibrium is stable furnishes a method for its experimental determination. If the body is suspended at a point, the centre of gravity must lie in vertical line through it ; and so, if the body is suspended in turn from two points, the centre of gravity must be the intersection of the two corresponding vertical lines. Compound Pendulum. — If a body is suspended free to turn about a horizontal axis, it is called a com- pound pendulum; and if it is set vibrating through an infinitesimal amplitude, it will have harmonic motion. Let the cut be a section through the centre of gravity of the body (7, and perpendicular to tin- axis at P. Call the length of the line and the angle it makes at any instant with a vertical line through N; this is then the angular displace- ment. The force mg acting vertically down through Q- has a moment about the axis equal to ?////// sin X. which is in the opposite direction to the disp in. -nt ; tli' :ie angular acceleration is pendulum. PUoeMetkm ofarlffklbodyplTOtodby • flx«dbortxoatalav 70.- to Uw axU r. ud toclnd Inff tbo ocotra of frmrlty. n N. wl, / the moment of in. : ihe body O. ;il... nt the a\ix. If th.- amplitii'i ill. sin \ . .in he re- plar.-d b) .V. the angle itself; and th.- acceleration has the Y i ,iis.'qii.-nlly the motion is hannome; and the peri !n-±-. The period maj be observed, and in pendnlui! ipes, m, A, and / may all be i: therei nay be determined. Calling th« p« 136 MECHANICS T, g = 4 7T2 . A simple pendulum is a special case of a compound one, in which there is only a particle vibrating ; m h l so / = mh2, and the period becomes 2 ir\- . A simple pen- dulum cannot actually be constructed ; but one can be imagined of such a length that its period equals that of any compound pendulum. 9 By suspending the compound pendulum described above so as to vibrate in turn about different axes, all parallel to the original axis, one may be found on the opposite side of the centre of gravity, so placed that this point lies on a line perpendicular to the two axes and such that the period of vibration about it is the same as that about the original axis. This fact was deduced by Huygens as early as 1673. The distance between the two axes may be shown to be equal to the length of a simple pendulum having the same period as that of the compound one. Therefore, although it is impos- sible to construct a simple pendulum, a compound one may be made provided with two axes of vibration and so ad- justed that the periods of vibration about them are the same. If T is this period, and I is the distance apart of the axes, T= 2 ir^\|-, or g = -j^-. Galileo, in 1583, first called attention to the apparent isoch- ronism of a pendulum ; and he made use of this fact in cer- tain observations. He also, in 1641, described a plan of using a pendulum to regulate a clock, and had a drawing of his invention made. This fact was not generally known, how- ever; and in 1656 Huygens independently invented a pen- dulum clock, which came into immediate use. Historical Sketch of Mechanics Although the main facts in regard to the historical devel- opment of mechanical principles have been stated in connec- GRAVITATION 1;>>7 tion \vitli them, it may be well to give a brief review. Up to the time of Newton the fundamental property of matter was thought to be weight ; and the only forces considered were those produced by weight. Archimedes and da Vinci had stated the laws of the lever and Stevin had explained the equilibrium of a body on an inclined plane, making use of what we call the parallelogram of forces; the "proofs" in each case were made to rest upon certain assumptions which appealed to the philosopher as being fundamental and which could not be proved themselves. Galileo made a great step in advance, because he undertook the experimental study of • ///////////'•a, and formulated certain statements in regard to the properties of matter in motion. He assumed that if a body were free from any force, it would continue to move in a straight line with a constant speed ; he showed that the acceleration of a falling body is constant ; and he deduced many well-known theorems; he further assumed what is equivalent to saying that a force acting on a body produces tion independently of the existing motion of the body. (ialileo's experiments on Mechanics were published in 1638. Newton's attention was attracted to a property of matter different from weight, and to other forces than that of weight, by his conception of the explanation of the motion of the planets. In his />/•///'•///»/. published in lt'»87, he proposed three laws of motion which are equivalent to the following: 1. A h»»dy left to itself will maintain its velocity constant. •_'. It a l.od\ i- BOted on by an external force, it will r< < an acceh -uch that f— nm \ forces act independent 1\ ,,!' each other. 3. Action and reaction are equal and opposite. se laws are in accord with the principles enunciated on . and the\ have served for over two hundred years as the basis of all work in Mechanics. \e\\toii t hus int induced the ideas of ni;i^s ,iiid of the proper measure of an\ force. I'.- fore NeWton'> ideas \\cl-e accepted, there \\as a dispute as to the value to be assigned "quantity of motion of mar 138 MECHANICS Descartes maintained that the proper value was wv, where w was the weight of the body and v its velocity, while Leibnitz adopted wv. It was shown by d'Alembert, in 1743, that both were correct ; mv measures the momentum or the impulse of the force, £rav2, the kinetic energy or the work done by the force. The greatest of Newton's contemporaries was Huygens, whose treatise, De Horologio Oscillator io, published in 1673, is equaled in importance only by the Principia. In this he discusses the motions of pendulums, simple and compound; the laws of centrifugal force, etc. He had no conception of mass and used less elementary assumptions than did Newton. He adopted in his work, as the fundamental property of matter in motion, wv2, and showed the great importance in questions of rotation of the quantity which we call the moment of inertia. Since the publication of the Principia progress in Mechanics has been (1) in a philosophical study of the nature of the postulates and definitions of Dynamics, (2) in deducing from Newton's laws other principles which are more useful for particular classes of motion. BOOKS OF REFERENCE MACH. Science of Mechanics. Chicago. 1893. This gives an interesting history of the progress of Mechanics, together with a critical study of the principles on which the science is based. POYNTING. The Mean Density of the Earth. London. 1894. This contains a full account of the experimental investigation of Newton's Law of Gravitation. ZIWET. Mechanics. New York. 1893. This is a more advanced text-book than the present one, but will be found most useful for purposes of reference. PERRY. Spinning Tops. London. 1890. This is a most interesting series of lectures on the mechanics of spin- ning bodies. WORTHINGTON. Dynamics of Rotation. London. 1892. This is a most useful book of reference for elementary students. CHAPTER IV PROPERTIES OF SIZE AND SHAPE OF MATTER Solids and Fluids ; Liquids and Gases. — The most obvious property of a material body is that it has a certain shape and size, both of which can be changed by suitable forces. As has been explained before, the name "solid" is given to a body that keeps its si/c and shape under all ordinary conditions; and the name "fluid," to a body that yields to any force, however small, that acts in such a manner as to make one layer move over another. Fluids are divided into two classes, accordiug as they
can form drops or not; the former are called "liquids"; the latter, "gases." See Introduction, page 16. Elasticity. Viscosity, etc. — Some bodies when deformed slightly by a force will return to their previous condition after the force is removed; they are called "elastic." Thus, glass, steel, ivory, etc., and all fluids are elastic. Certain solids, however, when deformed in a similar manner, remain so after the force ceases ; they are called " inelastic " or " plastic." Such bodies continue to yield to a force as long as it is applied. In the deformation of all inelastic bodies tin-re is a sliding of portions of matter over each other, as when a piece of lead is bent or hammered; and consequently there is what has been called "internal" friction between these parts. There is a sliding of this kind whenever any actual solid, however elastic, is deformed, all hough its amount may be very small. This is shown by the fact that, if a body as elastic as a glass rod or a steel tuning fork is set in vibra- tion, the motion soon ceases, and the temperature of the body 140 MECHANICS is raised slightly. Similarly, when currents are produced in a liquid or a gas by stirring them in any way, the motion soon ceases, and the temperature is found to be increased. A fluid which offers great frictional opposition to the relative motion of its parts is said to be " viscous," while those which flow easily are said to be "limpid." As will be explained later, when a fluid is made to flow through a long tube, the quantity that escapes from the open end is independent in most cases of the material of which the tube is made. This proves that the fluid layers on the inner surface of the tube stick to it, and so the fluid actually flows through a tube of the same material as itself. Consequently, in this flow the velocity is zero at the surface of the tube and increases toward the axis ; and so layers of the fluid flow over each other. It requires work to accomplish this ; and the quan- tity of fluid escaping under a given force or "head" measures the viscosity of the fluid, being inversely proportional to it. Similarly, when a solid moves in a fluid, there is a layer of the fluid attached to it, which moves, then, over other layers of the fluid. Consequently, if a pendulum vibrates, or if a disk is supported by a vertical wire which twists to and fro around its axis, making torsional vibrations, the rate of decrease in the amplitude of the oscillations measures the viscosity of the surrounding fluid. In this way the viscosity of various fluids has been measured ; and it is found that it varies greatly for different fluids and with the temperature of any one fluid. Rise in temperature decreases the viscosity of a liquid, but increases that of a gas. Diffusion. — Whenever any two gases are brought together, they mix ; and after a short time the mixture is homogeneous. This process is called "diffusion." If two liquids like water and alcohol are brought in contact, one will diffuse into the other ; and, even in other cases like mercury and water, where there is no apparent mingling, it may be proved that at the surface of contact there is a slight diffusion. OF -///•: .i.v/> >//.!/'/•; OF .\/.i /•//•:/; 111 The most important investigation of the phenomena of diffusion was carried out by Graham (1850). He was led t«> divide substances into two classes — "crystalloids" and "coll The former diffuse much more rapidly than the latter, and can as a rule be obtained in a crystalline form, while the latter are amorphous. The mineral acids and salts arc crystalloids; the gums, starch, and albumen are colloids. If the former are dissolved in water, the solutions have prop- erties most markedly different from the water ; while if the latter are dissolved in small amounts in water, they have little, if any, effect, in some cases the colloids being merely suspended in the water in a very finely divided state. If colloids are mixed with not too much water, they form jellies OF membranes ; and crystalloids are able to diffuse through many of these with almost as much ease as through pure water. This process is called "osmosis," and one case of it will l)e discussed later. (This evidently offers a method for the separation of crystalloids from colloids if there is a mix ture of them. Osmosis was first observed by the Abbe Nollet in 174s, who used a piece of bladder as the membrane. Parchment paper is often used.) Similarly, gases can pass through a thin sheet of India rubber; the latter absorbs the gas on one side and gives it off on the other. Many gases can pass through metals with ease if the latter aiv red hot; thus, hydrogen Can pass through red-hot platinum, oxygen through red-hot silver, etc. If two soft solids like lead and gold are brought into c t.ict. experiment! show that alter the lapse of suilicient time there has been diffusion «,f ,,ne into the other; and the same is believed to be true to a certain extent at the surface sepa- rating B -olids, or in fact any two portions of matter. I,' :•• inured by placing two bodies in OOntad OV< r a known area and determining the (plan: of either which pass this surface in a given time and the 142 MECHANICS distances they permeate. There are great differences in the rates of diffusion of different bodies, and these rates vary with the temperature. Solution. — One of the most important phenomena deal- ing with the properties of matter is illustrated when some common salt is put into a basin of water : the salt as a solid disappears, it is said to be "dissolved." The salt is called the " dissolved substance " or " solute," and the water the "solvent." Mixtures which are homogeneous and from which the constituent parts cannot be separated by mechani- cal processes, are called "solutions." The formation of a solution is evidently closely connected with the process of diffusion. Similarly, we can have solutions of other solids in liquids, of solids in solids, of liquids in liquids, etc. In the case of salt dissolving in water, it is found that, if the temperature is kept constant and more and more salt is added, a condition is reached such that, if more is intro- duced, it does not dissolve, but remains as a solid pre- cipitate: the solution is said to be "saturated." If the temperature is lowered, salt will be precipitated if some solid salt is already present; otherwise, this does not in general take place. If the liquid thus contains in solution more salt than would saturate it at a given temperature, it is said to be "supersaturated "; and its condition is unstable, for by adding a minute piece of salt, all the salt in solution in excess of that required to produce saturation is precipi- tated. Similar phenomena are observed in many other solu- tions, but not in all. It is found also that, as one substance dissolves in another, there are temperature changes ; thus, as common salt is dis- solved in water, the temperature of the water falls, while if sulphuric acid is dissolved in water, the temperature rises. These changes will be discussed later. Kinetic Theory of Matter. — It is impossible to explain these facts of diffusion without assuming that the minute /'/,'o /'/•:/,'•///•> OF SIZE AXD SHAPE OF MATTER 1 1 '. portions of material bodies — their molecules — are endowed with motion of translation; while, if they are free to move and are moving, the general explanation of the phenomena is at once evident and needs no statement. To account for tiit- differences between solids, liquids, and gases, it is only necessary to assume different degrees of freedom of motion (•I the molecules. Since solid bodies offer great opposition. in general, to changes in size and shape, we assume that in them the molecules are held together as if by a " framework." so that they can vibrate, but cannot move from one part of the body to another unless the "framework" breaks down : this may happen with difficulty or with ease, thus causing the difference between elastic and inelastic bodies. The word ••framework" is used to describe a condition, not an actual tiling; we mean simply that there are forces between tin- molecules which hold them together exactly as the frame- work of a building or bridge holds it together. In a liquid assume that the molecules are so free that they can and do move about limn one point to another, but yet that the forces are sufficiently strong to prevent them on the whole from getting far apart. Of course, if a molecule strikes tin- surface with suffici cut velocity it may escape, and thus evapo- ration is explained. In a gas we assume that the forces between the molecules are so minute that the freedom of motion is practically perfect; they can move freely from any point to another in the space open to them; and we think of tin; molecules as having rapid motion to and fro through this space. We shall show later h»>\v simple it is to ;in in general terms all the properties of a gas as due to notion nf its moleeules. It is not kno\\n whether these M b.t ween the molecules which are so evident in the case of solids and liquids are due to gravitation or not, but ist possible. The phenomenon of viscosity is at once explained by this assumption of moving molecules; for, if one layer of a fin id 144 MECHANICS i> moving over another, molecules will pass between the layers, and each one that passes from the layer flowing more slowly into the other retards the latter ; while, if one moves from this layer into the former, it accelerates it. Thus, owing to the continual interchange, the two layers finally have no relative velocity; and, if one of them is at rest being in contact with a solid, all the fluid comes to rest. (This is a case where a force is explained in terms of the motion of particles of matter. See page 68.) Coefficients of Elasticity ; Hooke's Law. — When an elas- tic solid is subjected to a force, it will, in general, yield slightly and then come to rest, e.g. bending a bow, stretch- in
g a wire. This means that the changes in the molecular forces which are called into action at any point by the defor- mation are sufficient to neutralize the action of the external force. There is thus at any point of a deformed elastic solid a change in the position of the molecules immediately around it and a corresponding "force of restitution." When there is equilibrium between this internal force and the external one, the former may be determined from a knowledge of the latter. The elasticity of the body is measured by the ratio of this change in the internal force at any point, which is produced by the deformation of the matter near it, to 'the amount of this deformation. These internal forces between portions of the body are called " stresses " ; and their numeri- cal value is defined as follows : let the internal force between two portions of the body whose area of contact is A have the value F, then the limiting value of the ratio — as A is made smaller and smaller is that of the stress at that point. (If the stress is uniform, it equals the force per unit area.) Owing to the deformation the internal forces are changed ; and calling the change in the force A jP, the ratio — — is the stress corresponding to the deformation. The deformation F -A A F A. OP ^///-: AND SHAPE OF MATTKIi 1 !."» which produces this stress is called the "strain"; and its numerical value is defined differently, depending upon the kind of deformation. Thus, if the volume of each minute portion of the body is changed, without there being at the same time a change in shape, let v be the value of the origi- nal volume of a minute portion of the body at any point, and Av be the change in this ; then the value of the ratio — v is defined to be that of the strain at that point. Similarly, if the length of a wire or rod is changed by stretching or compression, the strain is defined to be the ratio of the change in length to the original length. If the shape of a solid body is changed, the measure of the strain may be defined also, as will be shown later. It is found that, if the strain is small, the corresponding stress is proportional to it: this is called Hooke's law, and \vas fust stated by Robert Hooke in 1676, in the form "Ut tensio sic vis." Tin- ratio of the stress to the corresponding strain in any elastic body is called the " coefficient of elasticity" of that body with reference to the type of strain. Hooke's law. then, states that all coefficients of elasticity are constants for a given body ; or, in more common language, the amount of the deformation of an clastic body is proportional to the force applied. "nee this proportionality between internal force of rexti- tution and displacement is t rue. and since one is in the oppo- site direction to the other, the elastic vibrations of any body must l.e harmonic; because, when in the course of its vibra- tions the body has a certain strain, i.,-. display-menu the elastic force of restitution is, in accordance with Hooke's proportional to it. and therefore the acceleration is I Again, if I lookers law is true, the elastic force corresponding to any displacement must !»«• directly propt.rt ionul to it, as has just been said; HO, if/ is the value of tin nd x that of the displacement/^ or, where c is a factor ..f pro] tionality depending upon the nature and dimensions of the AMES'S MMHICI- ]o 146 MECHANICS strained body and the character of the strain. As the dis- placement, then, increases from 0 to some value xv the aver- age force of restitution is £ cxv and therefore the work done in producing the displacement x1 is the product of xl by £ cxv or £ cxf. A solid body can undergo two independent deformations : a change in shape and a change in volume ; and correspond- ing to these an elastic solid has two coefficients of elasticity. If the coefficient with reference to change in volume is large, the body is said to be nearly "incompressible " ; while if the one with reference to a change in shape is large, the body is said to be " rigid." A fluid, 011 the other hand, has only one coeffi- cient of elasticity, that corresponding to a change in volume. Gases are very compressible ; liquids are not. These kinds of matter will be discussed separately. Density and Specific Gravity. — Before, however, proceed- ing to do this, a physical quantity should be defined which must be used often in the following pages. This is the " density " of a body. It is that property of a body which expresses its denseness, using this word in its ordinary mean- ing. If the body is homogeneous, and if m and v are the values of its mass and volume, the ratio — is defined to be v 931 the value of the density. But, if the body is heterogeneous, the density at any point is defined to be a quantity whose value equals that of the ratio in the limit, where Av is the volume of a small portion around the point and Aw is the value of its mass. On the C. G. S. system, the den- Ai> sity of pure water at 4° C. is almost exactly one ; for the kilogram was so constructed that its mass almost perfectly equaled that of 1000 cu. cm. of pure water at 4° C. ; i.e. the mass of this volume of water is most approximately 1000 g., and therefore the density of the water is as stated above. .* Of >///•; AM> >//.!/»/; OF MATTEL 1 17 The ratio of the density of a body to that of a standard both is called its " specific gravity" with reference to the latter. Thus, it water at 4° C. is chosen as the standard sul »Maiice, its density is — , where m is the mass of a volume v\ and, if M is the mass of an equal volume of another substance, its (Unsity is — , and its specific gravity is therefore — v m Thus "specific gravity" is independent of the choice of units in terms of which the mass and the volume are measured, but some standard body must be selected. In order to measure the density of any body — solid, liquid, or gaseous — one of two methods must be followed: eitlu-r its volume and mass must be measured directly, or its spe- cific gravity must be determined with reference to some body whose density is known. The details of these different pro- cesses may be found in various Laboratory Manuals. In the following table are given the values of the densities of some substances in ordinary use. DENSITIES SOLIDS Aluminium (cast) . . 2.58 Brass (about) . . .8.5 Copper .... 8.92 ond . . . . - Glass (common) . . 2.6 Mint) . IcestO°C. . . 0.9107 I nm (wrought) . . . 7.86 Iron (gray cast) . . .7.1 Lead UJO Platinum .... 21.50 Silvr 10.5 Tii 759 /in.- 7.15 Ktl.yl alrnl.,.1 at 0°C Kthyl ether at 0°C. 0.791 l 1SJM Water at utli.-r t. mperaturea, see p. 148. 148 MECHANICS GASES AT 0°C. AND 76 CM. OF MERCURY Air (dry) Argon . Carbon dioxide Chlorine . 0.001293 . 0.001700 . 0.001977 . 0.003133 Helium . Hydrogen Nitrogen Oxygen . . 0.00021 . 0.0000895 . 0.001257 . 0.001429 WATER AT DIFFERENT TEMPERATURES 0°C. . 1° 2° 3° 4° 5° 6° 7° 8° 9° 10° 11° 12.999878 . 0.999933 . 0.999972 . 0.999993 . 1.000000 . 0.999992 . 0.999969 . 0.999933 . 0.999882 . 0.999819 . 0.999739 . 0.999650 . 0.999544 . 0.999430 . 0 999297 0.999154 16° C. . . 0.999004 17° ... . 0.998839 18° ... . 0.998663 19° ... . 0.998475 . 0.998272 21° ... . 0.998065 . 0.997849 23° ... . 0.997623 24° ... . 0.997386 . 0.99714 25° ... . 0.99686 26° ... . 0.99659 27° ... 28° ... 29° ... 30° ... . 0.99632 . 0.99600 . 0.99577 0.99547 2 ° 2 0 ° 2 ° 5 31° CHAPTER V SOLIDS General Description of the Strains of a Solid. — A solid body, being eharacteri/.ed 1>\ a definite shape and size, can, as has been seen, be deformed in two independent ways; and, in general, under the action of forces both the size and shape are changed. These changes, if small enough, will disappear in the case of an elastic body when the force is removed; hut, if the force is too large, certain permanent effects are experienced. These will be described in one particular case, that of a vertical wire whose upper end is fastened to a fixed support and to whose lower end is attached a scale pan into which weights may be loaded. So long as the stretching force is not too great, the elongation varies directly as the load ; and, if this is removed, the wire returns to its original length. If the load is increased, however, a point is reached, known as the "elastir limit." such that if the force exceeds this in value, the wire acquires a permanent elongation or "set" which does not disappear when the load is removed. It tin- load is increased still more, the elongation becomes greater; and at length a condition is reached sueh that. greater force is applied, the extension increases v< i \ rapidly and the wire becomes plastic, because the amount of the ex- tension now varies with the time the load acts. This point is called the "yield point/' If the load is increased beyond . the cross section of wire will eontraet until the •• break- s reached. Changes similar to these go on when the shape of a body is altered by twisting it. In what follows we shall discuss mily those chants \\hich take place below II'.' 150 MECHANICS the elastic limit; so that we can consider the strain as proportional to the stress. Change of Volume. — In this case the strain is, as has been explained, — , where Av is the change in volume of a portion v of the body whose volume originally was v ; and, if the cor- responding stress or force per unit area is Ap, the coefficient of elasticity is A# • — • Av In order to produce a change in size of the minute portions of a solid without changing their shape, it is necessary to immerse the solid in a liquid and then to compress the liquid. This is done by having the liquid inclosed in a stout trans- parent cylinder, one end of which is closed by a piston which can be screwed in or out. Such an instrument is called a "piezometer." When the piston is pushed in, the liquid presses against the immersed solid and compresses it, the volume of each minute portion of the solid being decreased proportionally. To measure the change in vol
ume of the solid, the latter is as a rule made in the shape of a rod ; two fine parallel lines are scratched on it, one near each end ; and by means of a comparator the distance apart of these lines is measured before and under the compression. If 1Q is the original length and I the length when the stress is increased by an amount Ap, experiments show that I — 1Q = — eZ0Ajo, where c is a factor of proportionality and is extremely small. If Ap is measured, the value of c may be determined. If the body is homogeneous, we may assume that similar changes take place in a plane at right angles to the length of the rod. So, if a cube were subjected to the increase in stress A/?, its change in volume would be found as follows : the original volume VQ is Z03 ; when the stress is increased by Ajt?, the volume is Z8, or Z03(l - <?Ap)8, which equals Z08(l - 3 <?Ap), since c is so small that terms involving c2 and <P may be neglected ; hence, the change in volume is — 3 Ifa&p or SOLIDS 151 - 3 v0<?Ap. So, in the original experiment the strain corre- sponding to the stress A/?, i.e. the ratio of the decrease in volume to the original volume, is — 3 c Ap ; and the coeffi- cient of elasticity corresponding to a change in volume is tl irii — - • The stress Ajo is due to the change in the internal o c forces, but, since there is equilibrium between it and the forces due to the pressure of the liquid, it is determined by measuring the force per unit area with which the liquid presses on the solid, as can be done in a manner to be described presently. The result of subjecting a hollow solid body like a bottle to such a liquid pressure as this is to decrease its volume exactly as if there were no hollow spaces, provided the liquid has access to them ; for each minute portion of the body has its volume decreased exactly as if there were no ies. It is as if, before the compression, the solid were built up of large "bricks"; while, under compression, it is made up of the same number of smaller ones. This coefficient of elasticity is sometimes called the " hulk modulus." Change of Shape. — An illustration of the < -han<je in shape large solid without any appreciable change in volume is given by the following experiment. Make a rectangular l.lo.-k of wood; to two opposite faces fasten two boards; and push these boards sidewise in their own planes, but in opposite directions. The shape will be changed as shown. tin- edges of the block being now <»l»lique; and the angle veen the two positions of an edge, that is (BAB1)* is the measure of the strain. It the sidrwise force on • •iihi-i I ><> nd is t\ there is an equal but opposite force of restitution; and, it tin- an -a of the cross section at the block 1 the stress corresponding to the strain is — • This stress A i% acting across all the planes of the block that are parallel to the boards. The coefficient of elasticity for a change in shape is, then, the ratio <>f this ipiantity to the angle referred t<> j." 152 MECHANICS r 7i above. This coefficient is also called the "coefficient of rigidity"; and the particular kind of stress that arises when one layer is moved parallel to another is called a "shearing stress," because it is like the force produced by a pair of shears. / // C B / 1 \ L= / / / 7 When a rod or wire is twisted on its axis of figure, each minute por- tion of the body experi- ences a shearing stress, because different plane cross sections of the rod or wire are turned through different amounts ; and, therefore, this is an illus- tration of a pure change in shape. B'B C'< \ \ FIG. 71. — Change of shape or shearing strain. The deformation is produced by pushing the two boards in opposite directions. If one end of the rod or wire is held firm, and the other twisted by a moment L around the axis of figure, the angle through which the lower end will be turned is given by the following formula, which may be "=§• deduced by the help of higher mathematics : where I is the length of the rod or wire, n is the coefficient of rigidity, and B is a constant for any one rod or wire depending upon its dimen- sions. For a circular cylinder of radius r, B = - ; and so, for a wire of circular cross section, 21L N = If the wire is maintained in this state of torsion by this moment, and if it is in equilibrium, there must be in each plane cross section two internal moments equal to L, due to the elasticity of the wire; one moment acts on one face at the cross section, the other, on the opposite i r n r 4 SOLIDS 153 and the two moments are equal and opposite in amount. It is exactly analogous to the case of a wire or cord under the action of a 4 force ; at each of its points there are two equal and opposite •ns, each equal to the stretching force. If such a wire is twisted and then set free and allowed to make torsional vibrations, the moment tending to oppose the motion at any instant, due to the reaction of the . will be M the angle of torsion is N. Hence, if a disk whose moment of iiu-rtia around the axis is / is fastened to the free end of the wire, its angular acceleration at any instant of the vibration will be — or — wn • so the vibrations will be harmonic, and the period of one complete vibration will be The elasticity of a helical .spring, that is, a wire coiled nj» in a spiral, is due to the fact that, when it is ]ni lied out, the wire is twisted, not elongated. Actually there is a sliding of one layer over another when a wire is twisted, or, in general, when there is a shearing stress; but in an ideal solid there would be no such sliding, if the shape were changed; and in sueh bndies as steel, glass, ivory, etc., the slipping is very small. Young's Modulus. — When a rod or wire is elongated. <>r when a pillar is compressed, both the shape and si/.e <>f its minute portions are altei-.-d ; and so both the coefficients previously discussed are involved. Since, however, the case is such an important one in all practical work, a new coefficient is defined \\ hi.-h refers to the elasticity shown w lien ire or rod is stretched or compressed. Let I be the original length ; A/, its increase when there is an increase &F in the stretching force ; A* the area of the cross section. Since the 154 MECHANICS wire or rod comes to equilibrium, the change in the internal force equals A.F, too ; and so the stress is - - • The corre- sponding strain is — ; and the coefficient is - - -s- — • This I A. I • is called " Young's Modulus " ; and, writing its value E, A -A/ E=l'^F. This is found to be a constant for any one kind of matter, e.g. a definite kind of brass, iron, etc. ; and, if its value is known, the elongation of a particular wire or rod under the action of a given force can be calculated, because If a pillar supports a building, its length is less than if it carried no weight ; but, if the weight supported becomes too great, the pillar will buckle and give way. It may be shown by a more elaborate discussion that a circular pillar pressed between two supports, and having both its ends fixed, becomes unstable when the force of compression equals -71" J , where r is the radius of the pillar and I is its length. If a beam is supported horizontally by resting on two horizontal knife edges placed near its ends, and if a weight is hung from its middle point, it will be bent. It is obvious that the lower side will be stretched, while the upper is shortened. There will there- fore be a layer of lines in the beam about halfway through, F.«. 73 -Standard form of metre bar, with the which are neither lengthened nor shortened ; this is called the " neutral section." This fact is made use of in construct- ing standards of length, such as metre bars. One form adopted has a cross section like the letter H< and the upper divided scale along its neutral section. OW 155 sinf.tr. • of tin- "bridge" is the " neutral section"; on this the scale divisions are marked. It is evident, further, that the coefficient of elasticity involved in flexure is Young's modulus. Formulas giving amount of bending t.»r beams of different dimensions may be found in books of reference. Impact. — \Vhen t\\o solid bodies in motion meet, there • •lunges in their velocities that obey certain laws. The principle of the conservation of momentum, of course, holds. This states that, if tin- two centres of mass are moving along -t might line joining them, and if in and M &TQ the two masses, -i-j and }\ the velocities before impact, and va and V% those after, proper attention being given the algebraic sign, 1111-^ + MV^mv^+Ml^. The relative velocity before impact - l\: after impact, V^ — v^ ; and experiments show that wo spheres the ratio of these quantities may be regai as a constant, depending only on the material of the two bodies. This statement may be written J^ — t>a = e^ — Fi), re e is a constant of proportionality. It is called the "coeilicieiit of restitution/' (More accurate experiments show that c is ma constant, but varies slightly with the If the impinging bodies are not spherical, there will be rotational acceleration also, in general. ;he bodies are inelastic. fvO = V^\ that is, after impact the two bodies proceed on together. If the bodies '.\ clastic," e = 1 and I", — V, = t'j - J",. If oinbined \\ith that given by the principle of the :;ion of momentum, it will be found equivalent t» ssion of the i mr,« + 4 M V* = J mrf« + \ MV*. The met h..d by which two elastic bodies affect each other's motion duriii'j impact can be compare, 1 with what h:ip\|i6IUI (led with \|p ,lTrrs col' The sprin-s are .Lrradnally compressed up to a certain p 156 MECHANICS and then expand, pushing the cars apart. So, when two spheres collide, they are deformed, producing elastic reac- tions which increase in intensity as the centres approach, and then decrease as they cause the bodies to separate. The laws of impact were investigated and stated at about the same time, 1668, by Wren, Wallis, and Huygens. In their present form they were first given by Newton
. Energy Relation. — The potential energy of a solid in a state of strain may be deduced at once if its elastic coeffi- cients are known. (See page 110.) As an illustration con- sider a stretched wire. Using the same symbols as before, which means that the force required to produce an elonga- tion AZ is — - — ; consequently that required to produce an elongation x is - -; and the potential energy of the wire when it is stretched this much is - — — 2 I 1 EAx* If a heavy body is attached to one end of a vertical spiral spring, the other end of which is fastened to a fixed support, it can be set in vertical vibrations which are harmonic. The change in the potential energy during the motion is due in part to alterations in the spring and in part to the to-and- f ro motion of the body with reference to the earth ; and the kinetic energy is due to the motion of the body and to that of the spring. The velocity of different parts of the spring varies ; and it may be proved by higher mathematics that the kinetic energy of the system is the same as if the spring had no inertia and the mass of the suspended body were increased by one third the actual mass of the spring. CHAPTER VI FLUIDS General Properties. — Fluids have been defined as those bodies \vhieh yield to any force, however small, which acts in such a manner as to cause one layer to move over another; that is, they yield to shearing forces. There are two classes of fluids: liquids and gases. The former have definite vol- umes, to change which requires great forces; and. if left to themselves, they form drops, but if placed in a solid vessel, assume its shape. The latter are easily compressible and assume both the shape and size of the containing vessel. When a fluid is not flowing, it is said to be at "rest," although this does not imply that the molecules are not moving : it simply means that there is no motion of portions of the fluid over each other. We shall discuss in turn the two eruditions : that of rest and that of flowing. Fluids at Rest Thrust. — The fluid exerts a force against the walls of the 1 that eontains it; and, conversely, the wall reacts against the fluid. For instance, if a toy balloon is inflated, the _r;ts presses outward against the rubber envelope, and this presses inward, tending to compress the gas; a dam holding baek a river is pressed against by the water; a t containing water presses inward \\ith sufficient force to with- stand the outward force of the water, otherwise it bursts. Similar! \, there ire forces against any foreign body im- 1 in tin- fluid. If the fluid is at rest, this force 167 158 MECHANICS between the fluid and the wall or immersed body is per- pendicular to the separating surface at each point ; for, if there were a component parallel to the surface, the fluid would flow. The total force acting on the surface is called the "thrust." Fluid Pressure. — The properties at various points of a fluid are best described in terms of what is called the " pres- sure." If any small portion of a fluid is considered as in- closed in a solid figure with plane faces, there is a stress across each plane face due to various causes; but, if the fluid is at rest, this force is perpendicular to the face, as just ex- plained. The limiting value of the ratio of this perpendicu- lar force to the area over which it acts is defined to be the value of the " pressure " at the point considered, in the direc- tion of the force. If the pressure is uniform, it equals, then, the force per unit area. We thus speak of the pressure at the bottom of a tank of water, etc. Pressure at a Point. — At any point of a fluid at rest the pressure has the same value in all directions ; for, consider a small portion of the fluid inclosed in a tetrahedron, ABCD, and express the con- dition that it shall be in equilibrium FIG 74 —A portion of a un(ler the action of the forces on its faces. fluid inclosed in a tetrahedron, The SUm of the Components of these or triangular pyramid. <• • -i. .. forces in any direction must equal zero. Choose as this direction that of the line AB ; then the forces on the two faces BAD and ABO have no components parallel to this line, because they are perpendicular to it. Call the area of the face A CD, Av and the force acting on it Fl ; the area of the face BCD, Av and the force acting on it F2. The component of Fl parallel to AB is Fl multiplied by the cosine of the angle between AB and a line perpendicular to the plane ACD. Imagine a plane perpendicular to AB\ the projection on this plane of the triangle ACD is another tri- angle whose area may be called A, and which is the same 169 as the projection on this same plane of tin- trian^. l Therefore - - equals the cosine of the angle between the line AH and a line perpendicular to the plane ACD; ami component of the force Fl along AB is, accord ii . Similarly, the component of F2 along Afl is F^—\ and -M AI .since tliere is equilibrium, Fl —- + F^ — = 0, or=-L = — ±i. Al AI Al AI refore, in the limit, when the tetrahedron becomes intini- nal. tlie pressure in the direction of F1 is equal numeri- cally to that in the direction of F2. Hut these directions may be any two : and consequently the pressure at the point around which we have imagined the tetrahedron has the numerical value in all directions-. Work done when the Volume of a Fluid is Changed. — If a fluid is contained in a cylinder one end of which 1 bv • vable piston, the work done on the fluid in order to c.impivss it may he calculated at once. If the ssure of the fluid against the piston is uni- ; its value be p (if the pressure is not uniform, let j> be its mean value): then, if A MC area <,f th.- piston, the force that mu.st be over- .1. Let the piston be displaced in\\ itance r : the work done on the fluid /•.I: the decrease in volume is ./.I: ami. ., ' ;" ' . the work done in compressing the fluid e.jiiaU the product of the pressure a'_r;miM the pi.st.in by the decrease in volume of the fluid. Simi- • rk done by the fluid US it expands C<1' product .,f th.- pressure by the increase in volume, provided tin-re ix no Rooelention. The work done by a fluid, or on it. as its volume change can I .'Taphically provided the changes are so slow that there is practically a uniform c throughout 160 MECHANICS E A F B P c * \ D^ p / the fluid at any instant. Lay off two lines, Ov and Op, making a right angle at 0 ; let distances along one corre- spond to values of the volume, and along the other, to values of pressure. The former is called the " axis of volume " ; the latter, "the axis of pres- sure." From any point D in the plane of the two lines drop a perpendicular DA upon Ov. The point D repre- FIG. 76.— Diagram illustrating work done by SCntS the Condition of a fluid a fluid as it expands ^^ volume ig equftl ^ Q£ and whose pressure is equal to AD. If the pressure remains constant while the volume expands from OA to OB, the new condition of the fluid is given by the point O where BO is parallel to the axis of pressure and DO to the axis of volume, and the work done ly the fluid is equal to the area of the rectangle ABOD. If now the pressure suddenly increases to a value BC', no work against external forces is done, since the volume does not change. The condition of the fluid is given by the point 0' ; and, if the fluid again expands until the volume is OE, its condition is given by the point F, and the work done by it equals the area of the rectangle BEFC' . It is at once evident that, if the changes in pres- sure and volume occur, not in a discontinuous manner as from D to 0 to O' to F, etc., but continuously, as represented by a smooth curve PQ, the work done during any change in volume will be the area included between this curve, the axis of volume, and two perpendicular lines marking the initial and final volumes. If the fluid is being compressed, the changes may be represented by a curve in the opposite direction, from Q to P\ and the area just described gives the work done on the fluid by external forces. If the curve describing the changes in the fluid is a closed one, it means that after a series of operations the fluid returns to its initial condition of pressure and volume ; it is said t«> d through a "cycle." If the curve is a right-handed that is, if the series of change- ii that if a man were to walk from point to point along the curve the inclosed area would lie on his right hand, this inclosed area gives the total nt-t work done by the fluid during the cycle. For. consider two portions of urve AB and CD which are intercepted by the same two ndicular lines through /•' and /,'; during the process represented by the curve AB, the fluid does an amount of < \gnm Rhowtn* work don« by • •. ^.. . . during the process represented by the curve CD, work is done on the fluid equal in amount to the area DCEF\ con- sequently the excess of work done by the fluid equals the i ABCD. In a similar manner other pairs of portions of the curye may be considered ; and, in the end, the entire work done 1)\ the fluid in excess of that done on it equals the area in< -loM-d by the curve. Conversely, if the curve is described in the opposite direction, that is, if it is a left -handed one, its area repre- sents the net work done ,,n the fluid by external forces. Cause of Fluid Pressure. - The pressure at any point in a fluid at ri-Nt is due to two causes: (1) the reaclion in\\ be walls of the vessel that contains tin- fluid : (-J) n nal foi.-es, such as gravit\ — this is, in fact, the only such force which we need, in general, consider. Illustration the former muse, t* sho\\ii by balloons, dams, and have .re : hut to have one where the prcs is due entireU to the containing walls, we must imagine a AMES'S 1'iiYBics — 11 162 MECHANICS fluid in its vessel carried to some point where gravity and other " external " forces cease to exist ; for instance, to the centre of the earth or far off in space. (There is also, of course, another pressure at a
ny point in a fluid due to the forces between the molecules. This we cannot measure ; but we can form an estimate of its value for a liquid by measuring the amount of work required to evaporate it, assuming that we can neglect this pressure for a gas.) Pascal's Law. — Let us, then, consider the properties at the centre of the earth of a fluid inclosed in some vessel which it fills; for instance, a cylinder closed with pistons of different areas. To maintain the fluid in a definite condi- tion, forces Fl and F2 must be applied to the pistons : — FI from without ; and there is, therefore, a corresponding pressure throughout the FIG. 78. — A fluid is inclosed in a vessel closed _ . _ _ . , _ . , by two pistons of different areas. Neglecting mild. But, Since the fluid gravity, ^1=^, Pascal's Law. is at rest, this pressure must be the same at all points; for, if it were not, the fluid would flow from a point of high pressure to one of low, there being no force to counterbalance the difference in pressure. Therefore, the fluid pressure due to the reaction of the walls of the contain- ing vessel is the same at all points throughout the fluid. If this pressure is p, and if the area of one piston is Av the force necessary to keep it from moving outward is pA1 ; and, if the area of the other piston is Av the force acting on it is pAv Thus we have a " machine " by means of which a force F1 which equals p A1 can balance one F2 which equals pAz ; and so a small force may produce a great one. If the fluid is a gas, a great pressure, and therefore large forces, cannot be secured unless the volume is made very small; but if it is a liquid, the pressure may be made as great as desired without any marked decrease in volume. Thus, FLUIDS water or some other liquid as the fluid in a cylinder provided with two pistons, a small force acting on the smaller piston may produce a great force by means of the larger one. This is the principle of the " hydrostatic press," which is shown in the cut. A pump forcing a small piston do\vn produces an upward motion of the large piston; and thus a force is exerted as much greater than the original one at the pump as the area of the large piston is greater than that of the other. (Of course, since this press is used at the surface of the earth, there are additional pressures due 79. — Hydrostatic press. avity ; but they are nearly the same on the two pistons, and in any case their effect can be neglected when compared with that due to the pistons.) I n a piezometer (see page 150) the force per unit area on the immersed solid Imdy equals the pressure throughout the liquid: and, if the force acting on the piston and the area "f the latter an* known, the value of the pressure equals their niiio. 1 law that tin- pressure is the same at all points in a fluid when not under 1 In- adi«»n of external forces was t stated byF . :• (felled M Pascal's Pressure due to Gravity. — The external force to which all fluids on the surface of the earth are subject is that of 164 MECHANICS gravity ; and since, when the fluid is at rest, the lower layers have to maintain the weight of the fluid above, there are dif- ferences in pressure at different points in a vertical line in the fluid. Thus imagine two horizontal planes at a distance // apart, and consider in them two portions having equal c D areas, of value A, one vertically above the other. Let their traces on the paper be AB and CD. The pressure over the plane CD due to gravity depends upon the fluid above it ; similarly for the pressure over the plane AB ; consequently the excess in the upward force across AB over that across CD equals the weight of the fluid contained in a cylinder planes in a fluid, at a whose cross section equals A, the area of either of the planes, and whose height is A, their distance apart. If d is the density of the fluid, this weight is dhAg, because hA is the volume and dhA the mass. Therefore the excess of pressure due to the vertical height h is dgh. It follows that the pressure at all points in the same horizontal plane of a fluid is the same ; because, if it were different, there would be a flowing of the fluid from the point of high pressure to that of low, as there would be no force to oppose the motion. Archimedes' Principle. — If a solid body (or a body with a solid envelope) is immersed in a fluid at rest, or if a liquid drop or bubble is immersed in a gas, it is acted on by the pressures against its surface ; and these produce a resultant force in a vertical direction upward, as can be easily seen. For if the solid body were replaced by a portion of the fluid having the same size and shape inclosed in a massless envel- ope, there would be identically the same surface pressures on this envelope as there were on the solid. But under these conditions the fluid in the envelope is at rest ; and this shows that the forces due to the surface pressures have a resultant FLl'llx 165 vertically upward whose amount equals the weight of tin- fluid in the envelope and whose line of act inn is through tin- centre of gravity of this portion of the fluid. Therefore, when a solid is immersed in a fluid, the surface forces due t<> the weight of the fluid combine to form a buoyant force which is equal in amount to the weight of the fluid displa he solid and whose line of action is vertically upward through the centre of gravity of the displaced fluid. (If the solid hody is homogeneous, its centre of gravity coincides with that of the fluid displaced ; otherwise, it will not in general.) This statement, having been first expressed in words by the great philosopher of Syracuse, is called " Archi- medes' Principle." It is illustrated by floating balloons and soap bubbles; by all solid bodies here on the earth, whose apparent weight is therefore less than their true weight; by bodies immersed in water ; etc. I f the solid is denser than the fluid, the buoyant force is less than its own weight, and it sinks; while, if its density is less than that of the fluid, it will rise. In both cases the in takes place in such a manner as to make the potential energy decrease. Tin- j.rinri],!,- furnishes a method for the determination of the specific gravity of a solid body in terms of any liquid which does not dissolve •i ise affect it If the weight of the solid is measured when Tee, its value is mg or dvg, when d is its density and v ito volume, due allowance being made for the bu<> If it is weighed again, hanging imineraed in the liquid, the difference in weight is is the density of tin- liquid. Thus the ratio of the on t to this loss in weight is ^, or the specific gravity of the solid. If at be the other is known, may Unit of Pressure. — The unit of pressure on the C. G. S. system dyne per square centimetre "; hut, since pree- s are as a rule produced and measured by using columns :«juids. a more convenient practi.-il unit has been chosen, pressure due to a vertical height «•• calculated once. 166 MECHANICS metre of mercury at the temperature of 0° Centigrade, under the force of gravity which is observed at sea level at 45° latitude." This unit is called a "centimetre of mercury"; and its value in terms of dynes per square centimetre may be calculated at once by substituting proper values in the formular p = dgh. On the C. G. S. system, the density of mercury at 0° C. = 13.5950 ; and the value of g at sea level at 45° latitude is 980.692. Consequently the pressure of one " centimetre of mercury " is the product of these two quan- tities ; that is, it is 13332.5 dynes per square centimetre. The pressure of 75 cm. of mercury is called a "barie";" and its value in dynes per square centimetre is 75 x 13332.5, or almost exactly 106 dynes, i.e. a "mega-dyne." The pres- sure of 76 cm. of mercury is called " one^ atmosphere," because, as we shall see later, this is about the pressure of the atmosphere at sea level. Other units are " one pound per square inch," "one ton per square foot," etc., where "one pound " means the weight of one pound, etc. Atmospheric Pressure. — Owing to the smallness of the density of a gas, there are only slight variations in pressure at different points in a gas confined in a reservoir of any moderate dimensions. There are, however, marked differ- ences in the pressure of the surrounding atmosphere in which we live as one rises far above the earth's surface or goes up a mountain. This is owing to the large value of h in the above formula, which may in this manner be secured. Owing to the presence of the gases forming the atmosphere, there is a pressure exerted by it against every solid or liquid surface with which it is in contact. This is called the "atmospheric pressure." It may be measured at any point on the earth's surface by balancing it against a column of some liquid of known density, as will be shown presently. The fact that the atmosphere exerts a pressure on solid and liquid surfaces was first clearly understood by Torricelli, a pupil of Galileo's, and by Pascal. Conditions at the surface FLUIDS !»;; nf the earth were in their minds comparable with those at tlu- bottom of an ocean of water, so far as fluid pressure was concerned. Torricelli devised the famous experiment, which bears his name, of filling with mercury a long glass tube • •d at one end, inverting it, and placing the open end under the surface of mercury in an open vessel, care being taken to prevent the entrance of any air. The mercury column stands in the tube to a height of about 76 cm. (at the sea level) ; lu -ing held up in the tube by the downward pressure «>f the atmosphere on the mercury in the open vessel. The space above the mercury column is called a "Torricellian vacuum " ; and it is evident that the only matter present in it is mercury vapor, if the experiment has been carefully per- formed. This » \JM rim, nt was performed for Torricelli by his fiu-inl Viviani in 1643. Pascal varied it by carrying a rometer," as this apparatus of Torricelli's is cal
led, to ditlVrent heights and noting the change in the height of the column. Many experiments to show the effects of the atmos- phri -ir pressure were devised after the air pump was invented by Von Guericke (about 1657) and improved by Boyle. Liquid- are in general contained in open vessels which they only partially fill. The atmosphere presses against the free : ace of the liquid, exactly as if there were a piston press- down on it. Therefore in the case of liquids in open vessels the pressure due to the containing walls equals the atmospheric pressure, and, as said above, this pressure is the same at all points in the liquid. Fluids in Motion If there is a difference in pressure between two points in .1 tlnid. tin i, will be mot: 11 high to low pressure unless re vents it. We shall consider several cases. Uniform Tubes. — 1 uents sh . within certain limits, thr .pi. in :luid that MM, -h a tube under 168 MECHANICS a difference of pressure at its two ends is independent of the material of the tube. This proves that what actually hap- pens is that a layer of the fluid sticks to the walls of the tube, and the escaping fluid thus moves through a tube made of the same material as itself. There is therefore friction be- tween the moving layers of the fluid, not between the tube and the fluid; and the quantity of fluid that escapes under definite conditions varies inversely as its viscosity. If the tube is horizontal, the pressure is uniform through- out it, so long as there is no flow; this is called the "statical" pressure. But as soon as the motion begins, the pressure falls throughout the tube. If a constant pres- sure is maintained at one end A — as in water or gas mains — and if the other end B is closed, there is a uniform pres- sure, as just said ; but if B is open so as to allow the fluid to escape, owing to the friction the pressure will decrease uniformly from A to B if the cross section of the tube is the same throughout ; and, if the tube is sufficiently long, the fluid will barely flow out, however great is the pressure at A. If the tube is bent, the flow is still further decreased. If the fluid is flowing uniformly, the quantity that passes through a cross section of the tube is the same at all points along the tube. If v is the average velocity of the fluid over any cross-section whose area is A, and d the density of the fluid, the quantity that passes in a unit of time is vAd. The manner in which the fall in pressure along a tube and the quantity of fluid flowing through it depend upon its length and cross section, has been found as the result of numerous experiments, and is expressed in various empirical formulae. Irregular Tubes. — If the fluid is flowing uniformly through a tube of irregular cross section, the quantity passing all cross sections in a unit of time is the same; and that passing any one cross section is, as shown above, vAd, where v is the mean velocity over the section, A is the area of the section, and d is the density of the fluid at that point. If the fluid FLUIDS is flowing slowly, there is no change in density at different points, and therefore there is the following relation between tin* velocities at different cross sections, VjAj = v^Ay in which t-j is the vdority at a point where the cross section Lfl ami j-2 is that at a point where the cross section is Ay '1 MS that, a< the cross section diminishes, the velocity in- creases, and vice versa. (Compare the flowing of a i into a lake and out through a narrow defile.) If tin- velocity iic'i'eases from one point to another in the direction of motion of the fluid, there must be a fall of pressure in this direction so as to produce this increase in the velocity. II' nee, if the velocity increases in one direction, the pres- sure must fall in this direction; and conversely. When the velocity is greatest the pressure is least, and where t la- the pressure is greatest. This is illustrated by the atomizer, the steam injector, the hall nozzle, etc. Another illustration of this principle is furnished by the •• -"ball. If a ball were in motion of translation only, < 1 be symmetrical about its Hue of motion. As >ved through the air, a layer of gas would stick to it; and, owing to friction between • and the surrounding air, currents would be produced h were symmetrical on all sides. But, if th> 4 also as it moves, things are cliff- - advancing, as indicated in th. •• direction of the arrow and be spinning on an to the plane of the pap<M clock I I on the si.l- mark-d A is equal to the turn of r, the he ball, and rA, where r In the ball and A ..city of a point of the ball on the side marked B is th. wwn these. Consequently. th<> < - the air by friction are i greater on the former aide; and the relative Telocity between the i nd the ball is !•••; therefore the /»rr«ruir in the air is greater. iff to this fact the ball is pushed sidewiae in the direct A to*. 170 MECHANICS Solid moving through a Fluid. — As a solid moves through the air, there are forces that oppose its motion ; and many experiments have been performed to determine the connec- tion between this force and the velocity of the moving body. Newton proposed the law that the force varies as the square of the velocity ; but the accepted relation to-day is due to Duchemin : R = av2 + bv8, for speeds below 1400 feet per second, where R is the force, v the velocity, and a and b are constants. The forces acting on a solid moving through a fluid are the same as those it would experience if it were at rest and the fluid flowed past it in the opposite direction. If a board is placed obliquely across a current of a fluid, the pres- sure will be greater at the edge which is " up stream " than at the other, because the stream strikes it directly and then flows down along the board. There is thus a moment which turns the board directly across the current. Similarly, as a board or a piece of paper moves through a fluid, for instance the air, it turns so as to move broad face forward. This is illustrated by a sheet of paper or a leaf falling in air, by a flat shell falling through water, etc. Liquids and Gases As has been said several times, both liquids and gases are fluids; but liquids are distinguished by the fact that when contained in an open vessel they have a free upper surface in contact with the air, or if left to themselves they form drops inclosed in spherical surfaces, while gases completely occupy any space open to them ; liquids are comparatively incompres- sible, while gases can be easily compressed. In discussing, then, the properties of a liquid as distinct from a gas, its surface of separation from other bodies and its incompres- sibility form the features to be studied; while the corre- sponding properties of a gas as distinct from a liquid are its power to expand so as to fill any space and its great compres- sibility. CHAPTER VII LIQUIDS Compressibility of Liquids. — For many years it was thought that liquids were absolutely incompressible; but later it was shown that all liquids could have their volumes changed by the application of sufficiently great pressure. This is done by tin- piezometer, which has already been described. The liquid whose compression is to be studied is placed in a glass bulb provided with a fine-bored stem; this stem is open and contains an index which moves up or down as the volume of the liquid in the hull) is altered; the bulb is placed in piezometer, which is filled with some transparent liquid like water. As the piston is screwed in, the pressure increases, and Uoth the bulb and its contents are compressed. The me may be determined if the force acting on the pis- ton and its area are known. Owin^ t<> the compression of bulb itself the index would rise; but, if the contained liquid is in.. iv compressible than the bulb, the index . is lowered, the amount of the fall depending upon the difference in the compressibility of the glass bulb and the liquid. If the volume of the bulb and of eaeli unit length of its stem .•own, this total decrease in volume may be determined; and. if the compressibility «»f the glass in the bulb is known, lecrease in volume may be calculated because the « Imnge in pressure is known: and by adding it to the decrua> volume observed, the true decrease in volume of the liquid I'-termined. The strain, then, corn spending to t le- isured increase in pressure Ap is tin p in volume At; divided by the original \olinnr r. The Coefficient of 171 172 elasticity is then - or -. The "compressibility" of the v . liquid is the reciprocal of this, or — — • (For water at 8° C., the compressibility is about 0.000047, if the unit of pressure is "one atmosphere"; for ether at 10° C. it is not far from 0.000147 ; for ordinary alcohol at 14° C., 0.00010. At higher temperatures, the compressibility is much greater.) Similarly, liquids can be stretched. If a tube which is closed at one end and which is thoroughly cleaned inside, is partially filled with a liquid and then carefully inverted, the liquid will not run out, but will stick to the end of the tube. It is therefore under a stretching force owing to its weight; and experiments show that under these conditions its length is increased. (This experiment must be done in such a manner as to avoid the action of the atmospheric pivs sure, or this must be taken into account in the calculations.; Liquids at Rest Form of Free Surface. — All liquids with which we deal are under the influence of gravity ; and, therefore, if they are contained in large open vessels which they only partly fill and are at rest, their upper surfaces are horizontal, that is, are perpendicular to the force of grav- ity. For, if they were not, this force would have a component parallel to them, which would cause the liquids to flow. Similarly, if point of the surface of a liquid when the horizontal surface is disturbed, it is not horizontal. for instance by dropping in a stone, the liquid which is thus forced below the
horizontal level of the rest of the surface returns, and owing to its inertia continues to rise until it is above the level, thus mak- Fio. 32.— Forces acting at any x . . . 173 inur a "hump" or crest; this in turn sinks and makes a depression or " trough"; etc. These up-and-down motions affect the neighboring portions of the surface; and, as a result, what are called "waves" spread out from this centre iisturliance. Their energy comes from that of the origi- nal disturbance; and it is wasted away in friction between tin- moving parts of the liquid. These waves will be dis- cussed in detail later, but it should be remembered that they owe their origin, not to any compression of the liquid, but to the fact that under the force of gravity the surface of a liquid is naturally hori- zontal. If a liquid is contained in a vertical circular cylind.-r \\hich is spinning rabidly around its axis of figure, the surface is no longer horizontal ; in fact, it is a paraboloid of revolution; that is, lane section through the axis is a parabola. Th»- reason for this is clear. Consider any par- >f the liquid in the surface. It is under the i of two forces, that of gravity acting verti- do\vn ami that due to the reaction of the rest of ill.- liquid against which it presses and which act> j .«•! pendicularly outward from the sur- face. Th'- -in-far- must then have such a form iii- force of reaction and the force of grav- ity have a resultant in a hmi/ontal direction toward Fio.88.-F<MtMtli* at toy point on the tortae of * liquid when It U run- Ulned in a retae! rapidly r-.tatliur "t. a wrtir.il -niv for the actual acceleration of the parti.-!.- is in thi- dir- since it i- moving in a circle \\ith con-taut s\|-cd. If this statement is expressed mathematically, it leads at once to the fact that the section face is a parabola. Thrust; Centre of Pressure, etc. — If a liquid contained in Basel i- at rest, the pressure at any point depends upon two causes, atmospheric pressure and gra\ Written P, and it tin- \ntieal depth of the 1 in quest! -n In-low tli.- Mil-far,- is //. th.- d. \| the liquid being rf, the total pressure at the point is P + dgh. Thi> pressure is entirely independent of the shape or size of 174 MECHANICS the vessel, depending simply on the vertical distance between the point and the plane in which lies the upper surface. For, consider any point Q in a liquid at rest which is con- tained in a cylindrical vessel. The pressure is the same at all points in a horizontal plane through it, otherwise the liquid would flow, and the same is evidently true of a liquid contained in any other shaped vessel, e.g. a conical one. But the pressure at Q is P+dgh\ and therefore all other points in the horizontal plane through it have the same pressure, wherever they are. (The pressure at a point under the slope of the side wall in the vessel shown in the cut owes its value directly to the weight of the column of liquid vertically above it and to the component downward of the reaction of the sloping side wall against the thrust of the liquid.) Fro. 84. —Pressure at a point whose dis- tance below the free surface is h. The pressure at a point in the liquid increases uniformly as the point is taken lower and lower ; and therefore the mean pressure in the case when the vessel is a cylinder is the average of the pressure in the surface, P, and that at the bottom, P + dgH, where H is the vertical depth ; that is, it is p _j_ 1 dgH. The force on the bottom of the vessel, if it is horizontal and if its area is A, is (P-f dgH) A\ and the thrust on the side wall, if it is rectangular and vertical, is the average pressure multiplied by the area of the wall in contact with the liquid. If this area is A, the thrust is then (P + J dgH) A. Its point of application is found from the consideration that it is the resultant of a great number of parallel forces whose values increase uniformly from the surface down. In the FIG. 85. — A vessel with oblique walls. case of a rectangular wall, this " centre of pressure," as it is called, is at a distance of one third the depth of the liquid LIQUIDS 176 from the bottom. In the general case, when the wall is not rectangular, the total thrust is found by adding all the infinitesimal parallel forces which act on the elementary portions of the wall ; and the centre of pressure is found 1 > y expressing the fact that the moments of these infinitesimal forces around any axis must equal tin moment of tin- thrust when applied at the centre of pressure. These operations require, however, the use of the infinitesimal calculus. If a liquid at rest is contained in a vessel that has several vertical branches of different shapes and sizes, its upper surface is at the same horizontal level in them all, provided they are not of such small bore as to cause capil- lary phenomena. This is evident from the fact that the pressure at all points in a horizontal plane AB through the body of the vessel is the same — otherwise the liquid would flow; and therefore the free surface must be at the A... FIG. M. — Same liquid* In conntctln* tubes. AB and CD are horizontal plan*. same height above this level in all tin- branches. Similarly, the pressure is the same at all points of the liquid in the branches that lie in the same horizontal plane; e.g. in the 10 CD. Liquids in Connecting Tubes. — If two liqui do not mix are \|»la« ed in the same vessel, the denser will sink to the hot torn, because by so doing the potential energy becomes A heavier liquid may, however, rest upon a lighter one provided there is no jarring; but the equilibrium is unstable. 176 If a vertical U tube contains two liquids that do not mix, the levels of the upper surfaces of the two liquids in the two branches are not the same. The heavier of the two liquids will occupy the bottom of tlio tube and rise to a certain height in one of the arms, while the lighter one will stand to a greater height I in the other. The pressure at the fl surface of contact between the two I liquids is the same as at a point in '«! — -- the heavier liquid at the same hori- zontal level, from what has been said in the previous paragraph ; so, FIG. ST.- TWO liquids of different if the upper surface of the heavier liquid is at a vertical distance h^ above this level, and that of the lighter at a distance 7/2, P + djt/Aj = P + d^gliy where d1 and d2 are the densities of the heavier and lighter liquids respectively. Therefore, density which do not mix. . , . . d^h^dji^ and -^i = -,-?• This is, then, a method for the «2 "i determination of the specific gravity of one liquid with ref- erence to another; and, if the density of either is known, that of the other may be at once calculated. Barometer. — The pressure of the atmosphere is, as a rule, measured by balancing it against a column of mercury. The apparatus consists of a long, wide tube, which is closed at one end and which contains a column of mercury, but no air or other gas (except mercury vapor). The tube is placed in a vertical position; and either its open end dips into a basin of mercury or the tube is bent into the shape of the letter J. The space above the mercury may be entirely freed from gases by different means. (One is to hold the tube, closed end down, fill it with mercury, cover the open end with the finger, invert it carefully, and place it upright in a basin containing mercury with the open end under the LIQUIDS 177 surf; i (•«-.) The pressure on the surface open to the air holds tin mercury in the tube and is the same as that at a point at its level in the mercury in the tube; so, if the surface of the mercury in the tube stands at a height h above this outer u surface, the pressure due to the atmosphere equals dgh, where <1 is the density of the mercury at the temperature of the air, because there is no pressure on the top of the column. and t lie pressure at any point is that due to . ity alone, (h is observed to be about •in. when the pressure is measured at sea level; and so the tube must have at : this length.) It should be noted that this height is independent of the shape or H section of the tube, provided only that it is so wide as to avoid capillary action. age 188. If a liquid other than mercury were used, it would stand at a height as much greater than this as the density of mercury is greater than its own. (Thus, since the density of mercury is about 13.6 times that of water, liquid would be forced up in a barom- eter by ordinary atmospheric pressure to a iit 1 .').»> x 7«i . MI. ) The pressure of the M ;_n\eii by a barometer is dgh dynes per square centimetre on the ('.(J.S. system. Flo gg..^ . however, ordinarilv expressed in "cen- b*r< ..... . Mid dlphon forms. timetres of merciir\. that is, in terms ol a unit pressure equal to that due to a vertical column of mercury at 0° C., 1 cm. in height, when g has the value it possesses at sea level and !."> latitude. (See page 166.) Calling this value of the deiiMt\ ..f mercury dQ, and that '. //4V this unit pressure equals d^g^ dynes per square S.i. it the observed height of the barometer is AMES'S PHYSICS— 12 178 MECHANICS h cm., when the temperature is t° C. and the latitude is Z, we may write the value of the pressure in dynes per square centimetre, P = dgh ; and its value in " centimetres of mer- cury," therefore, is p = This is a height ; and so we speak of a barometric " pressure of 76 cm.," meaning " 76 cm. of mercury." The relation between g and g^ is known (see page 131) ; and that between d and dQ is also known. For, as will be shown presently, if t is the temperature on the Centigrade scale, d0 = d(\ + 0.00018180- Therefore, if the temperature and latitude at the place of observation are known, the pressure in terms of centimetres of mercury may be at once calculated from the observed barometric height. In the formula, h is the height in centimetres. Sometimes, how- ever, the reading is made on a divided scale which is correct at 0° C. ; and in this cas
e the readings must be corrected in order to give A. If the scale is made of such a material that each centimetre increases in length an amount a cm. for each degree rise in temperature, two divisions which are 1 cm. apart at 0° are a distance (l+a"0 cm. apart at t° C. Conse- quently, if the observed reading is H scale di- visions, the height is H (\ + at) cm. ; and therefore h=H(\+af). Open Manometer. — In a similar manner, the pressure in a gas inclosed in any vessel can be measured. Let a bent tube containing some liquid be joined to If there is a difference in pressure FIG. 89. — Open manometer. the vessel as shown. LIQUIDS 17l» l)ct \veen the gas inside the vessel and the air outside, there will be a difference in level of the columns of the. liquid in tin- two arms. Call this difference A, and the density of the liquid d. Then, if the level is lower in the arm at i ached to the vessel, the pressure in the gas inside is P -f- dgh. (If the level is higher in this arm, the pressure if I' — dgh.) This instrument is called an "open manoni- The pressure, as here expressed, is in dynes per square centimetre, if the C. G. S. system is used ; its value in centimetres of mercury can be deduced, as has just been ained for a barometer. Floating Bodies. — There is one application of Archimedes' principle to liquids that is of special interest. It is to the case of a body floating on the surface of a liquid. If a solid of less density than a liquid is immersed in it and allowed to move, it will rise to the surface, but will come to a position quilibrium when, as it floats, it displaces a volume of the liquid whose weight equals its own; for, under these con- ditions, tin- upward buoyant force due to the liquid equals tin- downward weight of the solid. The line of action of the fon IK r is vertically through the centre of gravity of the di-jilaccd liquid: that of the latter, vertically through the <«ntre of gravity of the floating body. Therefore, when there is equilibrium, these two centres of gravity must lie in thcsamr vertical line; otherwise there would he a moment which would make tin- body turn around a hori/ontal axis. This equilibrium is stable if, when the body is tipped slightly, the resulting moment is in such a direction as to turn it back a.^aiu : it is unstable if this moment, under similar < onditions, is such as to tip it over. Thus, a board :HLT on its side is in stable equilibrium : but, if made to float upright, its equilibrium is unstable. Osmosis and Osmotic Pressure. — As one substance dissolves in another, it breaks up into small particles which diffuse through the solvent. These particles by their presence affect 180 MECHANICS its molecular forces, as is shown by many facts. One of these may be mentioned here. It is found that certain solid bodies allow some liquids to pass through them, but not others (see page 141) ; and it is possible to make a membrane that will permit the molecules of a liquid to pass through perfectly free, but will not permit the passage of any dis- solved molecules. Such a membrane is called " semi-perme- able." If now a solution is placed in a wide tube closed at one end with such a membrane, and is supported upright in a large vessel containing the pure liquid, which can pass through the membrane, it is observed that the levels of the liquid in the tube and in the outer vessel are not the same, as they would be if the membrane were absent; the height of the level in the tube is the greater. There is therefore on the two sides of the membrane a difference in hydrostatic pressure which is maintained by some force due to the difference in the conditions on these two sides. The molecules of the solvent can sure: inner tube contains pass freely through the membrane, and solution ; outer vessel, pure they Continue to do SO Until, when 60 ui- golvent. J librium is reached, the hydrostatic pres- sure prevents any further passage. (Of course molecules may still continue to pass through ; but, if they do, an equal number pass out.) There is therefore a difference between the pure solvent and the solution. If the density of the solvent is d, that of the solution does not differ much from this ; and if the difference in level of the two free surfaces is h, the hydrostatic pressure dgh measures this tendency of the pure solvent to pass through the membrane into the solution, that is, it measures the effect the dissolved sub- stance has upon the solvent in affecting its molecular forces. This passage of a liquid through a membrane or Fio. 90. — Osmotic pres- ., . . , . LKiflDS 181 ml porous partition is called "osmosis," as has been already stated ; and the above pressure, dgh, is said to measure the "osmotic pressure" of the solution. Experiments show that, as the solution is made more and more concentrated, the osmotic pressure increases. If m grams of a substance are dissolved in ml grams of a solvent, the ratio - - is called tin- l concentration"; and in certain simple cases the osmotic pressure varies directly as this, while in others it varies more rapidly. This law is the same as that for a gas ; viz., the pressure varies directly as the density of the gas, i.e. its concentration ; but if the gas changes its character by its molecules dissociating into parts, then the pressure varies more rapidly than the density. (See page 200.) We are thus led to believe that this abnormal osmotic pressure is due to a dissociation of the dissolved molecules into simpler parts. Liquids in Motion Efflux of Liquids. — If a liquid is contained in a large 1 which has thin walls, and if a small opening is made in cither the bottom or side, the liquid will escape. This motion is called "efflux" or "effusion." The velocity of escape may be at once calculated, because, since the wall is assumed to be thin, there is no friction, and since the opening is small, we may neglect any motion of the liquid except that of the escaping stream. Thus the phenomenon is the same as if a drop of the liquid disappeared off the surface and reappeared lower down with a certain speed. ic opening is at a depth // below the surface, and if the speed of ctllux is x, each drop of mass m loses an amount of potential energy mgh and gains an amount of kinetic energy % mt. Therefore these two quantities are equal, or * = V2 gh. This is, of course, the formula for a particle falling freely toward the earth; and therefore, if the jet 182 MECHANICS were turned upward, it would rise to the height of the level of the liquid in the vessel, were it not for the opposing action of the air. (This formula was first deduced by Torricelli.) The pressure in the liquid at the opening is P + dgh, while that on the outside is P ; so the difference in pressure caus- ing the flow is dgh. Calling this jo, the speed of efflux may be expressed in terms of it, viz., &= 2^-, or «=•%/—£• The direction of the jet depends, of course, upon the position of the opening ; and, unless this is on the bottom, the path of the jet is a parabola. a % a Other cases of motions of liquids will be discussed in Chapter IX. Capillarity and Surface Tension Fundamental Principle. — If a liquid is left to itself, free from external forces, it assumes the shape of a sphere ; and this is approximately the condition with falling drops of rain or of molten metal (like shot) and with soap bubbles. It is rigorously so if a small quantity of a liquid is immersed in another liquid of the same density with which it does not mix — Archimedes' principle. Of all solid geometrical fig- ures having the same volume the sphere has the least sur- face ; so this fundamental property of a liquid surface is that it becomes as small as it can. Thus the surface of a drop contracts until the resulting pressure in the liquid balances the contracting force ; it requires a force to blow a soap bubble, and, if one is left attached to the pipe and the lips are removed, it will contract. Again, if a glass plate is dipped in water (or any solid is dipped in any liquid that wets it), and is then raised slightly, the surface of the liquid near the plate is curved with the concavity upward. It has contracted from a rectangular shape, in doing which some of the liquid is raised above the horizontal surface ; and the liquid comes to rest when the weight of this elevated portion LIQUIDS 183 balances tin- contracting force of the surface. Similarly, if a glass plate is dipped in mercury (or any solid is dipped in any liquid that does not wet it), the surface of the liquid near the plate is curved s<> as to be convex. Since the liquid does not wet the plate, its surface continues around hel< >\v the plate ; and, as it contracts, it rounds off the corners, thus leaving a free space which the force of gravity would cause the liquid to till were it not for the contracting force of the surface. There is equilib- rium, then, when these two forces balance each other. This phenomenon in the neighbor- Fio. 91. -Capillary action when a solid plate is dipped in a dipping ill a liquid liquid: (1) when the liquid wets the solid ; (2) when the liquid is most marked doe8not when the former is a tube with a small bore. If such a tube is dipped in a liquid that wets it and is then raised slightly so as to leave a liquid surface on the inner and outer walls, the whole liquid surface includes that on the walls and that "f the liquid in which the tube dips. So considering the liquid surface inside the tube, it has the appearance of the inside of the finger of A glove. Tin- liquid is then raised in the tube, owing to the contraction of the surface; ami equi- librium is reached when this emit ract MILT force is balanced by the effect of gravity on the raised portion of the liquid. Similarly, when a glass tube is dipped in mercury, the surface in t he t Illic is de[i\| rssrd. Surface Tension. There is, therefore, a force produced by a liquid surface; and the simplest manner of defining it is to consider the force acting across
a line of unit length in the surface. This is a molecular force and is evidently du • to the fa.-t that a molecule in ih- a liquid is in a 184 MECHANICS different condition from one in the interior. For a surface of a given liquid in contact with a definite medium, then, no matter whether its area is large or small, this force is a con- stant quantity ; and unless it is stated otherwise, the sur- rounding medium is always understood to be ordinary air. The force acting across a line of unit length of the surface of a given liquid in contact with a definite medium is called its surface tension with reference to the medium, and has the symbol T. A simple direct experiment showing the amount of this tension is to construct a rectangular frame of wire, one side of which is movable, and to make a film of liquid _• fill the area. (This may be done by dip- ping the frame for a moment into soapy water and then removing it.) It will be found that a force must be exerted on the movable wire to keep the film from contracting. Calling the length of the movable wire Z, this force equals 2 Tl, because the film has two surfaces. If under the action of the force the wire is moved so as to make the film larger, work is done. If the distance the wire moves is called x, this work is 2 Tlx ; but the increase in area on the two sides is 2 Ix, and therefore the work done per unit increase of area is T. In other words, to increase the surface of a liquid by one unit of area requires an amount of work equal to T\ or, the potential energy of a surface of area A is numeric- ally equal to TA. As the surface of a liquid is increased, it is not stretched, as a rubber bag or toy balloon is, but new surface is formed by molecules coining up into the surface from the interior ; the new and the old surfaces being identically alike. (If in the experiment just described the movable wire be drawn out too far, or if a soap bubble be blown up too large, a point is reached when the two surfaces of the film come so close together that, if their area is FIG. 92. — A soap-film stretched on a wire frame, one side of which is mov- able. LIQUIDS 185 further increased, the interior molecules are no longer in the lition in which they are when the film is thicker; and all the properties of the film are changed.) Connection between Pressure and Surface Tension. — Let us n<>\\ examine imuv closely the illustrations of the contracting force of a liquid surface that were given above. A spher- ic al drop may be considered as made up of two halves touch- ing at an equatorial section ; they are held together by the ion in the surface, acting across the equator; and there is a reaction, as shown by the pressure in the liquid, acting over tin- equatorial section in which the two halves touch. If r is the radius of the sphere, the length of the equator is 2?rr, and the force of contraction due to the surface tension across it is therefore T2irr; the area of the equatorial section is Trr2, and, if p is tin- pressure in the drop, the force of reac- <»ver this section is ptrr. Since the drop is supposed to have ceased to contract, these two forces must be equal, or 2 T T - 77 /• : i.e. p = — — . This pressure is felt, of course, throughout the drop, and is in addition to the pressures due to the atmosphere or to gravity. The formula states that, if a liquid whose surface tension is T has a portion of its sur- face (or all of it) curved in the form of a >phere of radius r, there is a pressure - - toward the centre of curvature, ami •_' '/' i IK- surface is at rest, there must be an external pressure •' '/' — acting on it in order to hold it in equilibrium. Thus, if a soap bubble is hlnwn out, the pressure required .) y 2 '/' r the iras h,. side, in excess of that outside, is - - + - rl rl where r, and /-._, are the radii of the interior and exterior sur- faces. Since the til in is very thin, the radii may be considered '!, and the pressure of the air inside must be greater than l I y 'hat «"it bv an amount — . If 1»\ BK • pipe with two 186 R2 openings two bubbles of radii Rl and 722 are blown simulta- neously and are then left connected, the pressure — - will be 1 4 T 4 T required in one in order to maintain equilibrium, and — in the other. So there will not be equilibrium. If & > 7L, 4 T 4 T -— < — — ; therefore the bubble with the larger radius re- RI RI quires the less pressure, and it will become larger while the other contracts. It is seen that the smaller the drop or the bubble the greater is the pressure that is required to maintain it in equilibrium. In the limit, as the radius r approaches zero, the correspond- ing pressure p becomes infinitely great ; it is therefore im- possible to form a drop or a bubble with an infinitely small radius. Thus, drops of rain or dew always form around nuclei or points ; and bubbles, like those seen in a boiling liquid, start from points or small bubbles of dissolved gases. (If the nucleus is electrically charged, a drop can be formed with a diameter smaller than would be possible with one that is uncharged.) Capillary Phenomena. — Having deduced this value for the pressure corresponding to a certain radius, we can at once find the connection between the height to which a liquid is raised in a tube and the dimensions of the tube. Let us consider the case of a tube that is wet by the liquid, and let its cross section ;i( the top of the column of the liquid be a circle of radius r ; then the shape of this upper surface of the liquid is that of a hemisphere of radius r. The pressure in the liquid at a point inside the tube PIG. 98.— Liquid in a capillary tube On a level with the horizontal sur- whlch It wets, e.g. water In glassr.reatly magnified ) ™CQ outside is the same as that at LIQUIDS 187 ;i point in this surface, viz., P, the atmospheric pressure : lnit it is due to three independent causes : 1, the atmospheric pressure P on the top of the column ; 2, the vertical height of tin- column (this pressure amounts to dgh)\ 3, the contracting of the curved surface (this is equivalent to a pressure _' y — , and opposes the other two). Thus the total pressure is - T P + — . This, as has just been said, must be equal O m -o rn to P; and therefore dgh = - — or h = - • r dgr (If h is the distance from tin- bottom of the curved surface t<» the free horizontal surface, the above formula is not cor- . since allowance must be made for the portion of liquid above the bottom of the curved surface. It may be shown 1 ^ T without (litliculty that the correct formula is A + -r=- o hut, since as a rule r is small in comparison with A, the approximate formula is generally used.) It is thus seen that the smaller the radius of the tube the greater is //. ..r tin- higher does the liquid rise. (Phenomena dealing with tee tension are usually called "capillary," because the bore of tubes which show marked effects as just describe! is ble with the size of a hair, the Latin name fur which is capillus.) Illustrations of this action are given by a lump I_,MI- win n dipped in water, a blotting paper absorbing a dmp «,f ink, etc. It should l>e noted particularly that, in the above discus- ui'l f.. i -mn la, no reference is made to the cross section of the tube except at the point where the top of the column «.f liquid comes ; so the tul>e elsewhere can have any size. The liquid will nut rise in the tuhe of it*, If unless the bore is 1 throughout and the inner wall is \vet witli the liquid. if the liquid is sucked up in the tube and then allowed til, it will mine to rest at the height given by the .ula. 188 In a perfectly similar manner it may be shown that the depression of the surface of mercury in a glass tube (or of any liquid in a tube which it does not wet) is given by the same formula. This formula can be used to measure the surface tension of a liquid ; for 7&, d, g, and r can all be measured with accuracy. There are, however, other 'methods which in some respects are more satisfactory. The values of the FIG. 94. -Capillary action be- surface tension of a few liquids in con- tween mercury and glass. ta(jt ^^ ^ jn dynes per centimetre, at about 30° C., are given in the following table: Water. Mercury Alcohol TABLE 72.8 513. 22. Olive oil Turpentine Petroleum 34.6 28.8 29.7 Another mode of considering the curvature of a liquid surface near a solid wall is as follows : imagine the liquid to have a horizontal surface clear up to the wall; a particle of the liquid surface near the wall will be under the action of three forces (apart from gravity), viz. : 1, one owing to the molec- ular forces of the rest of the liquid ; it is represented in the cut by F\ 2, one owing to the molecular forces between the liquid and the solid; provided the latter is wet by the former, it is represented by Fl ; 3, one owing to the forces between the upper medium — the air, in general — and the particle of liquid; this is neglected here. The resultant of these is represented by R; and therefore the surface of the liquid must be so curved as to be at right angles to its direction. FIG. 95. — Forces acting on a par- ticle of a liquid in its surface at a point near a solid wall. If the surface of the liquid is not spherical, the formula for the pres- sure may be shown tobe/>= T( 1 ), where Rl and R2 are the so- \Rl R.2/ called principal radii of curvature. Thus, if the surface is a cylinder of LIQUIDS 189 T radius r, Rl = 0 and R = r ; so p = — • This formula may be proved di- rectly by considering a cylindrical drop instead of a spherical one as done on page 185. It may be shown further that, if a cylindrical por- i«f liquid (as, for instance, a jet of water issuing under considerable pressure) has a small radius, it breaks up into spherical drops. The ex- act statement is that if / is the length of the cylinder and r its radius, the surface is unstable when I > 2 TIT. Effect of Surface Impurities. — If a liquid is homogeneous, the surface tension of all points of its surface m
ust be same ; and therefore it is impossible to blow a bubble of such a liquid or to stretch a film of it on wire frames ; for it is evi- dent that, if a vertical film exists, the tension in the upper portions of the surface must be greater than in the 1< i use the former have to support the weight of the latter, and such inequalities in tension cannot exU in a homogene- ous surface. If bubbles or films are to be formed, thm. tin- liquid >urface must be made heterogeneous, e.g. soapy water be used. In these films the upper portions are observed to be purer than tin; lo\\»-r ones, showing that a pure film has a greater surface tension than a contaminated or hetero- geneous one. If iiiimit.' portions of camphor ar 1 upon a clfan surface of water, they are seen t.. .lari '• iW\\ar«l \\i\\i a m. i H stopped by an extremely small trace of oil on the surface. A piece of camplmr is never exactly symmetrical and there- fore dissolves in the water more My at one point than at Jg. lluton the surface :.- makes < n elsewhere ; and so the face < drawn away 1-y th.- he opposit .i .IT-..J. it spreads the surface under the action of the surf art* tensions of the various re are three of these forces acting on any portion of the edge of the drop: the tension 7, between the water and the air, T9 190 MECHANICS between the oil and the air, Tj between the oil and the water. The oil spreads because 7'3 is greater than the resultant of T2 and Tr This thin layer of oil will prevent the dissolving of the camphor, and its motions will cease. If a drop of alcohol is poured on a glass plate that is covered with a thin layer of water, the tension of the surface of the solution of alcohol is so much less than that of the pure water that the latter surface con- tracts, tearing apart the former and leaving the glass quite dry. The surface tension of a liquid varies with the tempera- ture, decreasing as the latter rises. This may be shown by many obvious experiments. Ripples ; Effect of Oil upon Waves. — It requires work to increase the area of a liquid surface, and so if the surface is increased slightly by some disturbance, there is a force of restitution, and waves will be propagated over the surface, which are quite distinct from those due to gravity. These are due to surface tension, and the crests come so close to- gether that they are called " ripples." These may be seen if a fine wire in a vertical position and dipping in a liquid is moved sidewise. They will be discussed later. As the wind blows over a liquid surface it will soon magnify ripples into regular gravitational waves, and it is evident that the less surface there is exposed to this action of the wind so much the less is its effect. If a thin layer of oil is spread over a liquid surface, the wind blowing over it will tend to gather the oil in the same regions where it would heap up the water ; this excess of oil over one region produces a scarcity of it over others, and the surface tension in the latter is therefore greater than in the former, and so it opposes the action of the wind in causing the water to form waves. CHAPTER VIII GASES General Properties Dalton's Law. — The fundamental properties of a gas as (list in. i from a liquid are explained by assuming that its molecules are so far apart and have such freedom of motion that it distributes itself uniformly throughout a vessel of any shape and may be compressed with ease into a much small -r 1 he aetnal space occupied by the particles of the gas must l>e «-\tremely small, because if two different gases are inclosed in the same vessel, they mix uniformly, and each hehaves almost perfectly as if the other were al^ent : the pressure at any point (or on the walls) is the sum of the pressures \\hieh each gas by itself would produce. This is km.un as Dal ton's Law of (iases. ha\in^ heen discovered by him in ISM-J; hut careful experiments show that it is not utely exact. Effusion. — If any gas is inclosed in a vessel that is sur- rounded l>y a d i fferent gas, and is allowed to escape into the ior all practical purposes as if it were escaping ;i vacuum, for each gas acts independently of the other, and the only difference enters from the fact that a longer time is required for a gas to diffuse into another gas than ; ilmte itself in an empty space. If the in the vessel f-.-m whieh the gas is escaping is small, and if \alls are thin, the same formula applies as for the efllux i.piid. vi/.. f.y±£, where * is the sj d of the 191 I (I "2 MECHANICS ing gas, d is its density, and p is its pressure. (JP is really the difference in pressure of the escaping gas, inside and out ; but if the outside space is very large compared with the vessel, the pressure outside may be neglected.) While the one gas is escaping, the surrounding gas is entering, and the two processes go on independently. This relation between the velocity of escape of a gas and its density sug- gested to Bunsen a method for determining the relative den- sities of two gases, because the quantity of gas that escapes in a given time can be measured with ease. If the openings through which the gas escapes are extremely fine, a different law is obeyed, which was investigated by Graham. Sensitive Flame. — If a gas escapes through a tube with a small opening under considerable pressure, it forms a "jet," which preserves its identity for some distance without diffus- ing into the surrounding gas. The jet is inclosed, as it were, in an envelope which prevents the two gases from mingling. This envelope is made up of particles of the escaping gas which are given a rotation owing to their rapid motion over the particles of the surrounding gas. If this envelope can be broken into, the two gases mix and the jet loses its character. If a jet of illuminating gas is formed in this way and is lighted, it forms a tall, narrow flame nearly cylindrical ; but if a whistle is blown, or a bunch of keys rattled near by, the jet breaks down into an ordinary fan-shaped gas flame. This is owing to the disturbances sent out by the sounding body in the foim of short waves or pulses in the air, which disrupt the gaseous envelope of the flame. For this reason a jet like this is called a "sensitive flame." It is used to study the many wave phenomena in connection witli Sound. Compressibility of Gases. — If a gas is inclosed in a cylin- der provided with a movable piston, this may be forced in or out, and the corresponding values of the pressure and volume may be measured. Or, if a tube is bent in the form of a let- a ASKS 193 ter J. and its lower end sealed, it maybe placed vertically, and some liquid like mereiiry may he poured in at tin- open end, thus trapping a quantity • if the air (or other gas) in the shorter arm. As nmre and more liquid is poured in, the volume and -me of the ^as vary, and hoth may be measured. (If h is the vertical dis- tuiiee he t ween the two sin-faces of the liquid, d its density, and P the atmospheric pressure, the ware of the gas is P + ,ljh.) Tin- ( -oetlicient of elastic- ity . - explained before, . where Av is the change in the volume !• produced by a change in -HIV equal to A/>. K\- mentS prove that this t, it depends greatly upon how rapidly the com- i expansion) is tor observing Boyle's f air la traji\|>«-«\| In tho meed. The reason i>> that, if the volume of a ga made smaller suddenh. its temperature is rai^-d. and this produccsan increase in pressure quite independent •hat which would accompany the same decrease in volun,,. if hn.n.rht ah., ut slowly. Two extreme cases are \ eoiisiden-d : «,Me, when the decrease in volume 1\ that there is no change in t empei'at itre : the other, \\hcn the change is made as rapidly as possible, so AMES'S PHYSICS — 13 194 MECHANICS that there is no time for the temperature effect to become weakened by diffusion. The former is called an "isother- mal" change ; the latter, an "adiabatic" one. Boyle's Law and its Consequences Boyle's Law. — The first philosopher to study experimen- tally the exact properties of gases was Robert Boyle, who, in 1660, carried out a most careful series of experiments on the variation in the volume of a gas as its pressure is changed. He discovered that, to a high degree of approximation, the pressure and volume of a given quantity of a gas are con- nected by the following relation : the product of the values of the pressure and volume remains constant during all changes, provided the temperature is unchanged. In sym- bols, that is, pv = constant, if the temperature is constant, where p is the pressure and v the corresponding volume of a given quantity of the gas. This means that if the volume is decreased to one half its value, the pressure is doubled, etc. This is known as "Boyle's Law for a Gas." Naturally, if there is twice the mass of the gas at the same pressure, its volume is twice as great, and writing m for the mass of the gas and k as a constant factor of proportionality, Boyle's law becomes pv = km^ or, writing as usual, d for the density, p = kd. k is, then, a constant for a given kind of gas at a definite temperature ; if either the gas or the temperature is changed, k takes a different value. As stated above, Boyle's law is not exact except for small variations in pressure. If the pressure is increased greatly, the product pv, instead of remaining constant, increases also. This fact was recognized by Boyle himself and has been con- firmed by more recent investigators, notably Regnault and Amagat. GAS i-:.- 195 Dalton's law may l>e expressed quite simply in terms of law. If several gases are inclosed separately in dif- :it vessels which have the same volume, but are at the same temperature, let their pressures be pv pv etc.; then, if tln-se gases are all put in one of the vessels, the pressure willbe /»=/»,+/»,+, etc., = y/i + v/2+,etc. Coefficients of Elasticity. --The value of the isothermal coefficient of elasticity may also be expressed in a simple t<»rm, because Boyle's law applies t
o the changes. Let p and v In- the initial pressure and volume of a gas; then, if tin- temperature is kept constant and the pressure is increased top + AJP, the corresponding volume v — A v must be such that pv = (p + A/>) (v - A »•) = pv + vA/> — p\v — A/>Av. Therefore, vA/> - p&v - A/>Ay =0. l)i it if the changes are extremely small, the last product may be neglected, and hence vA/> — />At> =0, or /• . ' =/>. Ao Ac Hut A/' is the dian^ein pressure corresponding to the change in volume Ai\ and the eoeiVicient of elasticity therefore equals />. S.» tin- isMtli.nnal coellicient of elasticity of a gas when under a pressure p numerically equals p. A ill l»e shown later tlmi ' lie ro.-mciaot "f ••l;i-t i.-ity for any gasisth- E the pmnire by a oomtanl whote ralna is aboai l.ll lie gases hydrogen, oxygen, ami nitm^ni, hut is different • Isothermal of a Gas. — !•«•> 1< s law may be expressed graj'li ieally li\ u^iiiLT ftXCB "' pr'-s^un- and vulnme as li,: done in j'lvvinii- . < e arl>itrarily some point /' in the plane of tin ;lir.,u^li il a line /Vt> parallel 100 to the axis of pressure; the corresponding pressure is QP, and the volume is OQ. To represent Boyle's law, a series of points must be chosen such that the products of the value of their pressures and volumes equal the product of QP and OQ. These points make up a curve which is called an "equi- lateral hyperbola," and it is shown in the cut. VOLUMES FIG. 98. — The curve Is an equilateral hyperbola, the equation for which is pv = constant. The formula pv = constant is known as the "equation of condition " for an ideal gas at constant temperature, or as the equation of an "isothermal" for a gas ; and, as has been said, it is only approximately true for an actual gas. Other formulae have been proposed which apply more exactly to ordinary gases over wider ranges of pressure. The most satisfactory of these is due to van der Waals, and has the form — constant. ( P + In this p and v have their usual mean- ing, and a and b are constant quantities for any one gas. This equation agrees fairly well with experimental results, when a gas is compressed from its ordinary condition until it is a liquid, as is explained in more advanced text- books. (See Edser, HEAT.) Closed Manometer. — A convenient method for the measurement of high pressures is afforded by Boyle's law. Some gas, such as air, is trapped in a closed tube by means of mercury ; its FIG. 99. — Closed manometer. QAStiS 197 volume is mca>urcd under atmospheric pressure; the pres- to IK.- measured is then applied to tin- mercurx , thus compressing the confined ^as, and the resulting volume is measured. Tin- ratio of these two volumes equals the reciprocal of the pressure expressed in "atmosphei This instrument is called a "closed manometer." (Such an instrument is used often in connection with a piezoni' in order to measure the pressure.) Kinetic Theory of Gases Fundamental Phenomena ; Temperature. - - The pressure that a e/as exerts on the walls of the vessel containing it is at once explained if we assume that it consists of a great number of minute particles which are in rapid motion. As any one particle strikes the wall, it has its momentum per- pendicular to the wall reversed, and therefore it exerts an impulse on it. The total force on the wall is the change in momentum produced in a unit of time; and the pressure is the force per unit area. It one portion of the wall is movable, we can imagine it yielding to these impulses, provided the external force is not sufficient to withstand the bombardment; but as it evident that the linear velocity of any particle rebniindin^ at that instant is less than if the wall did not move; and so the kinetic energy of translation of the par- ticles of the gas is decreased while wm-k is done by the in overcoming the external force. Similarly, if the movable portion of the wall is forced in. work is done on the gas; and it is evident that the linear velocity of a particle rehnundini: iiat instant is increased, and BO the kinetic energy of translation of the particles of the LTas is increased. K\ ; mints on .! tnal gases show that if one is allowed to expand its temper at a i, decreases, while if it is conipn ^rd, its temperature increases. Thus it is seen 198 MECHANICS that the temperature of a gas varies directly as the kinetic energy of translation .of its particles. Pressure. — We know nothing about the actual size or shape of a molecule ; but we can prove that, if we had inclosed in a vessel with rigid walls a great number of small, perfectly elastic spheres, moving at random but with great speed, this collection of particles would have many properties similar to those of a gas. For ease of calculation, let us assume that the vessel is a rectangular one, having edges of length #, 6, and c. At any instant a definite par- ticle has a certain velocity, but owing to impacts with other particles and with the walls, this changes frequently, and this is true of all the particles ; so, apparently, there is no regularity. But if things are in a steady state, there is a certain unvarying proportion of the particles — not the same individual ones, however — that have a given component velocity parallel to any one edge of the rectangle. Let N be the number of particles per unit volume that have the component velocity v parallel to the edge whose length is a; then the total number in the vessel that have this com- ponent velocity is Nabc. If each of these particles has the mass TW, the momentum of each parallel to the edge referred to is mv ; and therefore as each strikes the wall at the end, and its velocity is changed from v to — v, its momentum is changed from mv to — mv, or by an amount 2 mv. The time taken for a particle with the velocity v to pass from this wall across to the other end and return is - — (This is the time taken for the effect of the particle to be again felt at the wall, if, instead of moving over the whole distance and back again, it impinges on another particle and so hands on its momentum.) Therefore the number of impacts each par- ticle makes on the wall in a unit of time is - — Since at each impact of each particle there is a change in momentum •> GASES 199 . the change in momentum at the wall owing to one par- tiele during a unit of time is — • 2mv or ; and, as there a are Nabc particles having this component velocity v, the change in momentum in a unit of time, owing to the particles whose component velocity is v, is Nabc • — , or NmiPbc. This is, thru, the force on the end wall owing to the above particles; and since the area of this wall is fo, the pressure is NmiP. The total pressure is found by adding up similar terms for the particles having other velocities. The number of par- ticles per unit volume, however, which have the component velocity — v must also equal JVnn the principle of probabili- and so the number which have the same value v2 is 2N. Therefore, writing vv v2, etc., for different velocities and Nv NT etc., for corresponding numbers per unit volume, otal pressure is (2 N}v* + 2 .Vsy2a + etc.) m. Further, the mean value of ^ 2 JVy^ -f 2 JV>,2 + etc. (See page 30.) But 2 JVj + 2 AJ + etc. is the totoJ number nf particles in a unit volume; call its value N and write for the mean value of v2, v2. Then the total pressure equals mN\?. The tntal velocity V of any particle expressed in terms of its components parallel to three directions at ri^ht lc,s to each other, u, v, «\ is given by F3 = u2 4- v2 + «• l'>ut, since the nun ion of the particles is a perfectly random t he m. MII \ alaes « -f //-, »•-. and w2 must be equal ; so \\ i it - the mean value of F, P^-Sv2: and the tin d : for the pivssiire owiii'^ to the {.articles 18 p = JmA^M. Since tVis the t..tal niii: particles in a unit volume. and since each particle has a mass m, the density, d, or the 200 MECHANICS mass in a unit volume, is 7HJV; hence the pressure may be written i This states that, if the mean kinetic energy of translation of the particles does not change, the pressure varies directly as the density. This is Boyle's law, assuming that the tem- perature of a gas corresponds to the mean kinetic energy of translation of its particles. If this formula can be applied to an actual gas, the mean squared velocity of its molecules may be at once calculated, because V2 = —£, and the density of a gas at a certain pres- sure may be determined by experiment. The density of a gas varies with its temperature as well as with the pressure, and so does therefore V2. At the temperature of melting ice, V for hydrogen is calculated to be 1843 metres per second ; and for carbonic acid gas, 392 metres per second. At the temperature of boiling water, each of these is 1\| greater. Avogadro's Hypothesis. — Referring to the previous for- mula for the pressure, viz., p = ^mNV*, it is seen that, if there are several sets of particles inclosed in the same space, and if we can assume that they act independently of one another, the total pressure is p = J (m1N1 Vf + m27V"2 F22 -f etc.). It may be shown that, if two or more sets of particles are in equilibrium together, their mean kinetic energies of transla- tion are equal; hence, in this case, <m1 V£ = m^Vf— etc., .and therefore p = % (JYi + N% -f- etc.) m F2, showing that the pres- sure depends upon the total number of particles per unit volume, not upon their masses. The same statement is assumed to be true of gases, and is equivalent to " Avoga- dro's hypothesis" (page 201) ; and no known fact contradicts it, provided the gas is not too dense. It often happens that when a complex gas is raised to a high temperature, the pressure increases abnormally, which always corresponds to a dissociation of the molecules of the gas into simpler parts ; GASES 201 tin extent of the dissociation is calculated from measureni. of tli> -sinning tin- truth of the above formula. Again, if tin- re are two sets of particles that have the same -UK- and the same mean kinetic
energy of translation. and »ir;« Hows at once that ^ = N^i that is, that the two sets have the same number of particles in eaeh unit voln This statement when applied to gases would be: in two gases at the same pressure and temperature there are the same number of molecules in each cubic eentimetre. This •• A voLcadro's hypothesis," having been proposed thifl Italian chemist in isll.and it serves as the basis of lard methods lor the determination of -molecular Jits" in chemistry. The "molecular weight" of a gas number that is proportional to the mass of one of its rules. Thus, it' J/j and M., are the molecular weights s, W = — • Then, since ^ = NT under the above M., in., condition, the d. -nsitn -s of the two gases, d^m^N^ d^=in^NT :n the same ratio as the masses of tlie individual parti- 1. if an arbitrary number is assigned as the molecular weight of one gas, that of the oth.-r is found 1>\ mrasurin^ . of tin- drnsiti.-s of tin- two gases at the same tem- pera; d preavuie, and multiplying by the arbitrary numl.er. For ~1 - ^J - i nr M - M ''' ' c ~ accepted system of molecular - is based _r the nnmln < It may he I hy\|»«»tli«-si.s ix ii, .1 t rue if a gag •mpivxsrd. because, a^ has lir.-ii said in sjn-a! •l-eat. the i;a -aed according: t.. this law. and x,, then- dh-r 202 MECHANICS number of molecules in a unit volume than is required by Avogadro's hypothesis. But, if the pressure is small, there are no known facts contrary to it.) Viscosity and Diffusion. — As has been shown before, we can explain the viscosity of a gas and the diffusion of one gas into another, and we shall soon see that we can explain the method by which a gas " conducts heat " by assuming that its particles have such freedom of motion that they can move about uninfluenced by other particles except when they come very close together, i.e. when they have what may be called an "encounter." In the interval of time between two encounters the particle is moving in a straight path with a constant speed ; the length of this path is called the "free path," and its average value for all the particles is called the " mean free path " of the gas. When two par- ticles have an encounter, their centres come within a certain distance of each other and then separate ; one half of this minimum distance is called the "radius of the particle." Similarly, if we consider a set of minute elastic spherical particles, we can explain its viscosity and the manner in which any increase in the kinetic energy of one portion is distributed throughout the whole set ; and if we have two sets of such particles, we can explain the diffusion of one into the other. Further, we can calculate what the force of viscosity, the rate of distribution of kinetic energy, i.e. of conductivity of "heat," and the rate of diffusion of such sets of spheres are, expressing these quantities in terms of the mass of a particle, its mean energy, its mean free path, its radius, and the number of spheres in a unit volume. Then, if we assume that an actual gas behaves approximately like a set of spheres, we can measure its pressure and density at a given temperature, its viscosity and conductivity for heat, and its rate of diffusion, and, by comparison with the me- chanical formulae deduced for a set of spheres, obtain approxi- mate values for the various properties of a gas molecule. A OASES of these may be mentioned. At a pressure of 76 cm. of men Miry and a temperature of about 20° C. (i.e. 70° F.), the in. an free path of a hydrogen molecule is 0.0000185 (in., and tin- number of impacts it makes in a second is 9480 million ; for oxygen, these figures are 0.0000099 cm. and 0 million ; for carbonic acid gas, 0.0000068 cm. and 5510 million. By various processes the dimensions of a molecule and the number in a unit volume may be approximately determined ; the " radius " of a molecule is found to be of the order of a ten-millionth of a millimetre, and the number in a cubic centimetre is of the order of 2 x 1019, i.e. twenty quintillions. Fourth State of Matter. — If a gas is inclosed in a glass bulb which can he gradually exhausted by means of an air pump, as will be explained later, the most evident change produced is the decrease in density and the consequent increase in length of the mean free path. (If the exhaus- tion is (anied so far that the pressure in the bulb is that of one thousandth of a centimetre of mercury, the mean free path is 7630 times as great as it is at the pressure of 1 cm., or about 1.5 cm. for hydrogen.) The properties of matter in this condition are quite different from those of ordinary gases; and for this reason the matter is now said to be in a urth State." Its chief properties were investigated by \Villiam Crookes, and they will be described later when electrical phenomena are discussed. One purely mechanical property should, however, be mentioned here. It is illus- .straicd l.y the following experiment : a framework is made •>tin'_r of t\vo or more CM-OSS arms, which carry at each i small piece of mica blackened on one face and not on tin other: the plane of each mica vane is perpendicular to that of the cross arms, but includes the line of direction of the arm which carries it, and the blackened face of one vane is turned toward the polished face of the next one. This little wheel i> suspended in a Imlh in such a manner as to be 204 MKCHAMCS KIG. 100. — Crookc radiometer. free to turn alx>ut an axis perpendicular to the plane of the cross arms. If a hot body, like a burning match, is brought near the bulb, nothing noticeable happens if the gas inside the bulb is at ordinary pressure ; but, if the gas is exhausted to a few thousandths or hun- dredths of a centimetre of mercury, a stage is reached when the framework begins to rotate on its axis in the direction which it would move if the blackened face of each vane were repelled by the hot body. (If the gas is exhausted as completely as possible, this motion does not arise ; and at slightly higher pressures there are complications in the phe- nomena which need not be discussed here.) The explanation of this action is as follows: a blackened surface becomes hotter than a polished one when exposed to a source of heat, as is known to every one, and therefore those molecules in the gas that rebound from or leave the blackened face of a mica vane do so with a greater velocity than those that rebound from the polished faces. If the gas, however, is at ordinary pressure, the impacts are so numerous that the condition throughout the bulb is nearly uniform, and there is no mechanical action on the vanes ; but if the gas is so rarefied that the mean free path is a centimetre or more, the molecules leaving the blackened face give the vane a backward push which is not counterbalanced by that given the other face of the vane, and consequently the framework rotates in the direction described above. This instrument is called a "radiometer"; and its action was discovered by chance by Sir William Crookes in the course of a research which involved weighing small amounts in a vacuum. It has been modified recently so as to serve as an instrument for measuring certain quantities that are of importance in the discussion of heat phenomena. CHA1TKU IX HYDRAULIC MA( HIXKS: 1TMPS, ETC. Barker's Wheel. — This is n simple instrument, consisting of •>[ lixed tube to one end of which is attached by a pivot a framework consisting of a number of cross tubes, whose ends are bent at right angles t<> their length and to that of tin- fixed tube. Connection is made through the pivot from tin- fixed tube into the mova- ble ones. If now a stream of . either liquid or ,reed through thr li , tube and out the bent ends of the cross tubes, the Litter will rotate rapidly in a direct ion opposite to that in which the fluid escapes. This is then simply an illustration of the conservation of momentum. . he.-ls an- often seen connected with devices for 101. — Barker's wheel sprinkling lawn> with water. Turbines. — A turbine consists «>f a wheel that can rotate <>n an axis and that lias I'm- i; --s curved flanges SO I that as a lluid presses against them it exerts a uioinrnt annnid the a\i. ( ( 'ompare the action <»f the wind «'ii the fans of a windmill. ) In e, a water turbine is it the bottom of a deep well (or high cylindrical till).' ) \\ hi< 1 ihe wlie- under great pi -T is admitted through the Ml 206 MECHANICS turbine near its axis, flows out along the flanges, and escapes at the edges, so that the wheel is set in rotation by the pressure. In a similar manner steam can be used to drive a turbine, as is done in the so-called " turbine boats," in which there are several turbines fastened directly to the shaft of the boat. Hydraulic Ram. — In this instrument, which was invented by Montgolfier, in 1796, and is in such general use for forcing water from springs into tanks at a considerable elevation above them, the principle made use of is that a large quantity of water falling through a small distance may raise a small quantity through a great distance. A simple form is shown in the cut. The essential features of the machine are a large tube, down which the water flows and which is closed by the escape valve P opening inward, and by another valve Q opening outward into an air-tight reservoir called the "air cham- ber." This contains some air; and into it enters for some distance the outlet pipe which carries the water to the tank. The escape valve P has a weight which exceeds slightly the upward force against it due to the water when there is no flowing — this upward force is the excess of that on the lower side of the valve over the downward force on its upper side. Therefore, at the beginning of the operation the valve drops ; as it does so, the water escapes, and, since in moving water the pressure is less than in water at rest, the downward force on the upper side of the valve, over which the water is flowing, is diminished, and the upward force is FIG.
102. — Hydraulic ram. P and Q are valves opening automatically. HYDRAULIC MACH1NX8 1 I'l'.MPs. ETC. 207 sutlii -lent to raise the valve and close it, thus stopping the flow; the valve therefore again drops owing to its weight, and the operation is repeated automatically. When the water is at rest at the beginning of the operation, the other valve Q is down, closing the opening; it remains so as long the eseape valve is open, for the pressure on its lower side is now small since the water is flowing. When the escape valve closes, there is an immediate increase in pressure throughout the whole tube, the outlet valve Q is pushed up and some water enters the air chamber ; then the valve drops as the pressure is thus relieved, and the operation is re- peated. As more and more water enters the air chamber, a time is reached when the level of the water covers the open end of the outlet pipe which connects the chamber with the tank ; after this time, as the water enters, the air trapped above it is compressed and has its pressure increased. Water is thus forced up the pipe into the tank. This operation is more or less continuous; for, as the water enters the air chamber rapidly, the air is compressed and some water flows up into the tank : and then, during the interval of time which passes before some more water enters, the compressed air ;ids and continues the flow. Siphon. — This consists of a large tube or pipe bent into the form of a U •• it with its two arms of unequal h. This is placed in a vertical .on, with its shorter arm dipping belnw the surface of a liquid in a vessel, and its longer arm outside. l',\ is of suction applied to the open end. the siphon is now tilled with liquid ; and. if left to itself, the liquid in the vessel will How nut through the siphon until its sur- comes below the end of the short arm. The explana- tion is evident if one considers the conditions that exist 208 MECI1AMCS when the siphon is full, at tin- instant before the flow starts. The pressure in the liquid in the longer arm at a level with the liquid surface in the vessel is, of course, that of the atmosphere P; so, if the open end of the siphon is at a depth h below this, the pressure in the liquid at this point is P -\-dgh if its density is d. But the opposing pressure is simply that of the atmosphere P, and the difference of pressure dgh forces the liquid out. The shorter arm of the siphon must not be too long ; for if it is greater than the height to which the liquid would rise in a barometer, the pressure on the free surface of the liquid will not be sufficient to force the liquid up to the turn of the siphon. (If the tube is of fine bore, other actions than gravity and atmospheric pressure come into play.) Liquid Pumps. — These are instruments devised for the purpose of raising liquids from one tank or well into another at a greater height, or for forcing a liquid through a long pipe against friction. There are two types : the u lift pump " and the " force pump." The former consists of a cylinder in which fits an air-tight piston provided with a valve B opening upward, and whose lower end is closed by a valve A, also opening upward, where the pipe leading to the tank containing the liquid to be raised is attached. The vertical distance from the lower end of the cylinder to the level of the liquid in the well or tank must not exceed the barometric height of that liquid. Then if the piston is raised, some liquid is forced up through the lower valve into the cylinder by the pressure of the atmosphere on the surface of the liquid FIG. ID. - Lift pump. A in the tank; if now the piston is brought and B are valves opening up- , , 111 ^11 ward8. to rest and then pushed down, the lower .\fAfin\Ks: valve drops juul the one in the piston is lifted, the liquid ;n_r through it from below the piston to the space above. When the piston is again 1, the liquid on top of lifted and may escape through a side outlet into a tank ; at the same time more liquid is being drawn up through the lower valve into the cylinder, and the process may be repeated indefinitely. In the force pump the piston 10 valve, and an air cham- ber, like that of the hydraulic, ram, is attached to one side of the cylinder. The explanation of its action is self-evident. This pump is as a rule placed near the surface of the liquid which is to be pumped, and the upper tank may be as high as is necessary. Ki... 105. — Force pump. Air Pumps, a. Mechanical. — These are instruments de- i either to force more and more ijas into a given space, 01 to withdraw as much gas as is desired from a closed sel ; in other words, to increase or to decrease the pressure inside the vessel. The former are called "compression" pumps: the latter, "exhaust" punn The simplest form of exhaust pump is illustrated on pa^c -in. Its mode of action is essentially that of the lift pump :dy described, the main point of dilTerence bein^ that in the latter the valves open and close automatically, while in the air pump they mut be operated h\ mechanical means, the difference in pressure of the gas on the two sides of the \al\cs is nut snrtieient to move them. Such pumps as this are called "mechanical" ones. Other forms in general use are the Sprengel and the Geissler-Toepler. AMES'S PHYSICS — 14 210 MECHANICS b. Sprengel pump. — The action of this purnp consists in having drops of mercury so fall as to trap the gas between them and thus carry it away. There is an elongated glass bulb, to the side of which is joined a long tube, as shown in the cut, whose lower end is connected by a rubber tube with a reservoir, so that the mercury may be thus forced Fro. 106. — Mechanical air pump. The vertical rod, A, is held by the moving piston'with sufficient friction to move it up or down until brought to rest by the conical ends entering their sockets ; then the piston slips along the rod. In the cut the piston is moving down. FIG. 107. — Sprengel air pump. slowly into the bulb. At the lower end of the bulb is joined a glass tube of narrow bore and at least 80 cm. long, and at the upper end of the bulb is a connection with the space to be exhausted. The tubes at the side and bottom of the bulb are so arranged that, as the drops of mercury break off and fall, they hit the opening of the lower tube and pass down it HYDRAULIC Jf.-l(7//.v/>: I'UMPS, ETC. 211 in the form of short cylinders. The space between these cylinders thus formed is occupied by small amounts of the • 1 raw a in from the connected vessel; and so these drops act like a succession of small pistons forcing out the gas. The lower end of the long tube may dip into a basin of mer- cury, and the gas will bubble out at the surface, or it may be bent so as to form a "trap." As the exhaustion continues, the mercury will rise in the long tube, and will finally stand at the barometric height when the vacuum is as complete as it can be made. c. O-eissler- Toepler pump. — In this pump there is a large bulb to which are joined two tubes, — one at the top, the other at the bottom. The lower one is at least 80 cm. long and is connected at its lower end to a large vessel of mercury by means of a long rubber tube. The upper tube is bent over into a vertical direction downward, and dips into a basin of mercury, or forms a trap. Around the large bulb there is a branch tube connecting the upper and lower tubes just as they leave the bulb; and into this branch is joined a long ver- tical tube leading to the vessel which is to VESSEL TO BE EXHAUSTED Fio. 108. — G«toftler- Toepler air pump. be exhausted. (This tube is replaced often by a short ver- tical one containing a glass valve.) I f the large vessel of mercury is now raised, as it can be owing to the flexible rubber tubing, the mercury will rise into the hnlh and the connecting tubes, shutting off connec- tion with the vessel to be exhausted, and will drive out all t!i« -.ras in the bulb through the tube in the top, so that it will bubble out through the mercury in the basin at its end. 1 1 . now, the movable vessel of mercury is lowered, no air can enter through the tube at the top of the bulb, because it is "sealed" by the mm-nry in the basin, which \\ill rise in the 212 MECHANICS tube; but as soon as the mercury falls below the opening to the long vertical tube, the gas in the vessel to be exhausted will expand and fill the bulb and the connecting tubes. When the movable vessel of mercury is again raised, it drives out the gas in the bulb; and as the process continues, the exhaustion of the vessel proceeds rapidly. The tube leading from the top of the bulb around to the basin of mercury must be at least 80 cm. high, and the long vertical tube leading to the vessel to be exhausted must be still longer. BOOKS OF REFERENCE KIMBALL. The Physical Properties of Gases. Boston. 1890. BARUS. The Laws of Gases. New York. 1899. This contains Boyle's original paper and also Amagat's memoirs on the variations of a gas from Boyle's law. BOYS, Soap Bubbles. London. 1890. This is a description of many most interesting capillary phenomena. GREENHILL. Hydrostatics. London. 1894. A standard advanced text-book. TAIT. Properties of Matter. Edinburgh. 1885. A most interesting and useful text-book. RISTEEN. Molecules and the Molecular Theory of Matter. Boston. 1895. A popular, yet accurate, description of the kinetic theory of matter. HOLMAN. Matter, Energy, Force, and Work. New York. 1898. This is a philosophical discussion of the properties of matter. POYNTING AND THOMSON. Properties of Matter. London. 1902. This is an advanced text-book, and will be found most useful for reference. JONES. Elements of Physical Chemistry. New York. 1002. This contains an excellent description of the properties of solids, liquids, and gases. HEAT INTRODUCTION Tm: properties of matter that have been discussed in the previous pages are mass, weight, shape, size, elasticity, sure, etc. The mass of a body cannot be changed by any meehanical means, nor can i
ts weight at any one point on tin- earth's surface; but the other properties may be changed at will. One of the simplest methods of doing this is to alter the temperature of the body; and this process will be ;>sed in the following pages. Molecular Energy. — We have proved, in the discussion of <li tins i<>n. of viscosity, and of the properties of gases, that matter consists of minute parts which are in motion, the extent of the freedom of this motion varying with the con- ditinn of the matter. In a solid these minute particles as a rule only make oscillations; while in thuds they can move :i one portion of space to another. Thus these partu -les have both kinetic and potential energy, — the former owing to their motion, the latter owing to the fact that work is required to hrin^ two mulreules into a certain <1. -finite posi- tion with reference to each other. The molecules themselves of parts, and these have energy in both the kinetic and tli.- potrntial forms. '1 'hi- internal energy of a body is quite apart from its energy owing to its motion as a whole, or to its push inn with e to the earth, and may be increased, as is obvious, loing WMI! i^ainsi forces that act in connection with the inuleeiile8,e.^. bv niiiLT friction or 1>\ compressing a gas. 213 214 UK AT It may be varied in the case of an elastic body by setting it in vibration, or by sending waves or pulses through it ; for under these conditions the kinetic and potential energies of the molecules are altered. Thus a bell if struck by a hammer vibrates, and as a result waves are produced in the surround- ing air, the particles of which therefore are set in vibration. We may, therefore, consider two kinds of internal molecu- lar motions : one corresponding to a state of wave motion when all the particles or molecules are in similar vibrations ; the other, to a condition in which there is no regularity in the vibrations or motions. This last condition exists when a body is in its ordinary undisturbed state, and is altered when friction is overcome, when a gas is suddenly compressed, etc. The phenomena associated with variations in this in- ternal energy of bodies, owing to their irregular molecular motions, belong to that branch of Physics which is called " Heat." CHAPTER X UK AT PIIKXOMKNA Preliminary Ideas. — In describing and discussing mechan- ical phenomena the sequence of ideas was somewhat as fol- lows: by means of our muscular sense we experience certain sensations which we associate with matter, and we are led t<> distinguish certain properties of matter which we consider as independent of each other for the time being, viz., mass, weight, elasticity, etc.; we study these at first by means of our own muscles, but later discover physical methods for the same purpose. By means of our other senses we can also in vestigate other properties of matter and the corresponding -.sot* nature." Everyone knows what is meant l>y tin- words "hot" and "cold"; and if a man dips his hand in turn into two basins of water, he is as a rule able to dis- uish between them by means of his temperature sense, and so can say one is hotter or colder than the other. We experience this sensation of hotness when we stand in the nine or 11. -ap a fire, when \ve put our hands in a K of water on a stove, when we touch a body that has been rubbed violently against another, etc.; and we feel the sensation of coldness when we touch or stand near a block of ice, when we wet our hands and allow the water to evaporate, etc. If we expose inanimate objects to the same in ions, they undergo changes; and. in tact, in general all tin -ir physical properties with the exception of their mass and weight change. Tim-. M <»f iron is exposed to the Hun or put on a stove, its volume increases, its elasticity < -han^es, it feels hot 116 216 HEAT to our fingers. If a piece of ice is put in a basin on a stove, it changes its state, becoming a liquid ; this water gradually feels hotter and hotter to our fingers, and finally boils away in the form of a gas. If a gas is inclosed in a glass bulb and exposed to a flame, both its pressure and volume change, etc. All these changes could be produced equally well by friction. Similarly, if a piece of iron is put on a block of ice, its volume becomes smaller and it feels cold to our hands ; water can be frozen by making some of it evaporate rapidly, etc. These changes are called "heat effects." Nature and Cause of Heat Effects. — If we investigate the conditions under which these changes occur, we see that in them all work is being done either on the minute portions of the body or by them. We shall consider one or two of these conditions in detail. When two solid bodies are rubbed together, the force of friction is overcome, and the changes produced, in general, are increase in hotness, in- crease in volume, etc. In friction, however, the force owes its origin to minute inequalities in the two surfaces, which are leveled off or altered as the work is done. Similarly, in all cases of fluid friction, the work done in maintaining the motion is clearly spent, as has been shown, in giving energy to the molecules or minute moving parts. An illus- tration of the heating effect produced by friction is furnished by meteors and "shooting stars." As these pieces of matter enter the atmosphere of the earth, they are heated to incan- descence by friction against the air. Again, when a gas is compressed, it becomes hot and its pressure increases ; but, as we have shown on page 197, in this case work is done in increasing the kinetic energy of translation of the particles of the gas. If the gas is allowed to expand, doing work against some external force, it becomes cold and its pressure decreases ; but it has been shown that while this happens the molecules of the gas lose kinetic energy. In a flame, or any process of combustion, there UK AT rilKSOMENA -IT molecular changes going on which must necessarily in\<>l\v the energy of the molecules. It will be shown later that all bodies are emitting waves in the. ether; and, if those whose lengths art- comparable with the size of mole- cules are absorbed by bodies, they become hot, expand, etc. This is the explanation of the effect of exposure to sunlight, etc. Again, if we consider the effects themselves that are produced in bodies under the conditions described, it is seen that they correspond to what we would expect if they are due t«> work being done on or by molecules. When a body amis, the molecules are separated, and work is done against the molecular forces ; similarly, if a body contracts, these forces do work. When a solid is incited, the molecules iter freedom of motion, and therefore work must be done on them; similarly, when a liquid solidities, the molecules lose their freedom of motion, and in this change lose energy. As a gas becomes hot, owing to com- mon, its molecules gain kinetic energy of translation ; as the gas cools, owing to expansion against external forces, iis molecules lose kinetic energy of translation ; so that in tin- case of a gas, at least, changes in temperature correspond to changes in kinetic energy. Similarly, changes in elas- ticity, hardness, viscosity, etc., may be explained if we think of the energy ,,f the molecules as being aii'-'cted. It is natural, therefore, to believe that all these changes are ;lie fact that t: I the molecules ha-* been increased by the result -rn;il forces or that it has been decreased by the molecules doin^ work against external s. Then two question I )«> th.se changes depend ply upon the Y/////////V of cilery that the bodies receive ip«>ii the manner in which the work is doi \ml is the 'hat llie molecules receive exactly equal to the \\ork done on them'' The experiments that have been performed 111 0 ; hesc oi. \\ ill be described ', 218 HEAT but it may be stated here that there is every reason for believing that the changes depend upon the quantity of energy received by the molecules, not upon the manner in which it is delivered, and also that there is neither loss nor gain of energy in the process. Application of the Principle of the Conservation of Energy. — During most of these changes the volume of the body that undergoes them alters ; and, in this process, work is done on or by whatever external force or pressure is acting on the body maintaining its volume, and, in general, by the force of gravity also. Thus, if a pillar supporting a build- ing expands, the building is raised and work is done ; the air presses against the sides of the pillar, and this force is also overcome as it expands ; again, the centre of gravity of the pillar itself is raised, and thus more work is done against gravity. Consequently, when work is done against the molecular forces of a body so that it undergoes changes in temperature, in size, in state, etc., a definite amount of energy is given the body and this is spent in two ways : (1) in increasing its internal energy ; (2) in doing external work as just described. (During these changes the mole- cules are affected, and some of their kinetic energy may become potential or vice versa ; but in these internal changes there is neither loss nor gain of energy.) Similarly, when reverse changes take place and the .body becomes cool and contracts, the external forces of pressure and gravity do work on the body, the internal energy decreases, and the molecules of the body do work in such a manner as to give energy to external bodies; the amount of this last must therefore equal the sum of the work done on the body by the external forces and the amount of the decrease in the internal energy. Thus, when a piece of iron is placed in a basin of hot water, the latter loses a certain amount of inter- nal energy, and, since it contracts, the atmospheric pressure does work on it; and in return for these two supplies of in: AT ru }-:\<>\ii-;_\ A 219 energy the internal energy of the iron increases, and as it mis it pushes back the atmosphere
and so does work, i \\'e are neglecting purposely all losses of energy by radia- tion and conduction.) In any change, then, we may write the equation : quantity of energy received by work done on the molec-nlo = increase in internal energy + work done against ex- ternal forces, such as surface pressure and gravity, or the equivalent one : quantity of energy given up by the molecules doing work against external forces = decrease in internal energy -f work done on the body by external forces, such as surface pressure and gravity. Heat Energy. — The energy that is given a body when work is done on it against molecular forces is called "heat energy," and the effects that bodies experience when they gain or lose this energy are called "heat effects." In ordi- nary language the word "heat" is often used in place of heat energy, and we speak of k% adding heat to a body/ OK withdrawing it, of a "source of heat." etc., where the mean- ing is obvious. As has been said, all the properties of a body except its mass and weight in general change when heat energy is added to it or taken from it. The most obvious of these changes are the following: 1. ('hange in hotness, as perceived by our temperature sense. _'. Change' in volume, if the external pressure is kept const 3. Change in pressure, if the volume is kept constant, in case the body is a fluid. I . Change in state, such as fusion, evaporation, etc. Chan-.- in electrical or magnetic properties, such as electrical conductivity, magnetic strength, etc. UK AT Heat Quantities. — In the discussion of heat effects two physical quantities enter that have not hitherto been described with exactness. In the first place, we must explain what is meant by the word " temperature," which is used ordinarily as giving an idea of the hotness of a body, and must describe a method by which a numerical value may be assigned it. Further, in all heat effects we are concerned primarily with quantities of energy entering or leaving bodies; and some convenient unit must be denned in terms of which this energy may be measured. Changes in volume and pressure can be measured by means already described, and may be expressed in terms of the ordinary units — the cubic centi- metre and dynes per square centimetre or centimetres of mercury. In changes of state we have to consider alterations in volume, in elastic properties, etc.; but these require no new definitions or units. Temperature Preliminary Ideas in Regard to Temperature. — We have used, whenever convenient, the word " temperature " in its everyday meaning as a quantity describing the hotness of a body, and have said that a body which felt hot to us had a higher temperature than one which felt cold. This sensa- tion is due to some property of the molecules of the body, in virtue of which they affect our nerves. We saw in speak- ing of gases, page 197, that the mean kinetic energy of trans- lation of its molecules obeys the same laws as does its temperature ; or, in other words, the temperature or hotness of a gas is due to and is measured by the kine'tic energy of translation of its molecules. There are many reasons for believing that this is true also of solids and liquids ; and so, when the temperature of a body is raised, we believe that the kinetic energy of its molecules is increased. Temperature Scales ; Thermometers. — Evidently this prop- erty of a body cannot be measured, because it is impossible HEAT to conceive what is meant by a unit of hotness ; but we can assign it a numerical value. For when the temperature of a material body is changed, its physical properties all change; and so. instead of using our hands or bodies as instruments for investigating temperature, we can employ any material I unly and observe those properties which change as the tem- perature is changed. This body would then be called a • tliLTinonu't Thus we might use a homogeneous metal rod and observe its length. If the rod had the same length ID immersed in two different baths of oil, we should say that their temperatures were the same with this thermome- ter ; whereas, it the length were greater when the rod was in one bath than when in the other, the former would be said to have the higher temperature. Kxperiments show that, if two bodies at different tempera- tures are placed together in such a manner that heat energy (an pass between them, e.g. if two liquids are stirred up together, if a solid is immersed in a fluid, etc., they finally come to the same temperature intermediate between their initial temperatures ; the body at the lower temperature must therefore gain energy, and the one at the higher tem- perature must lose it. (The former also loses energy in general, but it receives more than it loses, and so on the whole gains. Similarly, the latter in general gains energy, but its loss exceeds its gain.) Thus, from a physical point of view, the difference of temperature between two bodies determines whirh is to lose energy, or it determines the ••direction of the How of heat energy." The temperature of a h«>dy. then, is a property defining its thermal relations with neighboring bodies. The general method of assigning a number to the tem- perature [fl as follows: two stand, in 1 thermal conditions •elected, and numbers are given them arbit rarily — let •• lie f} and /., ; then some definite p of a definite . which can be ni.M>uied and \\hich changes with the •2'2-J. HEAT temperature, is selected ; its numerical value is determined in the two standard conditions and in the one to which a number is to be assigned — let these values be ar «2, and a ; finally, the number for the temperature is obtained from those arbitrarily given the standard conditions by simple proportion between the numbers thus obtained, viz., calling it £, t-tl:t2-tl = a-al:a2-al. Experiments show that the temperatures of a mixture of pure ice and water when in equilibrium, and of steam rising from boiling water, are constant and the same the world over, provided the external atmospheric pressure is the same ; that is, a definite metal rod always has the same length if it is put in a bath of water and ice no matter when or where it is done, a given quantity of mercury has the same volume, etc. For this reason, and because they are easily obtained and include the ordinary temperatures, these two thermal condi- tions when the external pressure is 76 cm. of mercury are selected as the standard ones ; and the numbers 0 and 100 are assigned them on the " Centigrade " or Celsius scale. The quantity agreed upon by physicists, the change of which is measured, is the pressure of a definite amount of hydrogen gas whose volume is kept constant. Let the values of the pressure of this gas at the standard temperatures be p0 and jt?100, and that at a temperature for which a number is desired p. Then, calling this number t, it is given by the equation : t - 0 : 100 - 0 = p - p, :Pm - Pot or t = 100 P-P. Pun - Po This is the " temperature on the constant volume hydrogen thermometer," using the Centigrade scale ; and, whenever hereafter the temperature of a body is referred to, its value on this instrument and scale is meant. This number is always ex- pressed as a certain number of " degrees," and is written t° C. HEAT PHENOMENA Several other thermometers are in more or less common use. In one the volume of a definite quantity of nitrogen (or of air), the pressure being kept constant, is the property measured as the temperature changes; in another it is the apparent volume (see page 234) of a quantity of mer- cury inclosed in a glass bulb with a fine stem that is measured. The former is called a "constant pressure nitrogen (or air) thermometer"; the latter, a " mercury-in-glass thermometer " ; in them both the number for the temperature on the Centigrade scale is given by t = 100 v ~ "o , where the symbols have obvious meanings. Another form of instrument is one in which the elec- trical resistance of a piece of platinum wire is measured. If /,'„, 7?100, and R are the values of this quantity at the tin. •• temperatures, the number for the temperature is given by R-R. t = 100 Such a thermometer as this is called a "platinum resist- ance" in>trnment. The numbers obtained by these last three instruments for the temperature of the same body will not agree with each other, or with that obtained with the standard thermometer, except at 0° and 100°; but careful com- parisons have been made by different observers of the readings on all four instruments at the same temp through wide limits; so that, if a number is obtained for the temperature of a body using a mercury thermometer, the value which would have been obtained if the standard hydrogen instrument had been used is known: this is tin- temperature of the body. (The cor- 'ii to I..- applied to the scale reading <»f an ordinary MI -y thermometer, such as is used in laboratories, is small.) other scales than the Centigrade are in use for non- , title purposes. In the Fahrenheit system the num- bers 32 and 212 are assigned the standard temperatures of melting ice and sl.-.im. and in the Keaumur the num- bers 0 and 80 are used. < Thu>. 1>0°C. = 68°F.= \>> B 'O.-Mer- • The ordinary laboratory thermometer is a men-ury-in-glass ment. who.-- stem i, divided into numbered parts. The maker ,.f the 224 HEAT instrument aims to have these numbers come at, such points that, \\licn the division corresponding to the top of tin- nn-ivury column is read, it gives the temperature on the mercury scale, i.e. the numbers correspond to equal increments in apparent volume. If the instrument is to be used for accurate measurements, however, the readings at the standard tem- peratures must be noted, and the volume of different portions of the stem must be measured, in order to determine exactly the error of each division as marked by the maker. Moreover, a glass thermometer is subject to an error due to two facts : a glass bulb whose temperature is raised from one value to an
other, and then lowered again to the former value, has a larger volume at the end than it had at the beginning ; and this increase is not permanent, but disappears gradually after the lapse of weeks or months. This is owing to the heterogeneous character of glass ; the molecular changes produced by raising the temperature persist after it is again lowered. Thus, if a glass thermometer reads 0°.02 C. when put in melting ice and is then, after being heated to 80° or 90°, again put in melting ice, it may read — 0°.01 C., showing that the volume of the glass bulb has increased. This is known as the " depression of the zero point." If the thermometer is kept in ice for some months, the readings will gradually rise. There are numerous other defects in the mercury ther- mometer which must be carefully guarded against. To give a number to extremely low temperatures some substance should be used whose properties are the same in kind as at ordinary temperatures. Thus, mercury should not be used, for it solidifies at about — 39° C., and the changes in volume of the solid mer- cury cannot be compared with the similar changes of liquid mercury at ordinary temperatures. Hydrogen gas at a small pressure may be used, or a platinum resistance thermometer. Similarly, to give numbers to extremely high tem- peratures special precautions must be taken. The best Fro. 110. —Rutherford's maximum and minimum thermometers. The former contains mercury, the hitter alcohol. in-: AT /•///•;vo.v/-;v.i 2-25 •ical methods depend upon certain laws of radiation which will be issed later; but for standardizing purposes a hydrogen thermometer mu>t be used. There are many special types of thermometers devised for particular purposes. Among these it may be worth while to describe briefly one that registers the extreme temperatures which occur during ;i CHI tain tl of time, and one that is used by physicians for clinical purposes. Rutherford's "maximum and mini- mum thermometers" are two instruments, as shown in ut, which are supported with their stems horizontal : one contains alcohol, the other mercury. Just inside the • f the alcohol column is a small glass rod with rounded ends, as shown on an enlarged scale in a portion of the (lit ; so, as the temperature falls and the alcohol contracts, this rod is drawn back by the curved surface of the alcohol ; when the temperature rises sixain, it remains station- ary. Similarly, just outside the end of the mercury column is a small iron rod which is pushed forward by the mer- as it expands and remains stationary when the mer- contracts. Thus both the lowest and the highest temperature- readied are recorded. The glass index can b«- jarred bark in place again, and the iron one can be dra\\n hack by a magnet. (In Six's form of instrument these two thermometers are combined in one.) :iical thermometer is shown in the cut. Its stem has a tin.- bun- \\ith a sw. -llin^ at its lower end separating in the bulb containing the mercury : so that, as the tem\|Krature rises, the mercury does not reach the diviile.l stem until a temperature of 90° F. is reached. At the swelling then- -triciimi in the tube; and if the ills when the mercury is above this, thecol- umt the mercury in the stem. Thus. if the instrument is inserted in the month ami tin- mer- ! to a certain divi-ion. it will -.till stand at this K... 111.— I'Hnl- r«l thermometer. point aft«T the thermometer i withdrawn. The mercury may be driven into the bulb by -liakiu- the instrument in the proper manner, •rifiiijal fore«- is usually applied.) fari thermometer \\a> made by (ialileo, as early as 1598. It nded upon the expansion of air. and was not devised in such a !"\|-eudeiit of ban-met rie changes. I'.otilliean. in in-glass instrument. The credit of showing that i MY8IC8 — 15 226 UK AT melting ice and boiling water furnish "fixed points," and of proposing that they be adopted in a thermometric scale, belongs to Huygens (1665). Many improvements were made by Fahrenheit Units of Heat Energy. — In all heat effects, as has been said, we are concerned with the quantity of energy that must be added to or withdrawn from a body in order to produce a given change ; and so a convenient unit must be chosen, and suitable methods of measurement must be devised. The scientific unit for the expression of amounts of all kinds of energy, including therefore heat energy, is the erg or the joule, i.e. 107 ergs (see page 112); but heat effects are not as a rule produced by direct mechanical proc- esses in which the amount of work done can be measured by a dynamometer. The standard method of producing a heat effect in a body is to immerse it — if it is a solid — in a quantity of water at a different temperature ; the temperature of the water falls or rises because it loses or gains the heat energy that enters or leaves the body, allowance being made for external work and for the influence of surrounding bodies. The natural unit of heat energy is, then, either the amount required to raise the temperature of a unit mass of water through 1°, or one nth the amount required to raise its temperature through n°. (These two quantities of energy are not in general the same.) By stirring a paddle rapidly and continuously in a known quantity of water, the amount of work (measured in ergs) required to raise its temperature through a known number of degrees (on any scale) may be determined by a dynamometer; and so the value of the practical unit of heat energy may be expressed in ergs. Experiments show that the number of ergs required to raise the temperature of a definite quantity of water through 1° is different for different temperatures, i.e. it is not the same when the temperature is raised from 5° to 6° as if the limits were 10° and 11°, etc. ; but the difference is very small. The work required, however, to raise the in-: A -i temperature of a definite quantity of water from 0° to 100° C. most exactly 100 times as much as that required to raise it from 15° C. to 16° C. For this reason the practical unit of heat energy is defined to be the "amount required to raise the temperature of 1 g. of water from 15° to 16° C."; this is called a "gram calorie at 15° C.," or, simply, the calorie. Its value in ergs, as determined by Rowland, Callendar and IJarnes, and others, is 4.187 x 107; that is, it is 4.187 joules. The great disadvantage in having as a "heat unit" one that depends upon a range of temperature (other than from 0° to 100° C.) lies in the difficulty of determining temperature accurately, and in the fact that so many arbitrary quantities and ideas enter into the definition of a temperature scale. If it were practicable, it would be much better to take as a heat unit the amount required to melt 1 g. of ice at 0° C., or to produce some other change in state, because during these changes the temperature does not vary. Transfer of Heat Energy. — Before discussing the various heat effects in detail, a few words should be said in regard to the various methods by which heat energy is added to or taken from a body. These are three in number, and are illustrated in the following experiment : if one's hand is held above a heated stove, it feels hot. and at the same time one 'iiscious of an ascending current of air. Similarly, if a body is held in the upper portion of any fluid whose lower portimi is maintained at a hi^h temperature, it will receive heat energy from the ascending currents of heated tluid. This process is known as "convection." If one end of a metal nnl is put into a lire, its temperature rises, and that of •r neighboring portions of the rod also. In this process there is n<> actual displacement of the matter, and therefore there is no convert ion : hut the energy is handed on from niolreuh- to iii"l,Tiile ,l,,\vn the rod. This is called ''con- duction, ii, if a l»od\ ,^M! to the sun or is held at one side ot ,t hot stove, its temperature iii general rises, it 228 HEAT is receiving heat energy — not, however, by convection or conduction. This process is called "radiation," and will be shown later to consist in the absorption by the body of waves in the ether. All these pro- cesses will be described in detail in a later chapter. Convection and conduction cannot take place through a vacuum, and radiation is almost entirely prevented by having the sur- face of the body or vessel covered with a highly polished metallic layer. In his experi- ments on liquid air and hydrogen, Dewar has used a flask, called by his name, which con- sists of a double-walled glass vessel, the space between the walls being exhausted as completely as possible. Traces of mercury vapor are left in this space ; and at low temperatures this, freezes, forming a metallic surface over the glass walls. Fio. 112. — Dewar flask. CHAPTER XI CHANGES IN VOLUME AND PRESSURE Introduction. — The fact that all bodies, with the excep- tion of water below 4° C. and one or two unimportant sub- stances, increase in volume when their temperature is raised, is most familiar to every one; and numerous measurements have been made of these changes. Naturally the amount of the increase in volume depends upon the external force act- ing, and upon whether this is constant or not. The mechan- ical force required to influence the expansion of a solid or a liquid is so great that ordinary changes in the atmospheric pressure have no measurable effect ; but this is not so in the case of a gas. It is therefore necessary, in studying those variations in volume of a body which accompany changes in temperature, to describe the external conditions, if one wishes to be definite. The condition which is always assumed, unless the contrary is stated, is that of constant external pressure. Solids Linear and Cubical Expansion. — In measuring the change in volume of a solid it is, as a rule, easier to measure the changes in length of certain linear dimensions of the body and from these to calculate the change in volume. It tin- body is isotropio,
i.e. has the same properties in all direc- tions, this calculation is most simple. Imagine the body in the form of a cube, the length of whose edges at any one trm p. Mature tf is /, and at the temperature fa° is /3, then tin- voluin.- ;it /, ii /,:! and at tj is lf\ so the change in volume is I* — I- It the body is not isotropic, but has no 230 HEAT different properties in different directions, three directions in it (which depend upon the arrangement of the molecules) may be determined, such that the changes along these are independent of one another — these are called "axes." (In crystals they are the crystallographic axes.) If, then, a rectangular solid is made of this body with its edges parallel to these directions, and if lv mv n^ are the lengths of the edges at t^ and ?2, m2, n2 their lengths at £2°, the correspond- ing volumes are l^n^ and Z2w2w2; and the change in volume is ?2W2W2 ~~ liminr The change in length of any straight line in the surface of a solid or of any straight edge may be measured by various means. The body is immersed in a bath of some fluid, whose temperature may be varied ; and the lengths are determined by a comparator of some kind. (Reference may be made to any laboratory manual.) Experiments show that, to a sufficient degree of accuracy, the change in length varies directly as the original length and as the change in temperature, but is different for bodies of different materials. If, as above, Zx is the length of a certain line of the solid body at ^°, and Z2 that at £2°, these facts may be expressed by the formula /,-! = af, ft - Q, in which a is a factor of proportionality, which is different for different substances, a is called the "coefficient of linear expansion of the body referred to the temperature ^V Ordinarily, the temperature to which these coefficients refer is 0° C. ; and if 1Q is the length at 0° and I that at £°, the relation is I — Z0 = #(/o^' where a0 is the coefficient of linear expansion referred to 0° C. This formula may be written I = 1Q (1 + a0£). (It is evident, then, that a in the first formula is connected l-f-#0& with aQ by the relation a = ^ — - — ; and if aQ is an extremely small quantity, as it is for all solids, we may neglect the term a^t in comparison with 1, and write a = a0 approximately.) 1\ VOLl'MK AM) PRESSl'llE 231 If the solid is isotropic, and if we write v0 for the volume at 0° C. and v for that at t°, v0 = /0« and v = /» = /0» (1 4- <V)» = »0 (1 + a,/)'. IJut (1 +- a003= l + 3a0« + 3n0V + aoV; ancl if aois a 8ma11 quantity, tin* last tun trims may be neglected in comparison with 1. So v = u0(l 4- 3 aQt) ; or, finally, writing 60 for 3 a0 b0 is called the " coefficient of cubical expansion referred to 0° C.v ; and its numerical value is, from what has just been proved, three times that of the linear coefficient. Similarly, if the body is not isotropic, and if a, a', a" are the coefficients of linear expansion along the three axes, we may write for the lengths of three lines parallel to these at t° C. : ro = m0(l + a'), n = n0(l Consequently, multiplying these three equations, we have the formula, v = v0(l + at) (1 + 00(1 +a"0 = »0[1 + (a + a' + a")G» if thf i\|ii:mfitirga, a', and a" are all small. If we write b0 = a + a' + a", v = v0(l + &„/), UH before. In < •« -i -tain crystals one of the coefficients of r\p;iii-ii.M i> !i.--aiiv«', i.e. lengths parallt-1 to one of the as the temperature rises: ami. it its value excee<ls niimrrically ii "f ih.. other two coefficients, the volume \\ill contract with Increased temperature. If a In.llow body has its temperature changed, it .\pan.ls or contracts exactly a.s if it ha<l no cavities; these spaces increase in -\/>- or <liminisli according to the same law as \}\>- ^.li-l ;.. .:-ii..n>. This is at once evi«l--nt if \\.- n-^anl the body as built, up of solid blocks, and consider the change !/.«• of thes,. M,,cks as the temperature varies. An iron i; nit on n ious coatings on a large gun, by first Dg th.-ir ' im until they \\ill just slip in place, and then cooling them. 232 HEAT The following table contains the values of the coefficients Brass of linear expansion of a few familiar solids : ' . 0.000019 . 0.0000168 . 0.0000083 . 0.0000121 0.0000090 Copper Glass . Iron Platinum Steel (annealed) . . 0.000011 Steel alloyed with 36% nickel . . . 0.00000087 Zinc , 0.0000292 Illustration of Expansion. — The fact that the coefficients of expansion of different bodies are different is often made use of to neutralize the expansion which ordinarily follows rise in temperature. Thus, the period of a pendulum clock would naturally increase with rise in temperature owing to the expansion of the pendulum rod ; but this may be avoided in several ways. One is to make the pen- dulum as shown in the cut, which illustrates Harrison's gridiron compensated pendulum. In it the rods which are shown as single black lines are of iron and the others of brass. It is seen that, as the temperature rises, the expansion of the iron rods lowers the pendulum bob, but that of the brass ones raises it. The linear expan- sion of brass is about 1£ times that of iron ; and so if the combined length of the two iron rods on each side and the middle one is 1* times that of the two brass ones on each side, there will be no change in length as the temperature varies. (A simpler method is used in the clock in the tower of the Houses of Parliament, London. The pendulum consists of an iron rod sur- rounded by a zinc cylinder, which in turn is surrounded by an iron one carrying the pendulum bob. The lower ends of the iron rod and the zinc cylinder are attached, and the upper ends of the two cylinders. Since the linear expansion of zinc is about 2£ times that of iron, the combined lengths of the iron rod and cylinder must equal 2i times that of the length of the zinc cylinder.) Graham's compensation pendulum consists of an iron rod whose lower end screws into a cast-iron cylindrical vessel partially filled with mercury. The quantity of mercury is so chosen that by its expansion combined with that of the iron rod and cylinder the centre of gravity of the whole does not move when the temperature is varied. FIG. 113. — Com- pensation "grid- iron " pendulum. The single black lines represent iron rods ; the double lines brass ones. Again, the period of a watch or of the ordinary spring clock is regu- lated by the vibrations of the balance wheel. This consists of a small Cll . 1 \ ' . ES IN VOL I M /•; - 1 V It PRESSURE 233 wheel attached to a flat coiled spring, which coils and uncoils, making harmonic vibrations. If the temperature rises, two changes occur: the elasticity of the spring is decreased, and the wheel expands. Owing to both these effects, the period of the spring would be increased were it not for the peculiar <a^<^ construction of tin* rim of the wheel. As is shown in the cut, this consists of two (some- times tlnve) parts or sections. One end of >tened to a spoke, but the other is free. Near this latter end are screws or mov- able weights. Each section of the rim is double, consisting of iron on the inside and brass on the outside. Consequently, owing to the greater expansion of the brass, when the Fro. iu. — Balance wheel of a temperature is raised each section of the rim watch ; the Inner P0100 of the rim is iron ; the outer la brass. is made more curved, and the weights are brought in nearer the centre. This decreases the moment of inertia, in i«l thus tends to decrease the period of vibration. Therefore, by suit- ably adjusting tin- \v rights or screws, the balance wheel may be made to r-'t.iin a constant period, however the temperature changes. ite recently Guillaume has discovered that an alloy of nickel and M«'-l in the proportion of 36 parts of nickel to 64 parts of steel has a coefficient of linear expansion equal to 0.00000087; and so its expansion l»e neglected in all ordinary apparatus or work. The change in length of this alloy with change in temperature takes place very slowly if tli. Hue in t. inj>«-rature is small; for a rod made of it does not reach its full exj>.in>i<>ii for one or two months. Liquids Apparent and Absolute Expansion. — In measuring the change in volume ••!' ;i liquid, mviu^ to any cause, a ditli- culty is met which has been referred to heiWe ; namely, the fact that the liquid must !•«• contained in a solid vessel, and whatever affects the volume of the former will also, iu general, alteri that of the latter. Thus, \\heu the contain- ing vessel is a hull) with a tube attached, if the liquid tills the hull. ;md pan ,,f the tube, it is observed that, when the l>ull> is heated suddenly hy immersing it in a basin of hot i «>r in any other \va\, the top of the column of liquid 234 UK AT in the tube immediately sinks and then rises gradually, ascending finally higher than it was originally. This is owing to the fact that the first effect of the application of the hot bath is to raise the temperature of the bulb, and it therefore expands before the liquid inside is affected ; as soon, however, as its temperature is raised, it expands and rises in the tube. If the bulb is chilled, instead of heated, the reverse of these changes takes place ; the liquid first rises in the tube and then sinks, falling below its original position. It is evident, then, that the apparent change in volume of the liquid is less than the real change by an amount equal to the change in volume of the containing solid. So, if the coefficient of expansion of the solid is known, the direct method for determining the change in volume of a liquid that accompanies a change in temperature is to inclose the liquid in a bulb with a finely divided stem whose volumes are known, and to measure the volume of the liquid at any one temperature and the apparent change in volume when the temperature is altered. Let vl be the initial volume at the temperature t±, and v the apparent increase in volume when the temperature is raised to £2° ; furth
er, let the coefficient of cubical expansion of the solid have the value b. Then the increase in volume of the solid is v1 [1 + b (£2 — ^)] ; and hence the true change in volume of the liquid is v + vl [1. + 6(£2 — tfj)]. Experiments show that fyor nearly all liquids — water is an exception, as will be explained below — the relation between the change in volume and that in tempera- ture is of the same form as for solids, viz., v2 — v1 = vj) (£2 — ^), where b is the coefficient of cubical expansion of the liquid, referred to ^°. If the initial temperature is 0° C., this becomes, as before, v = vQ(l + bQt~) ; and it is to be noted that b = Q — , and therefore only if 50 is small can it 1 + bQt replace b. The coefficient of expansion is found to be different for CHANGES IN VOLUME AND PRESSURE 235 different liquids, as is shown in the following table. It should b»- noted that the expansion of liquids is, in general, m in -h greater than that of solids. Kthyl alcohol . . between 0° and 80° C. 0.00104 Kthyl ether . . . between - 15° and 38° C. 0.00215 (ilycerine . . . 0.000534 •ury . . . between 0° and 100° C. 0.000182 Turpentine . . . between - 9° and 106° C. 0.00105 It is evident that, if the coefficient of expansion of a liquid is known, tfl of observations and measurements gives a method for '{••termination of the coefficient of cubical expansion of the solid that contains the liquid. This method therefore can be used for all solids that can be formed into bulbs, e.g. glass or quartz. Mercury is the liquid that is, in general, used, because its expansion is known, and for r obvious reasons. Measurement of Coefficient of Expansion. — A better method, however, for the determination of the coefficient of expansion liquid, which does not involve a knowledge of that of solid vessel, was devised by Dulong and Petit and im- proved by Regnault. It depends upon the fact that the titfl at whieh two liquids of different densities stand when they balance each other in a (J-tube is independent of the material of the tube. If 7/t is the height of the column of LUC liquid of density dr and 7/.2 that of the column of the liquid whose density is dT -* == -2 (see page 176). But the formula for expansion, 2 l l>e replaced by v = 0(1 + /, 4 = ''0 + V). ii the density at £°, and </,, that at 0°; for tin- density of any ln»dy whose mass remains constant varies inversely as its volume (m = dv). Therefore, if the ratio of the densities of " liquid at <» and /' is known, its coefficient of expansion can be deduced at once. 'I'!"1 simplest i.»nu of the actual experiment is as follows : A tul..- ifl made in the form of a W- with a cross tube at the 236 HEAT top, as shown in the cut ; this last has a small opening -4 near its middle point and is kept horizontal ; there is a branch tube joined to the apparatus at B, which is con- nected with a reservoir K containing air that can be compressed to a pressure greater than that of the at- mosphere ; a quantity of the liquid whose expansion is to be studied is poured in, just sufficient to flow out of the opening A in the upper cross tube, thus insuring the con- dition that the upper level surfaces of the two columns are at the same height. These two separate portions of the liquid are then surrounded ==H: FIG. 115. — Apparatus for the determination of the coefficient of expansion of a liquid. by baths, one at 0°, the other at t°. When equilibrium is established, the difference in level of the two surfaces of one portion of the liquid is not the same as that of the other. Call this difference for the liquid at 0°, A0, and that for the one at t°, h. Then dQghQ = dyh, or ¥& = —. Consequently, if hQ and h are measured, or d ' may be calculated ; for, ^ = 1 + bQt. .-. 1 + b0t = n, Expansion of Water. — When water is studied, it is found that as the temperature rises from 0° C. to 4°C., it contracts; but above 4° it expands. These changes in volume of water (or of any substance) which accompany changes in tempera- ture may be best shown graphically. Lay off two axes, one to indicate temperatures; the other, volumes of a definite CHANGE l.\ VOLUME .l.\/> 237 quantity of the substance. For water the curve giving the connection between v and t is as shown in the cut. (If the coefficient of expansion is a con- .t, the curve is a straight line sloping upward from left to right.) This fact, that the density of water is a maximum at 4° C., and that it decreases continuously from this temperature to 0° and to 100° is of great importance in nature ; for as the temperature of the water in a lake or a pond sinks in winter below 4° and reaches 0°, the water at 0° floats on top and does not sink to the bottom, as would be the case with other liquids; and consequently ice forms on the u]>]H-r surface, not at the bottom. TEMPERATURES CENTIGRADE 0 2 4 6 8 10 This jN»culiarity of water is explained by assuming that when ice melts the liquid formed is not at first made up of molecules that are all alike, but contains two kinds, "ice molecules" and «T molecules." The former are assumed to be lighter than th«« latter; and, as the temperature rises, they are transformed gradually .•han-,,1. Ki<;. 110. — Curve showing expan- sion of water as the temperature is he latter. Gases Expansion at Constant Pressure. -—When the temperature of a gas is increased, its volume increases also, unless special autions are taken; but, if the external pressure varies •luring tin- change in temperature, the change in volume is moditi. •«!. \Vt» shall then assume that during the change the rare is maintained constant. A gas is always inclosed in some solid vessel: and its expansion must a 1 \\ays be taken i i it «> account although its effect is small in mmpai -ison \\ith the - n of the gas. When this is done, experiments that the general formula for expansion is true. 238 HEAT This coefficient of expansion is constant for any one gas ; and, further, it is found by experiments to be practically the same for all gases. This is known as the " Law of Gay- Lussac." The value of this constant is very nearly 0.0036600, or 2^-j, using the Centigrade scale. It is to be noted, then, that gases expand more than liquids. (The apparatus used by Gay-Lussac in his investigation on the expansions of gases is shown in the cut.) General Law for a Gas. — Having thus determined the relation between the volume and temperature of a definite quantity of gas when the pressure is kept constant, and ° ^ A -T-L I A. M ( ( '-& &? B i i o o FIG. 117. — Gay-Lussac's apparatus. • knowing — as stated in Boyle's law — the relation between volume and pressure when the temperature is kept constant, Ti° we can combine the two and deduce an expression that will include them both. If we start with a given quantity of gas at 0° and at a pressure p, it will have a definite volume which we can call VQ. If now the temperature is raised to £°, the pressure being kept constant, the resulting volume v = vQ(l. 4- £0£). If the pressure is then changed to P, the temperature being kept constant, the volume will be altered and the new value F'must be such that PV=pv=pvQ(\ + #0£). Thus, if the volume of a gas is measured at a temperature t° CHANGES IN VOLUME AND PRESSURE 239 and pressure 1\ the volume whieh this same quantity of gas would occupy if its temperature were made 0° and its pres- sure y> ean be calculated; for ro-7 _p v (The "standard conditions" for a gas are a temperature of 0° C., and a pressure of 76 cm. of mercury ; and when the ••density of a gas" is referred to, it is always implied that the gas is in its standard condition, unless the contrary is stated.) This formula can be expressed in a simpler form, viz., PF or - 1 1 1 1 1 VQ is the volume of a given quantity of the gas at 0° and at pressure JD, consequently the product pvQ is a constant for PV this gas by Boyle's law. Therefore, •= - is a constant for \ any one gas, but has a different value for different gases. This m a y - - = 7 a °o i • / * any one gas. The quantity — + e, which, using the Centi grade scale, approximately equals 273 + 1, is called the i + fyabsolute gas temperature" on the Centigrade scale, and is written T. It is evident that, if different quantities of til- same gas at the same triii)><-r.ttinv and pressure are taken, their volumes will vary directly as the masses. Thus, writing M for the mass of the gas, the complete law is 240 I1KAT The constant R can be found for any gas by measuring its_ density at a known temperature and pressure. Thus, if t° C. is the temperature, p the pressure, and d the density, f -^ f°r hydrogen is, 4.14 x o -j- bo 107; for oxygen, 0.259 x 107 ; for nitrogen, 0.296 x 107. If the same law were to hold for a gas as the pressure became smaller and smaller and finally vanished without the volume at the same time being made infinitely great, the temperature of this condition would be given by T = 0, or t° C = - - = - 273° C. approximately. Similarly, if the same law could be applied to a gas as its volume was made less and less and finally vanished, the pressure remaining finite, the temperature would be given by the same value, viz., — 273° C. approximately. Of course the above law for a gas does not apply to one if greatly compressed, and the pressure of a gas can become zero only by its volume being infinite ; and so the above deductions have no physical meaning. We shall see later, however, by other reasoning, that this temperature -- , or C, 273° — can reach. For that reason it is called the "absolute zero." (See page 308.) universe lowest which marks body any our the in temperature Change of Pressure at Constant Volume. — Another fact is apparent from this formula. If the changes take place in such a manner that the initial and final volumes are the same, V= i>0, and therefore P = p(\ + b0t). Therefore the pressure increases at the same rate with increase of tempera- ture when the volume is kept constant, as does the volume when the pressure is kept constant. (Ac
tually, this is not exactly true of any gas ; but this discrepancy is a consequence of the fact that Boyle's law is not exact for an actual gas urn- is the coefficient of expansion a constant.) Laws of Gay-Lussac and Charles. — The fact that all gases have approximately the same coefficient for change of pressure when the volume is kept constant was discovered by the French physicist Charles ; while the corresponding one that all gases expand alike when the pressure is kept constant CHANGES IN VOU'Mi: AM) PEM88URM was discovered a few years later, ill 1802, by Gay-Lussac. se statements of fact are therefore called Charles's and -hussar's lav Other Forms of the Gas Law. — This formula for a gas, /' /'— HM'l\ ma\ be expressed differently and more simply if we assume the truth of Avogadro's hypothesis (see page I. If m is the mass of each molecule, and N is the num- ber in a unit volume, M= mNV\ consequently, on substitut- ing in the formula, we have ' P = RmNT, p RmT or N= - — - . But by Avogadro's hypothesis, if the pressure and temperature of two gases are the same, they have t In- sane number of molecules in each unit volume : that is, if P ami T air thr same for two gases, N is also. Therefore. fn»m the above formula, Rm must be the same for all gases; it is a constant of nature. Calling its value RQ, the formula becomes P = R0NT, which states that the pressure in a gas (or a mixture of gases) varies directly as the number of molecules in a unit volume and as the absolute temperature. This equation, in turn, may be expressed in a form which is nioiv us.-1'ul for practical purposes, l.rraiisr. of course, we have no accurate knowledge of the value of N. The " molec- ular weight " of a !_cas has hceii defined to l»e a number, char- acteristic of the Leas, which is proportional to the mass of one te molecules, and which is so chosen that the number for oxygen i> :\~2. Thus if //' is the ///••/,•,•///,//• //•./;//// of a gas, each of whose molecules has the mass 11,10= <•>//, when :it foi- all gases and has such a numerical value as to //• r.jiial :\'2 for oxygen. A numb' unsofasuh- I06 iMjual to its molecular weight (e.g. 82 gram- Q\| oxy- gen) is called uone gram-m«.l. "one mol" of that substance. Thus, if there are .V U of a gas in a unit vol- Miue. ['- I and M=Nic = N'cm in the general formula: hrnr.e, P=RmcN'T= R AMES'S PHYSICS— 16 242 HEAT The product R^c is a constant, the same for all gases ; call itR1. Hence P = R'N'T, or, the pressure varies directly as the number of mols in a unit volume. The value of R1 may be found by experiments on the densities of gases, because P is the pressure of a gas at temperature T when the density is such that there are N1 mols in a unit volume. It is found to be 8.28 x 107 on the C. G. S. system. Energy Relations during Expansion Mechanical Expansion. — When the dimensions of a body are increased, either by the addition of heat energy or by mechanical forces, there is a certain amount of external work done and there are alterations in the kinetic and potential energies of the molecules. In the case of the thermal effect this is evident from what has been already said ; change in relative position of the molecules, change in temperature, and external work occur together. Similarly, when a brass wire is stretched, its temperature falls ; when a gas is com- pressed, its temperature is raised, etc. We might have predicted that these changes in tempera- ture would be as just stated. (See page 104.) If the tem- perature of a brass wire should rise when it is stretched, it would be in unstable equilibrium. If the wire hangs verti- cally under the stretching force of a heavy body, and if a sudden downward blow is given this weight, it will move down, stretching the wire still more, then come to rest and move up, etc., making harmonic vibrations, showing that it was in stable equilibrium. But if, owing to this stretching, as the hanging body moves down, the temperature of the wire were to rise, it would lengthen ; and this increase in length would cause another rise in temperature, etc., so the equilibrium would be unstable. Therefore, only if the wire CHANGES l.\ VOLUME AND PRESSURE 243 cools when stretched, is the equilibrium stable. The general law is that, if a body expands when its temperature is raised, its expansion by mechanical means will cool it, while mechan- ical compression will raise its temperature. The converse is true of those bodies which contract when their temperature is raised, e.g. water between 0° and 4° C. As the sun radiates energy, it contracts; and owing to this cause its parts are slowly coming closer together, and therefore energy is being liberated to make up for that loss. Thus, in ordinary language, the sun owes its heat to its slow contraction. Undoubtedly also, meteoric pieces of matter are falling into the sun, and their energy is thus also added to that of the sun. If the expanding body is a liquid or a gas, or if it is a solid immersed in a fluid, the amount of external work done equals the product of the increase in volume by the pressure. (See page 159.) The internal changes consist of change in kinetic energy and change in potential energy. The first of • •is connected intimately with changes in temperature, as has been already shown. (See pages 197 and 220.) Changes in potential energy occur if there are molecular forces. The fact that these exist in solids and liquids and are large is evident from the obvious properties of these forms <»f matter, e.g. they retain definite volumes; but noth- ing is known in regard to the amount or cause of these forces. The case is different with a gas, for in it the forces are extremely small. (Nothing, however, can be said as to their ran Internal Work in a Gas. — This fact that the internal forces in a gas are extremely small was first shown by Gay-Lussac and latrr by Joule, working independently and also in col- laboration \\ith Thomson (now Lord Kelvin). The earlv exporinirnts «,f .Iniilr arr perhaps the simplest to consider. II apparatus consisted of l\\o strong metal cylinders con- bed b\ ;, tube in which was a stopcock. In one of these cylinders quantities of the gas to be studied were compressed 244 II EAT until the pressure was as high as the apparatus would permit ; while from the other the gas was exhausted. The whole apparatus was then submerged in a tank containing water, and this was stirred until the temperature came to a steady state. Then the stopcock in the tube connecting the two cylinders was opened; and, when the gas had redistributed itself, occupying a greater volume and thus coming to a smaller pressure, but doing no external work, the temperature of the water in the tank after being well stirred was again observed. In no case was there any meas- urable change. If there had been molecular forces of attrac- tion between the molecules, it would have required work to separate them when the gas increased in volume, this energy would have necessarily come from the molecules themselves, which would have thus lost kinetic energy ; and therefore the temperature of the gas would have fallen. If there had been molecular forces of repulsion between the molecules, their potential energy would have been decreased by the increase in volume, their kinetic energy would then have increased an equal amount ; and therefore the temperature of the gas would have risen. Either of these changes in the temperature of the gas would have affected the temperature of the surrounding water ; and this would have been observed by the thermometer, if it Fio. 113. —Joule's apparatus for study- Ing the free expansion of a gas: two cylinders, R and J£, are connected by a tube in which there is a valve ; the whole is immersed in a tank containing water. were sufficiently delicate. Consequently Joule's experiments prove that to the degree of sensitiveness of his thermometer, there are no forces between the molecules of a gas ; or, in other words, the internal energy of a gas is entirely kinetic. The later experiments of Joule and Thomson show, how- VBANQEb l.\ VOLUME A.\l> 246 ever, that gases do have measurable molecular fon-es, al- though they are extremely small, and that these in general are forces of attraction. (In hydrogen, at ordinary tempera- tures, the forces are repulsive ; but at very low temperatures they are attractive.) Expansion of a Gas when External Work is Done. — Joule modiiied hi> experiment by inclosing the two cylinders in sep- urate tanks <>t' water; and, when the expansion took place, he noted the temperatures of the water in the two tanks. He observed that the temperature of the one holding the cylinder in whieh there was the high -sure fell, while that of the other rose. The explanation is evident : the gas that stays behind in the high pressure cylinder after the expansion, has done work in giving kinetic energy to the escaping gas a% awhole, i.e. in producing a wind, consequently its temperature falls. This hlastof gas entering the other cylinder soon ceases, owing iscosity, and the kinetic energy of the moving gas passes into the energy of the molecules, and is apparent by a rise of temjM -rat u re. In other words, the gas in the first cylinder expands, doing work, and so its temperature falls; work is done upon the other gas in setting it flowing, and so its tem- perature rises after the motion is stopped by friction. 1 10. liy. — Jouie s second exporuueu Met that when a gas expands under such conditions as to do work its temp'-ratiin- falls in shown l»y all«>\\in- (lain)- air inclosed in a vessel to expand -udd.-nly. I f t li.-n- an- nuclei in tin- air. drops of water will !»«> loosed around them, thus forming a visil.li- mist and showing that the air has been chilled. (See page 186.) This is the explanation in many cases of the formation of clouds in the air. Expansion of a Gas in General. — When a compressed gas ilowed to expand out through a fine opening into a space 246 HEAT where i
ts pressure is less, changes in temperature take place owing to numerous causes. We shall consider two cases of practical importance. Let a cloth with fine meshes — a piece of cheese cloth or toweling — be folded over the nozzle from which the gas is escaping ; it will expand with violence through these numerous openings into the air, thus forming a wind ; the gas that does not have time to escape during any small interval thus does work on that portion which it blows out and so experiences a fall of temperature itself. This may be sufficient to liquefy or even solidify it ; and the cloth will in this last case be found to contain the solidified gas. (This is the ordinary way of obtaining carbon dioxide in the solid form.) Again, let the opening or openings through which the compressed gas escapes be so fine and tortuous that the outcoming gas has no kinetic energy as a whole, i.e. there is no wind, it does noi flow ; in this case its temperature is lower than when in the compressed con- dition (except with hydrogen at ordinary temperatures), as shown by Thomson and Joule, owing to the fact that there are minute molecular forces of attraction, and as the poten- tial energy of the expanded gas is increased, its kinetic energy, and therefore temperature, must be diminished. This fall in temperature varies directly as the difference in pressure of the gas in its two conditions, and so may be considerable. In any ordinary expansion of a gas from a small opening, both of these actions take place. The gas that is escaping at any instant has done work in pushing out the portion that was just before it, and so its temperature is lowered, quite apart from the influences of the molecular forces. (Some distance away from the opening, however, in the case of a jet or a blast, the temperature of the gas is increased, owing to the friction of the moving currents.) This method of securing a decrease in temperature by allowing a gas to expand through a fine nozzle is being used practically in recent machines for the liquefaction of gases. (See page 280.) CHAPTER XII CHANGES IN TEMPERATURE Energy Relations when the Temperature is Raised. — It is 1)\ tin- change in temperature of bodies when exposed to some "source of heat" that our attention is directed to heat phenomena ; and it is in terms of changes of temperature that heat energy is measured, as we have already seen in the definition of the "calorie." If, however, we wish to deter- mine lm\v much energy goes to producing the rise in tempera- ture, it is necessary to ascertain how much is used in doing : nal work, and how much in overcoming the molecular forces. If there is a uniform pressure p over the body, and it its volume increases from v1 to vv the external work done against this pressure is p (v2 — Vj), and so may be calculated. It is only, however, in tin- case of a gas that anything is known <iuant it atively in regard to the molecular forces. 'Ill- '.hen the temperature of a solid or a liquid is raised, although tin- external work may be calculated if the external pressure is known, it is impossible to separate into its parts the energy that is spent in internal work ; and so do not know how much goes to producing rise in tem- r.ut tin; case of a gas is different. We can date the amount of external work done as well as meas- ure it. and we know further that the molecular forces are SO 1 that they can be neglected. ( '..us. -.jin -ntly when the temperature of a gas is raised owing to the addition of heat _ry, if we deduct the amount that is used in doing exter- nal work, the d nice is all spent in raising the tempera; i.e. in increasing the kinetic energy of the mole. ules. The Ml 248 HEAT amount of energy that must be added to the gas, then, in order to raise its temperature a definite number of degrees, depends upon the amount of external work done by the gas ; that is, upon the external conditions under which the gas is kept during the change in temperature. If this change is the same in different experiments, the energy that is spent in producing the increase in the kinetic energy of the mole- cules is the same whatever the external conditions ; and so the difference in the amounts of heat energy added must equal the difference in the amounts of the external work done. Special Case of a Gas. — For various reasons, partly prac- tical and partly theoretical, the change in temperature of a gas is considered, as a rule, under two different conditions : (1), when the volume is kept constant, and so no external work is done ; (2), when the pressure is maintained constant, so the external work equals the product of the values of this pressure and the change in volume. If Q1 is the heat energy added in one experiment when the volume is kept constant, Q2 that in another when the pressure is kept constant at a value p while the volume is increased from v1 to Vy, the change in temperature being the same in both' e^o, =,(.,-,>, where, of course, (?2 and Q1 are measured in mechanical units; i.e. in ergs or joules, since the product p(v2 — v^) is so ex- pressed. Reverse Changes. — When a body cools, it gives out heat energy ; but the external work done on it by the external forces is not necessarily the same as is done by the body against these forces when the body is heated and rises through the same range of temperature, because the external forces may have changed, or they may vary differently dur- ing the two processes. If, however, the pressure remains unchanged, or, when it is variable, if the series of changes is CHANGES 1\ il-MI'l-HMTRE tly reversed during the two processes, the external work done by the body during the rise in temperature equals that • on the body during the opposite change. If this is the case, the heat energy added in the former process must equal that LTiven out during the latter; for at the end of the two diaii^rs tin* body is hack in its original condition, and so there is no change in its internal energy, and there has also been no gain or loss of energy owing to external work. This fact serves as the basis of all methods for the measurement of quantities of heat energy. Many simple experiments show that, to produce the same change in temperature in equal quantities of different bodies, requires different amounts of heat energy. One of these, due to Tyndall, is to raise to the same temperature in a bath of heated oil several spheres, made of different materials, but of the same weight and of the same size (natu- rally, some are hollow), and then to place them upon a cake of wax that melts at a low temperature. As the spheres cool down they melt <li tit-rent quantities of the wax, as is shown by their entering the wax to different dUtancf.s; and at the end of the process they all have tin* same tem\|x>rature. This proves that in cooling over the same range of tem- perature different ] • ••• out different quantities of heat energy. - experiment does not pro\.- this fact beyond question, l.e.-au- eomi'lir.ite.l 1,\ differ.-nws in the conductivity of the different bodies and l.y the rnn-eqii'Mit fact that some give om more quickly than <>th. -i •<. and so more of the wax is melted before it can conduct the energy away. There are other actions, too; l»ut in reality the main effect is that descried above, and so the experiment is a proper one to illustrate the question referred to.) Measurement of Heat Energy. — Tin-re are three methods in if.-in-ral use I'm- tin- III«M^II i viiHMit of the heal en- required to raiae the temperature of bodies, or of that given Ifhen a body cools : (1 » To put tin- body in a hath of r at a different temper, tt mv. and measure tin- change in perature of tin- lien equilibrium is reached. It the maSB of tin- \vatrr is in ^rams. and it^ dian^c of tempera- ture i.s / ('.. //// calories enter or leave the water, if \\«- neg- bhe slig] oni in the value of tin caloric at different 250 HEAT temperatures. (2) To put the body in a bath of ice and water, so that a certain amount of ice is melted. It will be shown later that it requires 80 calories, very approximately, to melt 1 g. of ice at 0° C. ; and so, if m grams are melted, the ice must receive 80 m calories. (3) To put the body in a bath of steam, so that a certain amount of this is condensed into water by contact with the cooler body. It will be shown later that 536 calories, approximately, are given out by 1 g. of steam when it condenses into water at 100° C. So, if m grams are condensed by the body at this temperature, it must receive 536 m calories. "Specific Heat." — The number of calories that must be added to a body whose mass is 1 g. in order to raise its temperature through one degree Centigrade, from t° to (t + 1)°, is called the "specific heat" of that body at t° C. It should be noted that the specific heat of a substance is a number that is independent, in a way, of the heat unit used or of the tempera- ture scale adopted. We could adopt as a heat unit that quantity of heat energy required to raise the temperature of a unit mass (on any system) of water from t° to (t 4- 1)° (on any scale) ; and define the specific heat of a substance at t° as the number of these heat units that is required to raise the temperature of a unit mass of it from t° to (/ + 1)°. Or, if we assume that the specific heat is the same at all temperatures, we may define the specific heat of a body as the ratio of the amount of heat energy required to raise the temperature of the body through any range of temperature to that required to raise the temperature of an equal mass of water through the same range of temperature. With solids and liquids, as has been explained, we assume that the pressure is that of the atmosphere ; but with gases we distinguish two special conditions, constant pressure or constant volume, and so have two corresponding specific heats. For most bodies the specific heat is practically con- stant for all temperatures that are far removed from the melting points; and so,
if c is the specific heat of a body whose mass is m, and if the temperature is raised through <2 — £j degrees, the number of calories added is mc(t^ — ^). C7/.l.V(;/:> /.V 'IKMrKHATURB 251 In iron, boron, carbon, and a few other substances the specific heat varies to a marked degree at different temperatures ; and with them c in the above expression is the "mean spe- cific heat " for the range from tl to tv Measurement of Specific Heat. — Corresponding to the three methods of measurement of heat energy referred to above there are, then, three methods for the measurement of specific heat of a given substance. We assume in each case that there is no loss of heat by radiation, etc. 1. Method of Mixtures. If iw, = mass of body, wij = mass of water, /, = original temperature of the body, fa = original temperature of the water, ts = final temperature of equilibrium, ~«ift-W OF m2(<3-<,) The water is contained in some solid vessel, called a " calo- rimeter " ; and its temperature is affected by the changes in that of the water. If mz is its mass and c1 its specific heat, and if we assume that the changes in its temperature are the same as those of the water, the number of calories it receives is w8c'(/8— £a); and, since this energy comes from the foreign body hit rod u« •« -d into the water, it is seen that, in the above formula t'.u- <?, wa must be increased by w8<?'. This quantity !lrd tlir M wuter equivalent" of the calorimeter. mrtlind was fust used by Joseph Black about 1760; and it is the one most generally used at the present time. Tli.-ivaiv in an v objections to it, however. Chief among these are tin- ditVn -ulties of determining the water equivalent of the calorimeter and of avoiding losses by radiation, etc. Most of these are overcome in a modification of the apparatus due to l'rof«->sor \Vah-nnan <>f Smith College. 252 HEAT 2. Method of Melting Ice. If w<! = mass of body, 7/i2 = mass of ice melted, t = initial temperature of the body, m^t = 80 w?2. This method was also first used by Joseph Black. The obvious difficulty in it is the measurement of the quantity of ice melted. This has been overcome most successfully in a form of apparatus due to Bunsen ; but great skill is required to use it properly. 3. Method of Condensation of Steam. If 7/jj = mass of the body, m2 = mass of steam condensed, /! = initial temperature of the body, t2 = final temperature of the body, which is never far from 100° C., mlc(f2 ~ ll) ~ 536 m2' With this method there is a correction for the water equivalent of the calorimeter; and one of the chief difficulties is to measure accurately the quantity of steam condensed. The apparatus used was invented by Professor Joly of Dublin. Specific Heats of a Gas. — Any one of these methods can be used to measure the specific heat of a solid or a liquid ; but with gases a difficulty enters owing to their small density — the correction for the calorimeter is the larger part of the heat energy. To measure the specific heat of a gas at con- stant volume, the third method may be used if. the gas is compressed into a hollow sphere; but this is not very satis- factory. To measure the specific heat at a constant pressure the best method is to pass a large quantity slowly through a spiral tube which is immersed in a bath at a high tempera- ture, then through another spiral surrounded by water which is cooler and whose temperature is thus raised, and finally r/M.y ';/•;> L\ I i-'.Mi '/•;/,'. i /T/;/-: 253 out into the air or into some large reservoir. If the gas is forced through very slowly, its pressure remains practically constant. There are two indirect methods by which the specific heat of a gas at constant volume may be determined : one depends upon a knowledge of the ratio of this to the specific heat at tant pressure ; the other, upon a knowledge of the differ- ence between these two quantities. It may be proved by higher mathematics that the ratio of the specific heat at constant pressure to that at constant volume equals the ratio of the adiabatic coefficient of elas- ticity to the one at constant temperature (see page 194). But this last ratio is a constant for any one gas, which may be determined by measuring the velocity of compressional waves in this gas, as will be proved in the next section of this book. (See page 337.) It will be shown there that this velocity is given by the following formula, V = where p is the pressure of the gas, d is its density, and c is the ratio of the two elasticities, and therefore of the two specific heats. It is not difficult, then, to determine <?, since v, p, and d can all be measured; and. if < \, and Ov are the two ilic heats, one at constant piv>surc the other at con- 1 volume, Cp=cCv\ so, if <: and ('/( arc known. Of can be calculate. 1. The difference (7p— (7, expressed in mechanical units, eqna Is. i what has been ^aid on page 248, the amount of external work done when 1 g. of the gas expands at constant pressure/?, the temperature rising through 1°C. This work eq luils p(y^— r{ ). wl id r} are the volumes of the gas at two temperatures dill'erin^ l.\ 1°C. We may calculate this l.y ii>in-- the -_ras f. -nil n I i '' = Rm. Iii this case m= 1, 254 HEAT hence pvl = ET^ and pvz = RTV where Tz- Tl = 1. There- fore p(v2 -v^ = fi. So that Q, - Ov = R. If Cp and (7, are measured in calories, and if J is the value of a calorie in ergs, «/( Op — Cv ) = .#. The value of R may be found by ordi- nary methods of gas measurement, as has been shown; and, therefore, if the value at Cp is known, that of Cv may be calculated. Ratio of the Specific Heats. — We may obtain a better physical conception of c, the ratio of the two specific heats, by interpreting it in terms of the kinetic theory of gases; i.e. by considering the properties of a gas as identical with those of a set of elastic spheres. The internal energy of a gas may be regarded as entirely kinetic; but this energy is not entirely energy of motion of the molecules themselves. The phenomena of radiation show that as the temperature of a gas is raised, the energy of motion of the parts of the molecules is increased. Let us assume, following Clausius, that the entire kinetic energy of a molecule — that is, the energy of its parts and its own energy as a whole — is proportional to its energy of transla- tion. We can write it then \ mbVz, where m is the mass of the molecule, V its velocity of translation, and b a factor, which has the value 1 if there is no motion inside the mole- cule, and otherwise is greater than 1. If ^Vis the number of molecules in a unit volume of the gas, the energy of the molecules in this volume is then \mW^N. But wiN is the mass of the molecules in this volume; so the internal energy of a unit mass of the gas is \ bVz. This can be expressed in terms of the temperature; for the pressure of a gas is given by the formula p = dRT, if d is the density; and on the kinetic theory p = ^dF2. Therefore V* = $RT\ and the internal energy per unit mass is J bR T. So, if the tempera- ture is raised one degree, this energy is increased by an amount f bR. This quantity, then, is the specific heat of the gas at constant volume expressed in mechanical units ; or, in CHANGES IN TEMPERATURE 255 symbols, Cp = f bR. But Cp - CV=R-, hence Cp = Cv + -f 1). Hence the ratio of the specific heats __ ~~ 21 + The least value of b is 1 ; and therefore the greatest possible value of c is 1 + f or 1.67. For all other values of b, c is less than t Ins. \| 6 It is a most striking fact that for certain gases, viz., mer-^ cury vapor, argon, helium, and a few others, the value of c found by experiments on the velocity of waves in them is 1.67, while for all other gases it is less than this, being about 1.41 for air, hydrogen, and oxygen, 1.26 for carbonic acid etc. Those gases for which c equals 1.67 are called by i-li'-mists "monatomic"; and, whatever value may be attached to the above assumptions, it is certain that a large value of c indicates an extremely simple construction of the molecule or a molecule whose internal energy is small; while a small value «»f /• indicates the contrary. A few values of specific heats are given in the following tabi AVERAGE SPECIFIC HEATS Alcohol . . 0°-40°C. 0.597 Aluminium . 0°-100°C. 0.2185 Brass .... 0.09 Copper . . 0°-100°C. <> (crown) . . 0.161 In.n . . 0°-100°C. 0.11:1 , 0°-100°C. 0.031 Mercury . Paraffin . Platinum Bihra Tin r, Water . Turpenti iii- . 20°-50°C. . 0°-100°C. . 0°-100°C. . 0°-100°C. . 0.o:W 0.0:^:5 0.467 1.00 Air .... j Chlorine . Carbon dioxide (JASKS . o.i»:J7 . . 0.1'Jl . . O.'JO'J Of 0.171 "iv.; RATIO . l.40i . 1.06 . ; l.:to 256 HEAT GASES — Continued Cp Cv RATIO Helium Hydrogen .... 3.40 . . 2.40 Mercury (vapor) Xitrogen 0.244 Oxygen 0.217 .(17 .41 .41 .41 Law of Dulong and Petit. — When the values of the spe- , cific heats of a great many substances are compared, a connec- tion becomes evident between them and the " atomic weights " of the substances. (For an explanation of this last quantity some book on chemistry should be consulted.) This was first noted by Dulong and Petit. It is found that the product of the value of the atomic weight of any solid substance and that of its specific heat is a quantity that is approximately the same for all substances, viz., 6.4, using the ordinary system of units; while the same constant for gases is 3.4. This means that the same amount of energy is required to raise the tem- perature of an atom, whatever solid substance it belongs to. This product is called the " atomic heat." Naturally, this law of Dulong and Petit is only approximately true ; for, as has been said, the specific heat of a substance varies with the temperature, and it is impossible to know when different sub- stances are at temperatures such that their conditions are comparable. CHAPTER XIII CHANGE OF STATE Introductory. — The fact that heat energy enters or leaves a body when it changes its state is familiar to every one. In order to melt ice or boil wa
ter it must be exposed to some source of heat ; if water evaporates from one's hand or from the surface of a porous jar, tin* latter is chilled, showing that heat energy has been taken from it ; as steam condenses into water in steam coils or " radiators," they are heated, showing they have received energy ; when an acid or salt is dissolved in water, its temperature is changed; when water freezes or dew is formed, the temperature of the surrounding air is raised slightly, etc. During these changes of state, not alone are there heat changes, but alteration in volume, and so external work is done ; and we shall see that the external conditions are of fundamental importance. We shall con- sider in detail a few of the most important cases of change of state : viz., fusion, evaporation, sublimation, solution, and chemical changes. Fusion Freezing and Melting Point. — •• Kusion " is the name «_ri\ en tin- prOOeSfl in which a solid lm.lv in. -Its and becomes liquid ; the reverse process is called ki solidification." It' a solid l>od\ that can form crystals, e.g. ice, i^ exposed to a smir. heat, its temperature will rise until a point is reached when it begins to melt; and then, so long as there is any solid to melt, the temperature of the mixture of the solid and its liquid remain^ unchanged; hut when the solid is entirch i m -i. * — 17 267 258 UK AT melted, the temperature again rises. Conversely, if the liquid thus formed is placed in such a condition as to lose heat energy, its temperature will fall until a point is reached at which some of the liquid solidifies ; then the temperature remains unchanged until the liquid entirely changes into the solid form ; and from then on the temperature falls again. This temperature which marks the transition from liquid to solid is the same as that which marks the reverse change. It is called the "melting point," or the "freezing point." If the solid and its liquid exist together in contact, and no energy is added or taken away from them, they will remain in equilibrium ; so the melting point may be described also as that temperature at which the solid and its liquid can exist together in equilibrium. Effect of Variations in the Pressure. — Experiments show that this temperature of equilibrium depends upon the external pressure, varying as it is changed. Thus, if ice and water are in equilibrium together, under ordinary atmospheric pressure and in a region where the temperature is therefore 0° C., an increase in pressure will cause some of the ice to melt, showing that the melting point is lowered and that heat energy flows into the ice from the surrounding region ; a decrease in pressure will cause some of the water to freeze, showing that the melting point has been raised and heat energy flows out from the water into the surround- ing region. The explanation of this variation of the melting point with the external pressure depends upon the fact that when a solid melts its volume changes, in some cases increas- ing, in others decreasing. If ice melts, its size decreases ; that is, the volume of a certain mass of water in the solid form is greater than in the liquid. (Blocks of ice float on the surface of lakes.) Those metals which are used to form castings also expand when solidified ; but in general bodies expand. Thus, gold and silver coins are stamped, not cast, because they expand when melted. When ice melts, the CHAXQB "/' -I ATE 259 change in volume is of the same kind as would be produced l»v an increase in the pressure by mechanical means. In other w.»r<ls, increase of pressure helps on the process of melting; so that if ice is being melted by the addition of heat energy, the temperature does not need to be so high in order to secure melting ; i.e. the melting point is lowered. If a hody expands on melting, an increase in pressure will, for similar reasons, raise the melting point. Regelation. — This pressure effect is small. In the case of ice and water, if the pressure is increased from one atmos- phere to two, that is, by 76 cm. of mercury, the change in the melting point is only 0°.0072 C.; and, consequently, ordinary barometric variations have no measurable effect on the melting point of ice. The fact, though, that the melting point is lowered by an increase of pressure is shown by many familiar illustrations. If two pieces of ice with sharp points are squeezed together, the pressure may be enormous because the area of contact may be extremely small ; so the melting point of the ice at the points where the pressure is great will be lowered and, if the surrounding bodies are at 0° C., some of the ice will melt, and the result- ing water will be pressed out, so that it is at atmospheric pressure, and its freezing point is again 0°. But in order to melt this ice, heat energy must be taken from portions of the body near it, and their temperature is reduced below 0° ; so the melted water, with a freezing point of 0°, is in contact with ice at a temperature below 0°, and it will there- fore immediately freeze again. This is the explanation of t he formation of snowballs ; and this action also plays a most important part in the motion of glaciers. The phenomenon is called "regelation." n<>ti<m <>f :i in : is due to the fact that the ice •• pressure of the edge of the skate, and so he is actually •!iin layer of water. As the skate moves on, this water freezes again. Another illustration is the formation of so-called " ground 260 in-: AT ice," which is ice formed at the bottom of streams where there are eddies. The ice crystals are whirled round in the current and stick against the bottom, owing to regelation ; then others stick to them, etc. Non-crystalline Substances. — Bodies that are not crystal- line, like waxes, plumbers' solder, etc., do not have a definite melting point ; but as they are exposed to a source of heat, their temperature rises continuously until they are entirely melted. They pass through a " pasty " condition ; and the temperature at which this begins is sometimes called the melting point. Similarly, as the liquid is cooled it begins to pass into the intermediate condition at a temperature called the freezing point. These two temperatures are not the same. These temperatures are affected by changes in pres- sure exactly in the same manner as those of crystalline bodies. Undercooling. — The transformation from the liquid into the solid condition does not always take place as described above ; for instance, if water is cooled gradually, its tem- perature will fall far below 0° and yet there is no ice formed. Its condition is, however, most unstable, because if it is shaken or if a minute piece of ice is thrown in, the liquid will solidify immediately and the temperature will rise to 0°. This phenomenon of a liquid existing below its freezing point, as above described, is called " undercooling " ; it was discovered by Fahrenheit. "Heat of Fusion. " — The number of calories that must be added to a solid body at its melting point in order to make 1 g. of it melt, is called the "heat of fusion" at that temperature. (It should be noted that this number has the same value if we define the heat unit to be such a quan- tity of heat energy as will raise the temperature of a unit mass of water one degree Centigrade, and define heat of fusion as that number of these heat units which is required to melt a unit mass of the substance, quite regardless of the size of the unit mass.) This energy is spent in overcoming molecular forces and in doing external work if the body expands on r//.i.v<;A' OF 8TATM 261 melting; if it contracts, the external forces also do work in overcoming the molecular forces. The exact way in which this work is done cannot be determined ; but, it is evident tiiat, if the reverse process is carried out, the same amount of heat energy is given out by the body as is received during the direct one. Thus, the temperature of the air in a closed room may be kept from falling far below 0°, if a tub of water is {-lured in it; for, as the water freezes, a definite amount of heat energy is given off to the air. Again, by placing a pail of water under a fruit tree on a cold night, the fruit may be kept from being injured by the cold. The heat of fusion of a substance may be determined by various methods, such as are used for the measurement of specific heats. Thus, a known number of grams of ice may immersed (not allowed to float) in a vessel containing a known mass of water at a known temperature, and the fall in temperature may be noted. If rw, = mass of ice, ms = mass of water, including the watt-r equivalent of the calorimeter, /, = initial temperature of water, /t = final temperature of water, L = heat of fusion of the ice ; Effect of Dissolved Substances : Freezing Mixtures. - freezing point of a liquid is affected it' tin-re i> ;i I'oivi-rn sub- stance dissolved in it. In every case the free/in- point is lowered : ami the change is, within certain limits, propor- tional t<> the amount of dissolved substance iii a jjiven quan- tity of the solvent, for en-tain substances. With others, the change is abnormally great ; and it is to l>e noted that these -'instances arc those which have an abnormal osmotic pressure. (See page 1M , 262 HEAT If the temperature of a solution is lowered to its freezing point, the solid formed is that of the pure solvent, in general ; so that the solution becomes more concentrated. (In certain cases some of the dissolved substance is caught in the meshes of the solid solvent ; but this is a mechanical process, not a thermal one.) Then, in order to freeze out more of the pure solvent, the temperature must be lowered still further ; for, as said above, the freezing point of the solution falls as its concentration increases. A condition is finally reached with certain solutions such that the solution is saturated ; if now heat energy is withdrawn, some of the solvent separates out in the solid form, and at the same time some
of the dissolved substance is precipitated; the temperature remains unchanged, and as more and more heat energy is withdrawn, equivalent amounts of solid solvent and dissolved substance separate out. This complex solid mixture is called the " cryohydrate " of the two parts. It is in equilibrium with the solution of the same concentration, as we have just seen, at a definite temperature ; so, if a cryohydrate is placed in a region at a higher temperature, it will melt. Thus, the cryohydrate of common salt and water has a composition of 23.8 parts by weight of salt to 76.2 parts of water, and its equilibrium temperature is — 22° C. ; so, if salt and ice are mixed thor- oughly and are at a temperature greater than this, the ice will melt and dissolve the salt. In this process the tempera- ture of the mixture and of surrounding bodies falls, because heat energy must be supplied both to melt the ice and to dissolve the salt. (See page 283.) If the salt and ice are in exactly right proportions, this process will cease when the temperature — 22° C. is reached. Such a mixture of two bodies as this is called a " freezing mixture " ; and the above description explains the use of salt and ice in lowering the temperature of surrounding bodies in "freezers," and also the effect observed when salt is thrown on ice or snow. A freezing mixture of solid carbon dioxide and ordinary CHANGE OF STATE 268 sulphuric ether, known as Thirlorier's mixture, allows one to secure a temperature as low as — 77° C. The fusion constants of a few substances are given in the accompanying table : FUSION POINT HEAT or Fusion 1100° C. Ice Iron 1400°-1600°C. Lead B2PC. -39°C. iry Sulphur Zinc 0°C. llf,°C. 415° C. 80 23-33 5.86 2.82 9.37 28.1 Evaporation Boiling Point. — If a liquid stands in an open vessel ex- 1 to the air, it is observed that the quantity of liquid continually diminishes ; it is said to " evaporate " ; the sub- stance passes from the liquid to the gaseous condition. The gas rising from a liquid is called a " vapor," and an exact di >tinet ion between gases and vapors will be made later. (See page 278.) This process of evaporation requires that energy should be constantly added to the liquid, as may be proved by direct experiment, or as is seen by the fact that the hand is chilled when any liquid evaporates from it. ie the process may be hastened by applying some int. -use source of heat to the liquid. If this is done, its tem- perature rises until a point IN reached \\hen bubbles of the : (mm in the liquid, rise to the surface, and break. When this stage of "boiling," or "ebullition," is reached, the temperature ceases to rise, and remains constant until all the liquid is boiled away. This temperature is known as the " l><>iliii'_r point," and it is found to vary with the pressure of the m en the surface of the liquid. This is 264 HEAT what we should expect, because in order that the bubbles may form and rise to the surface, the pressure of the vapor in them must be at least as great as the pressure of the air on the surface, and the pressure of the vapor as it rises from the surface must equal this; so, if this pressure on the surface is increased, the liquid must be raised to a higher tempera- ture before it will boil. Similarly, if the external pressure is decreased, the boiling point is lowered. The process of boiling is one, then, of what may be called kinetic equilibrium, depending upon the equality of the pressure on the surface and that of the vapor as it rises from the surface. Saturated Vapor. — If the liquid is contained in a closed vessel, it is observed that after a time the evaporation appar- ently ceases. The vapor above the liquid is now said to be "saturated." If the pressure and temperature of this vapor are noted, it is found that if the temperature is raised, the pressure increases, and more liquid is evaporated ; while, if the temperature is lowered, the pressure decreases, and some of the vapor condenses to form more liquid. If the tem- perature is kept constant, however, the pressure remains the same, entirely independent of whether there is a small or a large amount of liquid present. This condition may be called one of statical equilibrium. On the kinetic theory of matter it is easily explained. The molecules of the liquid may attain sufficient velocity to break through the surface, thus requiring work to be done upon them. Similarly, the molecules of the vapor may strike against the surface and become entangled, thus losing kinetic energy. So, if both these processes go on together, there will be equilibrium when the number of liquid molecules which escape in any interval of time equals the number of vapor molecules which are retained by the liquid surface in the same time. Spheroidal State. — A simple illustration of the kinetic nature of evaporation is afforded by what is called the " spheroidal state." If a small quantity of water is allowed CUAXGE OF STATE 265 t<» llow gently out of a tube or spoon on to a metal surface which is at a high temperature — far "above 100° C. — and which is slightly hollowed out so that the water will not run it is seen to collect in a flattened drop, which does not on the surface, and it now rapidly evaporates. The explanation is evident; for, owing to the rapid evaporation on the lower side occasioned by the heat energy received from the hot metal, the molecules are leaving tin- drop on this side with such velocity and in such quantity that their mechanical reaction holds up the drop. There is thus a layer of vapor between the drop and the hot plate. This spheroidal state can be noticed when water is spilled on a hot stove, and in fact the hotness of a stove or a flatiron is often tested in this manner by seeing if it can produce this state in small drops of water. (This condition of a drop is sometimes called " Leidenfrost's Phenomenon," because tin- first observation of it U attributed to him, 1766.) Vapor Pressure. - One of the simplest :i<»ds of obser \inir the phenomena of satu- d vapor is to intro- duce a small quantity of the liquid to be erimented <»n above the mercury column in a barometer which ! a deep basin. Some of :i 190. — Experiment* illtintrmtlnjr tl preMare of Mtnrated vapor : » umall u ll.ini.l. t.g. water, to Introduced there tl the liquid will evaporate, and equilibrium will lx» reached at a pressure d. -pending upon the temperature. This ma\ 266 HEAT be varied at will by surrounding the tube with a bath of some liquid whose temperatures can be regulated. The pres- sure of this saturated vapor may be measured in terms of the atmospheric pressure by the ordinary law of hydrostat it- pressure. (If h is the difference in height of the mercury column in this tube and of that in a barometer, the pressuiv of the vapor is less than the atmospheric pressure by dgh, where d is the density of mercury.) If the temperature is kept constant and the ba- rometer tube slowly raised, some liquid will evaporate, but the pressure remains con- stant; similarly, if the tube is slowly pushed down, some of the vapor is condensed, but the pressure does not change. (If there were a gas above the mercury column in- stead of the vapor, its mass would not L00° 200° Fm. 121. — Curve showing connection between the tem- perature and the pressure of saturated water vapor. change, but its pres- sure would decrease and increase during the above changes, obeying Boyle's law : pv = constant. But when there is a vapor in contact with its liquid, there is no change in pressure, but the mass of the vapor increases and decreases.) If, however, the temperature is increased, some liquid evaporates and the pressure increases ; and, if it is lowered, some vapor condenses, and the pressure decreases. (If CHANGE OF STATE 267 there were a gas above the mercury column, its mass would remain constant during the above changes in temperature, a n« I its pressure would change, but at a different rate from that of the saturated vapor.) There is thus seen to be a delinite pressure of the saturated vapor which corresponds to a definite temperature when there is equilibrium, and conversely; the corresponding values may be found by either the statical method or the kinetic one, in which the boiling point at different pressures is determined. The results may be expressed by a curve drawn with axes of temperature and pressure. This curve for water vapor is given. That the boiling point varies directly with the pressure is shown l>y the fact that the temperature of boiling water is much less than 100° C. on a mountain top ; by the high tem- perature in steam boilers where the pressure is great, etc. From the description given above of the statical method, it is seen that there are two general methods available for con- dt nsiii'^ a vapor into a liquid : one is to lower the temperature ; the other is to decrease the volume. These will be discussed in full in a later section. If a vapor is not saturated, it obeys hiiyle's law quite closely ; and Dalton's law is also approximately exaet for the pressure produced by a mixture of vapors. A curve may be drawn that will >s this law of a vapor in con- tact with its liquid when the tem- i tu re is constant. Lay off axes « of pressure and volume; then, since a vapor keeps its pressure constant so long as the temperature is con- VOLUMES . the isothermal is a straight Fi«.i«. line parallel !•• theaxis of volumes. Formation of Dew, Clouds, etc. — We have seen that, as the temporal UP- d, the corresponding vapor pressure of -a i u rated vapor becomes less ; that is, if there is a certain 268 HEAT amount of unsaturated vapor in a closed vessel, it will become saturated if the temperature is lowered sufficiently ; and then, if it is lowered still more, some of the vapor will condense. This is illustrated by the formation of dew, of clouds, etc. There is always a certain amount of moisture in the air, and the method of expressing it is as follows : we measure the temperature
of the air and then by experiment find what temperature some solid body — like a metal can — must have in order to make moisture condense on it. This is called the " dew-point." The vapor pressure corresponding to these two temperatures is found from tables or from the curve given on page 266 : one of these expresses the pressure that the water vapor in the air might have if it were saturated at the existing temperature ; the other gives the pressure that the water vapor in the air actually has. The ratio of the latter to the former gives what is called the "humidity." If we assume the truth of Dalton's law, we can easily calculate the mass of a unit volume of ordinary damp air. This equals the sum of the masses of the air itself and of the water vapor in the space. The quan- tity of water vapor in a unit volume corresponding to various dew-points is given in tables ; e.g. if the dew-point is 10° C., the mass per cubic metre is 9.3 g. The pressure of this vapor corresponding to 10° C. is 0.914 cm. of mercury ; thus, if the barometric pressure is 76 cm., the pres- sure due to the air is 75.086 cm.; and, if the temperature of the air is known, e.g. let it be 20° C., the mass of the air can be calculated, since the density of dry air at 0° C. and 76 cm. pressure is known to be 0.00129. Thus, using the gas law, the density at 20° C. and at 75.086 cm. pressure equals \|Z? . 0.00129, or 0.00119. Therefore the mass of the air in a cubic metre is 1190 g. ; and the total mass of the cubic metre of damp air is 1190 + 9.3 = 1199.3 g. If the air were perfectly dry and at 20° and 76 cm. pressure, its density would be \|^ x 0.00129, or 0.00120 ; and so the mass of a cubic metre of it would be 1200 g. It is seen, then, that the mass of dry air is greater than that of damp. Boiling. — As already explained, the process of boiling consists in the formation of bubbles of the vapor in the interior of the liquid. Nuclei of some kind are required in order for CHANGE OF STATE these to form, such as sharp points or minute bubbles of some foreign gas, like air. As a liquid boils, the supply of such nuclei is used up, unless in some way it is renewed constantly, and it becomes more and more difficult for the liquid to boil. The temperature rises above the boiling point until the molec- ular forces are sufficient to form the bubble; there is a miniature explosion; and the temperature falls back to the boiling point. If the nuclei are removed from the liquid as completely as possible and if the walls of the containing vessel are smooth, the temperature of the liquid may be raided far above the boiling point; but this condition is of course unstable. If the liquid were entirely free from nuclei, it would never boil; but, ! temperature were gradually raided, it would finally explode. Heat of Vaporization. — The num- ber of calorie.s that must be added to a liquid in order to make one i^ram of it evaporate, or boil at a • If finite temperature, is called the ••l!'-at of Vapori/.ation " at that temperature. (This number does not depriid upon the size of the unit of mass if the heat unit is suitably defined. See page 260.) This eneiLfv is spent ill securing inter- nal changes corresponding to the it inn from liquid to vapor and also in doinur external work; this la>t can he calculated because the pressure must be kept con- stant, inasmuch as the temperature nilnntluii <>f thr li.'.i; • ofall.j.ii.l. /? Is » ring-burner for hcatinir tlio ll.\|iii.t .-..ntnln.-d In tho v<— < ^undented In the colls In the lower veaseL is assumed not to vary. The same number of calories is also given out when one gram of a vapor condenses at that t< -\\\- 270 HEAT perature. This physical quantity is measured ordinarily by the method of mixtures ; a known mass of vapor is condensed by making it enter a quantity of its liquid through a spiral tube, or in such a manner that it gives up the heat energy entirely to the liquid. A simple form of apparatus is shown in the cut. If 7/ij = the mass of vapor condensed, m2 = the mass of liquid in the calorimeter originally, including the correction for the calorimeter, ! = initial temperature of the liquid, t2 = final temperature of the liquid, ts = boiling point of the liquid at the given pressure, c = specific heat of the liquid, L = heat of vaporization, then m^L + mtc (t3 - t2) = m2c (t2 - ^), or L = -t 1 _ c (ts - 2). - m2c (t2 - tj) . Fio. 124. — Two forms of cryophorus. Several illustrations have been given already of the changes in heat energy when a liquid evaporates or a vapor condenses, but one or two more may be described. In an experiment due originally to Boyle, a small quantity of water is placed under a bell jar attached to an air CIIAM.I: <>r >TATE 271 care being exercised to guard against any possible conduc- tion of heat to the water; as the air is exhausted, thus diminishing the pressure on the water and removing the vapor, the water evaporates so rapidly that the heat energy required is taken from the water left behind, and its temperature falls until it freezes. A somewhat similar experi- ment is one that involves the use of the ' cryophorus," an instrument invented l»y Wolluston. This consists of two glass bulbs connected by a bent glass tube, as shown in the cut in two forms, a and b. There is sufficient water inside to half fill one of the bulbs. The experiment con- sists in placing the instrument a in a vertical position, with the curved portion uppermost, or the instrument b horizontal ; the water is poured into one bulb, which is carefully shielded against heat loss or gain, and the other is surrounded by a freezing mixture of salt and ice. After a short time the water will be found to be frozen, owing to rapid evapora- tion at its surface, which is caused by the continuous condensation and ing of the vapor in the other bulb. The action, then, is the same as if a substance " cold " were carried from one bulb to the other ; hence th»- name •• rryophorus," which means "carrier of cold." Some forms of apparatus for making artificial ice depend upon the same fact, that when a liquid evaporates, heat energy is required. In most cases the liquid which is evaporated is ammonia. This is placed in the space between the two walls of a double-walled vessel which contains water, and as the ammonia is evaporated, the water is frozen. Steam Engine. — As a further illustration of the proper- ol a vapor, the steam engine may be mentioned — a dia- gram of a simple form of which is given in the cut. The principle of its action is as follows: Steam is produced in a "boiler" under hi^h pressure, and therefore at high tem- perature; this steam is allowed at regular intervals to enter the "cylinder." in which there is a movable piston, at the instants when the piston reaches one end of the cylinder, and in such a manner as to exert a pressure on this piston, pushing it away from this end. Steam continues to enter, and the pressure is that of the steam in the boiler as Imi^ as the connection is maintained; but after the supply of steam is cut off, the steam, as it expands, decreases in pres- sure. In the meantime the pressure in the cylinder on the other side of the piston has been made as small as possible 272 HEAT by one of three methods: (1) by opening it to the atmos- phere— this makes a kt non-condensing" engine; (2) by join- ing it to a large vessel, which is kept as nearly exhausted as possible by means of pumps and as cool as possible by means of coils or jets of water — this makes a "condens- ing " engine ; (3) by join- ing it to another cylinder, exactly like the first one, only larger, and allowing the steam ejected from the first cylinder to work the piston FIG. 125. — Steam Engine. CHANGE OF 8TATM 273 in the second — this makes a "double, triple, etc., expan- sion" engine. When the piston reaches the other end of the cylinder, connection is made with the boiler, as above described, at this end, and the expanded steam is expelled, as the piston retraces its path, in one of the ways just mentioned. In the case of the non-condensing engine the steam escapes to the air and is lost; in that of the condensing engine it is condensed into water in the con- denser and pumped back into the boiler; in the expan- sion or compound engine it is used over again, expanding more and more until, finally, it is condensed in a condenser and pumped back to the boiler. These changes are made automatically by certain valves ; the " sliding valve " opens and closes the inlet pipes and also the exhaust pipe to the condenser. In all cases a certain quantity of saturated vapor is received at a definite pressure (and corresponding temperature) ; this expands, doing work on the piston, until a certain lower pressure is reached, then — in the condens- ini: engine — it is condensed to water at the pressure corre- sponding to the temperature of the condenser; this water is pumped into the boiler, its temperature is raised until it boils, and the process begins again. There is t hus a " cycle " of changes. While in the boiler, the water receives heat energy; and when water is lu-ing formed in the condenser, heat energy is given out by the steam. The steam does work in pushing on the piston, and work is done on it and on the water formed from it when the piston performs its reverse motion. If H is the heat energy received at the temperature of the boiler, and W is the net external work W H. done, the ratio -= is called the "efficiency" of the process. The amount of work done may he measured by a simple mechanical device. We have seen (page 161) that on a pressure- \o] u in. diagram, a closed curve .describing the series of changes tlin>u<_:h whieh a tin id passes indicates by its area AMES'S PHYSICS — 18 274 HEAT the total net work done. The curve giving the cycle of changes just described for the steam leaving the boiler, ex- panding, etc., must be somewhat as shown in the cut. A marks the instant when the water in the boiler begins to b
e changed into steam ; B, when this process is finished, the pressure and temperature remaining constant; this steam can be imagined as formed directly back of the piston and exert- ing a pressure on it; so the instant marked by B is that of the " cut-off "; the steam now expands, and both temperature and pressure fall; at (7, connection is made with the condenser, in which the pressure is that marked by the horizontal line ED\ so the pressure falls to D, and the steam condenses to water as marked by E\ this water is forced by a pump into the boiler, and its temperature and pressure both increase until the point A VOLUMES v is again reached. Looked at from another point of view, this curve indicates very approxi- mately the changes in volume and pressure of the space in the cylinder which is open to the steam as the piston moves to and fro : when the piston is close to one end, the volume is small, and as steam is admitted, the pressure rises from E to A ; then it remains constant as the piston is pushed out, thus increasing the volume, until the cut-off is reached at B ; then, as the piston continues to move forward, the volume increases and the pressure falls; O marks the end of the motion of the piston ; as it moves back, the volume decreases, but the pressure remains unchanged, being that of the condenser ; etc. These pressure and volume changes may be recorded automatically by the engine in several ways : a piece of paper may be fixed to a drum and a pencil moved over it, motion in one direction being secured < I1AXGE OF STA 1 1-: 275 Attaching the pehcil to a spring gage which measures the pressure in the cylinder and moves out and in as this i IK leases and decreases, and motion at right angles to this ln-ing obtained by last* -n ing the drum carry - the paper to a cord which is attached to the moving piston, and so lengthens or shortens as this moves, and thus makes the volume in the cylinder i nerease or decrease. These curves are called "indi- cator diagrams," and are of the greatest assistance in studying the actual work- in LT of an engine. A form of indicator mechanism often Fio. 127. — Indicator mechanism. I is shown in the cut. The first successful steam engine was made by Newcomen in 1705; but all the great improvements, as shown in modern engines, are with one exception due to .James Watt. He invented the condenser in 1763; in succeeding years he con- ceived the idea of introducing the steam alternately on the two sides of the piston, and <>f cutting off the steam from the boiler so as to allow it to expand; In also invented the plan for surrounding the cylinder with a jacket into which steam « -ould be admitted, so as to keep the cylinder always hot. Hornblower was the first to construct a compound engine, i.e. one with two cylinders, so as to have double expansion. This was done in 1781. Effect of Dissolved Substances. — The vapor pressure of a liquid is alTeetrd it some substance is dissolved in it, being always lower « n t. mj>erature than that over the pure liquid. Then-ion- the boiling point of a solution is higher 276 // /•;. I / than that of the pure liquid at a given pressure. These changes in the pressure or in the boiling point vary with the concentration of the solution. They are proportional to it in some cases, but in others the change is abnormally great. These last solutions are those for which there is an abnormal lowering of the freezing point and an abnormal osmotic pressure. When a solution evaporates or boils, the vapor formed is that of the pure solvent unless the dissolved sub- stance is volatile ; consequently, when the temperature of the boiling solution is to be measured, the thermometer must be immersed in the liquid, not in the vapor. The vaporization constants of a few liquids are contained in the accompanying table : BOILING POINT AT PRESSURE OF 76 CM. OF MERCURY HEAT OF VAPORIZATION 78° C. - 80° C. Alcohol (ethyl) Carbon dioxide Chloroform .... Cyanogen .... Ether (ethyl) .... Hydrogen .... Mercury ..... 357° C. Oxygen - 184° C. - 20°.7 C. 34°.6 C. - 238° C. 61°.20 C. 444°.5 C. Water 100° C. 209 58.5 72 at - 25° 90 103 at 0° — — 62 ___ 535.9 Isothermal for a Change in State from Vapor to Liquid. - The various physical properties of liquids and vapors can best be expressed and studied by graphical means. One method is to draw isothermals on a diagram whose axes correspond to pressure and volume. Let us imagine the vapor inclosed in a cylinder that has a movable piston. We can then represent by curves the various conditions of the vapor, as the piston is forced in gradually, the temperature CBAJfOM <>F STATE 277 kept constant. At first, as the volume is diminished, i In- pressure increases according tu Boyle's law, since the vapor is unsatu rated, as shown hy the curve AB. But finally, a pressure is ivaehed such that th« vapor is now saturated for the given temperature; and, if the volume is still further decreased, the pressure remains unchanged, as shown by the curve BC. As this decrease in volume goes on, more and more of the vapor condenses and sinks to the bottom of the cylinder. When all the vapor is condensed, the cylinder is full of liquid, and to produce any further decrease in volume a great increase in pressure is Fio. 128. — Isothermal for change from a vapor to a liquid. 450780 VOLUMES IN CC PER GRAM. Fio. 1».— Isothermal* of carbonic ari.i («>,). n hy the curve CD, which is nearly vertical. Cun ijx kind were first determined and studied by working with carbonic acid Lfas. in the years previ- 278 HEAT ous to 1869. The results of his investigation are given in the accompanying cut. It is seen that, as the isothermals at higher and higher temperature are drawn, they have the same general shape, but the horizontal portions become shorter, until a temperature is reached, the isothermal for which has no horizontal portion. This is called the "critical" temperature. For temperatures higher than it, the isothermals approximate more and more closely to those of a gas. The Critical Temperature. — The only points on the dia- gram for which the matter is in the form of a liquid are along the horizontal portions of the isothermals, where there is a free surface separating the liquid and its vapor, and along the continuations of these lines to the left, where the liquid completely fills the vessel. Consequently, whenever we see a liquid partially filling a vessel, it is represented on the diagram by a point on a horizontal portion of an iso- thermal. In other words, if a vapor is to be liquefied, it must be at a temperature whose isothermal has a horizontal portion, that is, it must be at a temperature lower than the critical one. Bearing this fact in mind, all gases, with the possible exception of helium, have been liquefied. The name "vapor" may then be restricted to a body in the gaseous condition which is at 'a temperature below the critical one ; while the name " gas " may be limited to temperatures above this. The exact point on the diagram, marked by A, where the critical isothermal has its point of inflection, that is, the point where an isothermal infinitely near the critical one, but below it, has a minute horizontal portion, is called the "critical point." If we have matter in this condition filling a vessel, and if the temperature is lowered, even the least amount, the liquid will separate out and sink to the bottom, filling approximately half the space and leaving the rest full of vapor. At the critical point, then, the surface of separa- tion disappears, and the liquid and vapor dissolved in each CHAtfQM OF -/.I//-: 'J7'.' other make a homogeneous form of matter. The pressure corresponding to the critical point is called the "critical pressure," and the volume of one gram of the substance in this critical condition is called the "critical volume." In order to liquefy a gas, then, two steps are necessary ; the temperature must be lowered below the critical tempera- ture, and the volume must be decreased until the pressure is reached that corresponds to the state of saturation of the vapor for the temperature. After this, any further decrease in volume or temperature will cause the vapor to condense. The values of the critical temperatures for various gases are given in the accompanying table : CRITICAL TEMPERATUI:I - Alcohol . . . 243°.6C. Ammonia . . . 130° C. Argon . - 120° C. Carbon dioxide . . 30°.9 C. Chloroform 260° C. Hydrogen ... - 242° C. Nitrogen . - 146° C. Oxygen . . . - 119° C. Sulphur dioxide . . 156° C. Water . 365° C. Liquefaction of Gases. — The critical temperatures of such gases as hydrogen, oxygen, nitrogen, etc., are seen to be extremely low ; so that special means must be adopted in order to liquefy them. There are three methods in use for the production of low temperature ; application of freezing mixtures, rapid evaporation of a liquid, expansion of a gas from lii^h to low pressure. (See page 246.) The standard method for liquefying gases is a combination of these to a certain degree. The gas to be liquefied is compressed by pumps to a high pressure and is cooled by a freezing mixture or l»y evaporation of a liquid : it is then allowed to expand through a small opening, is again compressed ; and tin- process is repeated. This expanded • older than it was : iind 1). -fore iM-injr compressed again, as .is it expandfl through the opening, it i^> drawn i the mi IQ unexpand : hus chilling 280 HEAT it. As the process continues, the temperature of the com- pressed gas gets lower and lower, until finally, on expansion, the critical temperature is passed and drops of the liquefied gas fall to the bottom of a Dewar bulb. Two forms of apparatus are shown in the cut on page 281 ; one due to Linde, the other to Dewar. The most important parts of the former are the two-cylinder air compressor and the "counter-current interchanger." This last consists of a triple spiral of three copper tubes wound one inside the other. The cycle of operation
is performed in such a manner that compressed air at the temperature of the coil g and at about 200 Atm. flows through the inmost tube of the spiral from top to bottom, and passes out at the lower end through a valve a under a pressure of some 16 Atm., it returns upwards through the annular space between the inner and middle pipes, is again raised to a pressure of 200 Atm. by the smaller cylinder d of the compressor, and then begins the same cycle over. The larger cylinder e of the compressor pumps a small amount of air from the atmosphere into the suction pipe of the small cylinder, that is to say at 16 Atm. A similar quantity of air must therefore leave the cycle at another point so that the pressures may remain constant. This escape of air takes place at the lower end of the counter-current interchanger immediately after the discharge from 200 to 16 Atms. so that a controllable amount of air issues from 16 Atms. to 1 Atm. through a second valve &. Part of this air, when the apparatus is cooled down to the temperature of liquefaction, becomes liquefied and collects in a "Dewar flask" c. That part of the air issuing from the second valve which is not liquefied leaves the apparatus, es- caping through the space between the middle and outside pipes of the spiral into the atmosphere. An iron pipe in the form of a coil #, which is cooled down to a few degrees below zero by a freezing mixture of ice and chloride of calcium freezes out the small quantities of water vapor contained in the highly compressed air until only traces remain, and also cools the air. The properties of bodies at extremely low temperatures are entirely different from what they are at ordinary temperatures. The tempera- ture of liquid air when boiling at atmospheric pressure is — 182° C. ; under these conditions lead becomes elastic and iron and india rubber extremely brittle ; egg shells, leather, etc., become phosphorescent ; the electrical resistance of all pure metals decreases greatly; etc. By suitable methods, air, oxygen, and even hydrogen may be solidified. CUA\>.i: OF 8TA /'/•: 281 Apparatus of Llnde for the liquefaction of air and other gases. Apparatus of Dewar. A is a ryllmlrr containing the gas HIH\|IT lilirh pr A U a Dewar flask containing a freezing mixture »uch aa solid carl etlMr ; C Is a second flask containing some liquid which Is made to evsponit- rnpidly. .</., etl, - a thlrtl ring the g« thr»ui;li //. ' '. • Nrft\|K- tliniinrh hjr the rod f. As the gas escni <lrawn out through t a-« It rises it c<>. 282 HEAT The melting point of hydrogen is estimated at — 257° C. ; and the max- imum density of liquid hydrogen is 0.086. By allowing a quantity of liquid air to evaporate slowly, and using the methods of fractional distillation, Ramsay discovered three new constituents of our atmos- phere, which he called Krypton, Xenon, and Neon. Continuity of Matter. — It is seen from the experiments of Andrews that it is possible to make a body pass from the state of vapor to that of liquid, and vice versa, by a series of continuous changes, because a path can be drawn from a point on the pressure-volume diagram where the body is in the form of a vapor to one where it is liquid, which does not pass through the region where the isothermals are horizontal, and where therefore the liquid and vapor exist separately, but in contact. The isothermal has two points of discon- tinuity, at the two ends of its horizontal portion, which correspond to molecular • rearrangements ; but a series of changes can be imagined, as above described, during which there will be no sudden molecular changes, but by which a vapor can be gradually and continuously changed into a liquid. This is ordinarily expressed by saying that matter is continuous from the liquid to the vapor state. Similarly, matter is continuous from the solid to the liquid state. Sublimation In many cases a solid body evaporates directly without passing through the liquid condition. This process is called " sublimation " ; and it is illustrated by camphor, arsenic, iodine, carbon, many metals, snow or ice, etc. It is found by experiment that, if this process takes place in a closed vessel, there will be equilibrium between the solid and the vapor when a definite pressure is reached, which depends upon the temperature. If the latter is increased, the equi- librium pressure is higher, and conversely. The reverse of this process of sublimation is seen in the formation of frost. It is found that all these substances which sublime under • ll^^<,l•: or STA n-: ordinary conditions may be obtained in the liquid condition it' a suitable pressure ami temperature are applied. (This i> one step in Moissan's method of making artificial diamonds. ) The number of calories required to make one gram of a solid sublime at a definite temperature is called the "Heat of Sublimation." It equals the sum of the Heats of Fusion and of Evaporation at that temperature, in accordance with the principle of the Conservation of Energy. Solution Heat of Solution. — When a body — solid, liquid, or gas — is • Ivi-d in a liquid, both being at the same temperature, tin-re is a change in temperature, showing that heat changes are involved. If the temperature falls, it shows that work is required to make the substance dissolve, if we assume that tin-re are no secondary molecular changes such as the forma- tion of ne\v molecules or dissociation. The heat energy that is gained or lost is measured by the product of the mass of the solution, its specific heat and the increase or decrease of temperature. If the temperature rises, heat energy is said to be " evolved " ; if it falls, the energy is said to be "absorbed." If one gram of the substance is dissolved in a eertain quantity of solvent, the heat energy thus involved will, in general, vary with the quantity of solvent ; but, by continually increasing this, it is found that after a certain point the heat changes are independent of the quantity of solvent. The heat energy gained or lost, when one gram of a substance is dissolved in such a large quantity of the solvent as this, is called the " Heat of Solution " of the substance. Some values are given in the following table, the solvent being water. A plus sign indicates a rise in temperature, or "evolution of heat," and a mi mis si LTD, the opposite. Ammonia ffaa . +495.6 calories. Caustic potash . •• ones. Ethyl alcohol . + 55.3 calories. Sodium Chloride . - 1 XL' calories. Sulphuric ;i<i<l . + 182.5 calories. Silver chloride . —110. calories. IIKAT Effect of Rise in Temperature upon Solution. — We can from these facts predict whether raising the temperature of a solu- tion increases its solubility or not. Let us consider a saturated solution with an excess of dissolved substance precipitated ; and let it be one in which heat energy is absorbed when solution takes place. If now heat energy is added, the sol- vent will dissolve more of the substance ; and if heat energy is withdrawn, the solvent will precipitate some of its dissolved substance ; because in this last case, for instance, if the effect were to make the solution more soluble, this act of solution would withdraw some heat energy, etc., and the condition would be unstable. Therefore, with a solution of this kind, an increase in temperature makes it capable of dissolving more ; a decrease, less. Just the converse is true of those solutions in whose formation heat energy is evolved. Dissociation ; Ions. — The fact that certain solutions have an abnormally great osmotic pressure, and also have freezing and boiling points that differ abnormally from those of the pure solvent, can all be explained if it is assumed that in these solutions a certain proportion of the dissolved molecules are broken up into smaller parts. This has the immediate effect of increasing abnormally the number of moving particles due to the dissolved substance. If we assume, further, that these fragments of the molecules are electrically charged, we can explain the phenomena of electrolysis, as we shall see later ; and all solutions of this kind do conduct electric currents. The whole science of Physical Chemistry is based upon these two assumptions, and they may be regarded as justified by experiments. When such a solution is formed, some of the molecules dissociate into their parts since thereby the poten- tial energy is made less ; or, as we may express it, there is a force producing dissociation. The process ceases — or equi- librium is reached — when this "solution pressure" is balanced by the electrical forces that are called into action. The equilibrium is not one in which there is no further dissocia- tion, luit one in which for each molecule dissociated there is formed. The dissociated parts, called "ions," are mov- ing t<> and fro in the solvent, and combinations and dissocia- tions are taking place continually but at the same rate. Chemical Reactions Heat of Combination. — In all chemical reactions there are molecular changes, and consequent heat changes. If the ;in^ bodies are gases, these changes depend to a marked degree upon the external conditions that are maintained ; for these determine the amount of external work. It is entirely immaterial, however, whether the change takes place in one or more stages. Thus, if one gram of carbon in the form of a diamond is converted into carbon monoxide (CO), -11<» calories are evolved: and if this is converted into carbon dioxide (CO2), 5720 more calories are involved — 7860 in all. And, if the same amount of carbon is oxidized at once into carbon dioxide, the heat involved is the same. This is an illustration of the Conservation of Energy. A few illustrations of these heat changes may be given. : <j. of hydrogen L^as and 16 g. of oxygen, at 0° C. and 76 cm. pressure, combine to form 18 g. of water at 0°, the heat energy evolved is 68,834 calories. If 65 g. of zinc are •1 ved in dilute sulphuric acid, 38,066 calories are evolved ; while if 63 g. of co
pper are dissolved in dilute sulphuric acfd, 12,500 calories are absorbed. Dissociation. — One of the most interesting chemical changes is the dissoci.it inn of a '_r;ls i,,((, other gases. Thus some gases with complex molecules break down into others with simpler molecules when the temperature is raised to a high degree. In many cases it is observed that for a definite temperature equilibrium is reached at a definite pre- nd, if the temperature is increased. H Lfl the •ndiiiLf pressure. This condition of equilibrium is one of continual recombination and d "ii. CHAPTER XIV CONVECTION, CONDUCTION, AND RADIATION It has been shown that there are three general methods by which heat energy is added to or taken from a body : Con- vection, Conduction, and Radiation. Each will now be dis- cussed in turn. Convection When a vessel containing a liquid is placed on a hot stove, the upper layers of the liquid receive heat energy from the lower ones by the process known as " convection " ; the por- tions of the liquid in the lower layers have their temperature raised and therefore expand ; and, since their density is thus diminished, they are forced upward by gravity, the cooler upper portions sinking. Thus the temperature of the whole liquid is made uniform. The mechanics of the phenomenon is not difficult to understand, because the molecules of the hot upward-moving portions of the fluid communicate by their impacts some of their energy to the other molecules ; and thus the internal energy is distributed. It is evident that convection processes can occur only in fluids under the action of gravity, and when the heat energy is applied to its lower portions. It should be noted that the energy is gained by the portions at low temperature and is lost by those at a higher temperature. This method of distribution of heat energy is the one that forms the basis of the ordinary means of heating houses, — hot-air furnaces and stoves, hot-water systems, steam pipes, etc. Convection is of funda- mental importance in the economy of nature, as is explained in Physical Geography, in connection with winds, ocean currents, etc. The metal 286 CONVECTION, CONDUCTION, AND RADIAtlON 287 •in of a tea kettle (or of a steam boiler) does not become unduly hot so long as there is water in it, localise, owing to convection, the cooler portions of the water are being continuously brought down to the >m. Conduction General Description. — When one end of a inetal rod, like a poker, is put into a fire, the temperature of this end rises, and in a short time that of the other portions not too far from the fire rises also. The temperature of the end in the fire is the highest; and that of the other points of the rod decreases gradually as one passes from the fire, until a point in the rod is reached that is at the temperature of the sur- rounding air. If a thin transverse portion or slab of the rod is considered after it has come to a steady state, it is evident that its side near the flame is at a higher temperature than the one away from it; the molecules at the former section have more energy of motion than those at the latter. As a consequence, the former molecules give up some of their energy to the molecules of the slab ; and its temperature would rise were it not for the fact that the slab is losing lit at energy by convection in the surrounding air (and by radiation, also, a process to be described presently), and that tin- molecules in the cooler end of the slab are themselves handing on energy to the other portions of the rod. This process by which molecules give up some of their energy to iguous molecules, there being no actual displacement as in convection, is called "conduction." Thus, considering lah across the rod, we say that it gains heat energy at the hot face and loses it at the cooler one by conduction : and the difference between the «\|uantities gained and lost must e<\|iial that lost at the surface of the rod by convention 1 radiation), -iuce the rod i- in a steady state. It is important to note that the heat energy is conducted from the hotter portions of matter to the colder ones. When the >t in a steady state, e.g. immediately after one end is 288 BEAT put in the fire, part of the energy that enters the slab to raising its ti-inperature, to doing external work, etc. " Conductivity " for Heat. — If the rod is in a vacuum, there is very little energy lost, in general, from the surface, because there is now no convection ; and, when the bar is in a steady state the energy conducted in at one section of a slab equals that conducted out at the other. If t± is the temperature of one section and t2 that of the other, if A is the area of each section and a the thickness of the slab, ex- periments show that the quantity of heat energy conducted through from the former section to the latter, if t± is greater than £2, is proportional to — ^— , but is different for rods Sf £ \^ of different material. This fact may be expressed by the following equation, in which Q is the quantity of heat energy conducted by the slab, Q = k— 2A, where k is a factor of proportionality, which is different for different bodies. It is called the " conductivity " for heat. If the conductivity of one body is greater than that of another, it is said to "conduct better." Thus silver conducts better than copper; copper, better than iron ; all metals, better than wood and other non-metals ; etc. The conductivity of any one body varies slightly with its temperature. If the conductivity of a fluid is to be determined, the upper surface must be made the hotter, so as to avoid con- vection. All liquids conduct poorly, with the exception* of fused metals ; and all gases conduct still worse. Thus loss of energy from a body by the processes of conduction and convection may be avoided by inclosing it in a quantity of eider down, feathers, or loose wool or felt; because these solids are poor conductors and motion of the air inclosed by them is prevented, as it is contained in small cavities. The best method of all, however, for avoiding these losses is to have the body inclosed by another and to have the space CONVECTION, CONDUCTION. AND HA />/ J77OJV 289 between completely exhausted of air. I>\ using a Dewar tlask (sec page -'28), liquid air and hydrogen may be kept for hours in a room at ordinary temperatures. Illustrations. — The fact that metals conduct well is shown by count- less experiments. Thus, if a piece of wire gauze with fine meshes is lowered over a flame, e.g. one from ;i Bunsen burner, the latter burns below the gauze only ; because the molecules of the gas as they pass through the wire meshes lose so much of their heat energy by conduction to the outer portions of the gauze beyond the flame that the temperature of the gas as it rises through the -;ui/. • is lower than that at which if burns. However, the temperature of the gauze gradually rises, owing to the Maine, and as soon as the temperature of combustion is reached, t In- flame will Imrn on both sides of the gauze. Or, if the gas is turned on through the burner, but is not lighted, and the gauze is held close to the burner, the gas rising through the gauze may be ignited by a match, but the flame will not strike back below it. (This is the principle of the miner's safety lamp invented by Sir Humphry Davy.) Again, a bright luminous flame may be made smoky by bringing a large piece of metal close to it, so as to conduct off the heat energy and thus lower the temj>erature. The cracking of a tea cup or tumbler when hot water is poured into it is due to the sudden expansion of the inner surface before the outer one has time to be affected; this may often be prevented by putting a silver spoon (not a plated one) in the cup or tumbler and pouring in the water along it ; the silver is such a good conductor that it prevents the temperature from rising too high at once. Conduction in a Gas. — The process of conduction in a gas is evidently simply the redistribution of the kinetic energy of the particles. When the temperature is high at one point in the gas, the kinetic energy there is great; and SO, owiiiLT t() the increased velocity of these particles, this energy is communicated to the neighboring ones. On the assumption that a gas behaves like a set of elastic spin we can deduce a value for the conduct i\ it \ in terms of the mean free path, etc. (See page 202.) The conduetU ;i few bodies at 0° C. are given in the following tahle, in which the heat unit is a calorie and the C.G.S. system is \\«-<\: AMES'S PHYSICS — 10 HEAT Silver 1.096 Copper 0.82 Aluminium 0.34 Zinc 0.307 Iron 0.16 Mercury 0.0148 Water 0.0012 Radiation Radiation as a Wave Motion. — When one's hand is ex- posed to sunlight, a sensation of hotness is perceived ; simi- larly, if a body is brought near a flame, — even when not above it, — its temperature rises, or if brought near a block of ice, its temperature falls. There is neither convection nor conduction involved in these changes of temperature, yet heat energy is being gained or lost. The process is called "radiation." Boyle noted as early as the seventeenth century that it went on through a vacuum, and this fact is proved also by the heating action of the sun which we observe here on the earth. When we discuss, in Chapters XVI et seq., the phenomena of waves and show how wave motions may be detected, it will be proved that this process of radia- tion consists in the motion of waves in the ether, i.e. in the medium which occupies space when ordinary matter is removed and which permeates ordinary matter as water does a sponge or as air does the stream of motes revealed by a beam of sunlight entering a darkened room. Without going into details in regard to waves, several facts may be men- tioned which are familiar to every one from observations of waves on lakes or the ocean, or of waves along a rope. One is that in wave-motion we do not have the advance of matter, but the propagation of a certai
n disturbance or condition; each particle of matter makes oscillations about its centre of equilibrium, but does not move away from this as the wave itself advances ; and therefore by the " velocity of waves " is meant the distance this disturbance advances in a unit of CONVECTION, CONDUCTION, AND RADIATION 291 time. (Consider the waves produced in a long, stretched rope when one end is shaken sidewise.) Thus, in order to produce waves, there must be some centre of disturbance or vibration, and this centre is giving out energy, for it is evi- dent that a medium through which waves are passing has both kinetic and potential energy. Thus, as waves advance into a medium, energy is carried forward, owing to the action of the particles of the medium on each other. We say, then, that "waves carry energy," although of course this energy is associated with the material particles. If wave motion ceases gradually as the waves enter a different medium (e.g. if a stretched rope is so arranged as to pass through some viscous liquid, waves sent along it will cease when they enter the liquid), this medium is said to "absorb" the waves; it gains the energy which the waves carry. Further, waves t different lengths, depending upon the nature of the dis- t iii-bin^ vibration ; to produce short waves, that is, waves in which the distance from crest to crest is short, requires very rapid vibrations; while long waves are due to slow vibra- tions. We know, too, that waves suffer reflection, as is seen when water waves strike a large pier with a solid wall. If several waves are passing through the same medium at the same time, the resulting motion is the geometrical sum of idividnal waves. Production of Radiation. — When ether waves are discussed, it will be shown that they have a velocity of 8 x 1010 cm. per second, or about 187,000 mi. per second, and that they are known to have lengths varying from -m-fc^ cm. up to man\ kilometres, depending upon the frequency of the vibrating centre where they are produced. It will be shown presently that all portions of matter, whatever their temper- ature, are producing spontaneously waves in the ether, whose lengths are so small as to be comparable with the size <>f HIM], The exact mechanism of this is not kno\Mi : but it is clear that there must be some mechanical connection 292 HEAT between the ether and the minute particles of matter, and that these last must be making exceedingly rapid vibrations. If the wave length is called I and the velocity of the waves v, the number of vibrations in a unit of time is -, because L during each vibration the waves advance a distance I ; and so, if there are n vibrations in a unit of time, the waves advance a distance nl in that time, or v = ril. If, then, there are waves in the ether whose length is y^ol7 cm'> the num- ber of vibrations per second is 3 x 1010 x 104 or 3 x 1014, i.e. 300 trillions. Consequently these vibrating particles are thought to be the infinitesimal parts of a molecule. Our conception, then, of the structure of ordinary matter is as follows : it consists of molecules which are moving to and fro, vibrating about centres of equilibrium in solids, or traveling from point to point in a fluid, and at the same time the parts of the molecule are vibrating and producing the waves in the ether — this is similar to the case of a man moving a ringing bell, for the bell moves as a whole and its parts are making vibrations. Ether waves are also produced during certain electrical changes, as will be shown later ; and they are short if the body experiencing this change is small, but long if the body is large. Measurement of Radiation. — In order to study the nature of these waves in the ether and the connection between them and the material bodies which produce them, it is necessary to have some instrument which will detect their presence and measure the energy they carry. To do this some body must be found which absorbs the waves, for then some change which can be observed will be produced in it, depending upon its own properties and the length of the waves. Thus, if the waves are short, they may produce vibrations in the parti- cles of the molecules in accordance with a simple mechanical principle known as that of "resonance." This is illustrated by a boy setting in motion another who sits in a swing ; this CONVECTION, <<).\1>I < 770.V, AND RADIATION 203 has a natural period of vibration, and may receive a large amplitude if a series of pushes are given each time the swing passes through its lowest point, going in the same direction ; hut if the pushes are given at irregular intervals, one may neutralize another ; so the force applied must have the same period as the natural period of the swing. Similarly, if the waves of a certain period enter a material body, the particles <>t' whose molecules have a natural period the saint- as this, they will be set in vibration by the waves, and will therefore absorb them, gaining their energy. (If the waves have a much longer period, they may produce electrical changes in the body.) If the energy of the waves is absorbed by the particles of the molecules, further changes will occur, deter- mined by the nature of the molecules and its parts. Thus certain ether waves falling upon the eye produce changes which result in vision; again, if certain ether waves are absorbed by photographic films, molecular changes go on which may be detected later; but in general the energy gained by the particles of the molecules is diiVused among the molecules themselves, and is manifested by the appearance of heat effects. To measure the energy of the radiation, the absorbing instrument should absorb all the energy, and the corresponding change produced should be one which is in direct proportion to the energy added to the body. It is found (sec hfr-hiw) that polished bodies like bright metallic surfaces absorb very little energy, while i-on^li, blackened ones absorb nearly all the incident radiation, if the waves are not to«> long. So ordinary radiation produced in the • •ili.T by material bodies is detected and measured by instru- m.-iits which are sensitive to heat changes, and whose sur- faces are covered with a layer of lampblack, or copper oxide or platinum black (finely divided platinum). It is worth while to describe a few of thrs,. indium, nts briefly. An • •nlinary mercury ili.-riii..ni,.trr \\ ith its luilb l.larkmr.l has been used; i more sensitive instrument is a platinum ivsM;in<-«- thermometer, HEAT with its strip of platinum blackened; this is called a "bolometer," and was invented by Professor Langley of the Smithsonian Institution. Another instrument is the thermocouple ; in some cases the electric cur- rent produced, as the temperature of one junction is raised, is measured by a galvanometer, while in other instruments, called " radio-microm- eters," and invented by Professor Boys, the wire in which there is the junction, and which carries the current, is suspended between the poles of a magnet in such a manner that it rotates when there is a current in the wire. The most sensitive of all instruments, probably, is the " radiometer," which is a modification of the instrument invented by Sir William Crookes and described on page 204 ; these alterations are in the main due to Professor E. F. Nichols, now of Columbia University. In its present form this instrument consists essentially of an exhausted glass bulb closed at one point by a window of fluorite, — which is particularly transparent to ether radiations of all lengths, while glass is not, — and containing a fine vertical quartz fibre which carries a horizontal arm; to each end of this is attached a thin piece of mica, polished on one side and blackened on the other. The blackened face of one mica disk comes opposite and parallel to the fluorite window; so, if radiation enters this, it falls upon the blackened face of the mica, whose temperature therefore rises and which then moves backward. This motion twists the quartz fibre; and when the torsional moment of reaction of the fibre equals the moment due to the "repulsion" of the blackened disk occasioned by its rise in temperature, everything comes to rest. The angle of deflection of the horizontal arm measures, then, the intensity of the radiation. Another class of instrument must also be used in order to describe radiation from any source; this is one which ana- lyzes it, and so distributes it that waves having different wave lengths proceed in different directions and may be studied separately. This process is called " dispersion " and is illus- trated by the action of a glass prism on the light from a lamp. Radiation Spectra; Energy Curves. — Using these instru- ments, certain facts have been established. All material bodies in the universe, so far as we know, are producing waves in the ether. Solid and liquid bodies emit waves of all wave lengths between certain limits, whereas gases emit trains of waves of definite wave lengths. The emission of a solid or liquid depends largely upon the condition of its < »\VECT1ON, CONDUCTION, AND RADIATION 295 surface, other things being the same. A polished metallic sin face emits very little radiation, i.e. the energy of the radia- tion is small; whereas, rough or blackened surfaces emit a great deal. ( This is the reason why stoves, steam pipes, etc. are blackened.) Again, the amount of the radiation from a body depends largely upon its temperature. As this is raised, the energy carried by each train of waves of a definite wave length increases; but this increase is greater for the short than for the long waves. This fact can be represented by a graphical method. Let two axes be drawn, distances SHORT WAVE-LENGTHS LONG WAVE-LENGTHS Fio. 181. — Radiation from blackened copj.. r. along the hori/.ontal one to represent the wave lengths of the eonipMnent \\a\cs, vertieal distances to represent quantities "f energy carried by the individual trains of waves. Several
eurves are given of the radiation from blackened copper at different temperatures. It is seen that these curves are in •!-d with tin- statements made above. If we consider any individual wave length at the extreme ends of the curves, it marks evidently tin* limiting power of the measuring instru- ment used ; ;md therefore waves whose energy at one tem- perature of the l)od\ is so small that they cannot be detected may be so intense at a higher temperature as to permit of 296 HEAT observation. For instance, the longest waves which affect the human eye in such a manner as to produce the sensation of light are those that cause the sensation we call red; therefore, if the temperature of a solid body, e.g. a piece of iron, is raised, the experiment being performed in a darkened room, the solid is invisible (except for stray re- flected light) until the temperature becomes so high that the energy of the waves whose length corresponds to " red light " is sufficiently intense to affect our eyes. (Actually, the fact must be taken into account that the human eye is not sensitive to all colors alike, and that if the light of any color is feeble, the eye perceives " gray.") The body is now " red hot " ; and as the temperature rises still higher, its color changes continually, and finally it appears white and is said to be "white hot." Laws of Radiation. — Careful observations upon the radia- tion of a blackened body have shown a most intimate connection between the total quantity of energy emitted and the temperature. If Q is the energy emitted at a given temperature, and T the absolute temperature, i.e. T=t°C. +273, Q=cT^ in which c is a factor of propor- tionality. This statement is called " Stefan's law," having been first proposed by him. Again, there is a connection observed between the temperature of a blackened body and the wave length of the train of waves which carries more energy than any other train, i.e. the wave length which corresponds to the highest point of an energy curve for that temperature, as shown on page 295. This relation is due to Weber ; and calling T the absolute temperature of the body and Lm the wave length just defined, TLm = «, where a is a constant. These two laws, which have been verified over wide ranges of temperature by most painstaking investigations, offer con- venient means of obtaining the temperature of bodies when they are so hot as to render it inconvenient to use ordinary CO.\T7-:<T/M.\. CQNDUi rfOJT, -i.v/; HAhiATioy 297 means. If we assume that the laws are true for temperatures which are higher than those for which they have been veri- fied, we may assign them numbers. (In this way, making the above assumption, the temperature of the sun is observed to be about 5700° C.) Pre vest's Law of Exchanges. — It is thus seen that the radiation of any body is independent of other neighboring bodies, because it depends upon the vibrations of its own minute particles. So, if two bodies are associated in such a manner that one receives the radiation of the other, each radiates independently; and the temperature of either one will fall if it radiates more energy than it absorbs. This principle of the complete independence of the radiation of two bodies was first stated by Prevost (1792) in his "Law • •I' Kxchanges," which is equivalent to the above. Newton's Law of Cooling. — If a body is surrounded by an inclosure at a lower temperature, it loses more heat energy than it receives; and, if the radiation from the two bodies is that which is characteristic of a blackened body, this net loss may be expressed at once. Let Tj and T.2 l>e the absolute temperatures of the body and the inclosure: then the heat energy lost by the former diminished by that received from tin- inclosure is c(Tf — TJ). This equals and. if T{ is only slightly greater than 7!r it may be written f\\Tl — 7!,). So the net loss in heat energy varies as the ditl ere nee in temperature between the body and the inel(,siire. This is called "Newt«m's law of cooling." and it is trin- of other bodies than "black" ones for small differ- ences in temperature. Absorption Reflection and Absorption. When radiation falls upon a l"«dy, SOUL- i> tboorbed, BOUM i* n-lle.-trd. and some is trans- mit ted. A body which allows waves of a certain wave length 298 HEAT to pass through it is said to be "transparent" to them/ but no body is perfectly transparent to any waves ; if it is suffi- ciently thick, it will absorb them. In a thin layer, however, a body may absorb certain waves completely and may trans- mit others comparatively freely. Thus, ordinary glass permits those waves to pass which affect our sense of sight, but either absorbs or reflects other waves which are shorter or longer. This is the explanation of the action of the glass roof of a greenhouse. The " visible waves " from the sun are transmitted through the glass and are then absorbed by the black earth or the green leaves. The tempera- ture of these is raised, — but not sufficiently to make them self-luminous, — and they radiate waves which are so long that they are reflected by the glass. Thus the energy which enters through the glass is trapped and stays inside; consequently the temperature is raised. A body which transmits comparatively freely those waves which carry the greatest amount of energy is called " diathermous " ; but the word is not often used, because of its indefinite character. If we wish to compare the properties of reflection and absorption, it is best to consider a body which is so thick as to transmit no waves. It is then at once evident that if a body absorbs well it must be a poor reflector, and conversely. Thus, a blackened surface absorbs well and reflects poorly ; while a polished metal reflects well and absorbs hardly at all. In order to secure what is called "regular" reflection, as from a mirror, not alone must the body be itself large in compari- son with the length of the waves, but its surface must be smooth to such an extent as to have no irregularities so large ; otherwise the different portions of 'the surface reflect the waves in different directions and so scatter them. Under the above conditions a body reflects at its surface waves of all lengths to a greater or less extent ; but in every case certain waves enter the body, although their intensity may be very small. Absorption. — Let us consider the process of absorption more closely. When ether waves fall upon a body, certain CONVECTION, CONDUCTION, AND RADIATION 299 partirK's in the molecules are set in more violent vibration bv ivsnnaiK-e, and thus the waves lose energy. la some bodies t IK-SI- vibrating particles emit waves immediately, without the temperature sensibly rising. This is the case with pieces of fluor spar, thin layers of kerosene oil, and with a few other bodies, as will be shown later under Fluores- cence and Phosphorescence. In other bodies the absorbed • •ii'Tgy is distributed among the molecules and becomes ap- parent in heat effects. This absorption, where the energy goes into heat effects, is called "body absorption." Many I xxlies absorb only waves of definite wave-length, and trans- mit others. 'I' bey are said to have "selective" absorption. Metals and substances that have strong selective absorp- tion reflect certain waves more intensely than others ; they are said to have "selective" reflection. Thus, the reason why gold appears yellow to our eyes is because, when viewed in ordinary white light, besides the waves that are reflected at the surface and that would make the gold appear white, thriv art- certain waves, of such a wave length as to produce the sensation of yellow, that are reflected more intensely than the others. Those waves which enter the gold are absorbed in the surface layer of molecules and produce heat effects. It is evident, then, that if white light is reflected again and again from a series of gold surfaces, in the end the only waves which will leave the last surface will be those which produce in our eyes the sensation of yellow. The waves which leave the last surface after a great number of reflections from the same material are called the " residual " ones. It one looks at a bundle of nonll.- in white light, the points being t ii mod toward the eye, they appear black, because the waves are reflected to and fro from needle to needle, but are continually getting weaker and weaker and being deflected down the needle ; thus no waves come back to the eye, and the points appear black. Finely divided silver and plati- num appear black for the same reason. Connection between Radiating and Absorbing Powers. — abruption i> due to resonance, it is simply a restate- oOO HEAT ment of this to say that a body absorbs to a marked extent waves of the same period as those which it has the power of emitting. But we can say more, if we consider the intensi- ties of the waves absorbed and emitted, and if we assume that there are no chemical or other molecular changes in the body. This excludes fluorescence, phosphorescence, etc. If several bodies at different temperatures could be inclosed inside a vessel which absolutely prevents any heat energy from entering or leaving, and which keeps a constant volume (so that no external work is done), there is every reason for believing that equilibrium would finally be reached, but not until the temperature of all the bodies inside was the same as that of the walls of the vessel. When this is the case, each body must be absorbing and turning into heat effects as much energy as it emits, provided there are no chemical or other energy changes, otherwise its temperature would change. That is, the absorbing power of a certain body at a definite temperature exactly equals its emissive power at that same temperature; where by absorbing power we refer to body absorption. (In general language, a body which absorbs well, in the sense of transforming radiant energy into heat energy, radiates well; e.g. a blackened surface.) We c
an imagine, moreover, a body in the vessel described above, which is entirely inclosed by some envelope which allows to pass through it waves of only one wave length ; therefore, when equilibrium is reached, the body inside must radiate as much energy in the form of waves of this wave length as it absorbs. Consequently, the amount of energy of a definite wave length which a body emits at a given temperature equals exactly the amount of energy in the form of waves of this same wave length which it absorbs at that temperature. In other words, the absorptive power of a body at a certain tempera- ture equals both in quantity and quality its emissive power at that same temperature. If at ordinary temperatures a body appears black when viewed in white light, it is owing to the CONVECTION, COMJUCTIO.\. AM) i;Al>IATK)\ 301 fact that it absorbs those waves which affect our sense of : : and. if raised to such a temperature as will enable a body to emit such short waves, it will emit them and so shine iitly in a darkened room. Similarly, if a body appears ivd at ordinary temperatures when. viewed in white light, it is because it absorbs all waves except those which produce in our eyes the sensation of red ; these are either transmitted or are reflected out from the interior by some small foreign particles. (Thus, a colored liquid appears perfectly black j.t by transmitted light, if it is entirely free from small solid particles; but, if a minute quantity of dust is stirred in it. it appears colored when viewed from any direction.) Then, if such a red body is heated until its temperature is suiliciently high, it will emit all the waves except those which < -spend to the sensation of red, and so, if viewed in a dark room, will appear bluish green. This law connecting radiation and absorption was fir>t stated by Balfour Stewart, but was discovered inde- pendently by Kirchhoft'. The latter, however, in expressing it, did so in a more mathematical form. He took as a stand- ard of absorption a hypothetical body, which is called a "perfectly black body" and which is defined to be such a body as will absorb and turn completely into heat effects all radiations which fall upon it. (Any non-reflecting body, if sufficiently thick, is such a body.) We can approximate to such bodies experimentally by usin<j lampblack or platinum black as the surface layer. (It is seen, too, that the radiation inside an\ hollow inclosure, provided it is rou^h, is that which is characteristic «.f a perfectly black body at thai perat ure, because after a sufficient number of irregular reflec- 18 any train of waves will he totally absorbed, SO the walls ic inclosure finally produce the effect of a black body.) It will be shown later that radiation may be produced and 1.1 m. ins than by raising or altering the tem- perature of the radiating h..d\ ; but this law of Stewart and 302 HEAT Kirchhoff refers only to radiation that is due to the same cause which conditions the temperature of bodies, and to absorption that results in heat effects. Atmospheric Absorption. — One important case of absorp- tion of radiation is that of the solar rays incident upon the earth. As these pass through the earth's atmosphere, a cer- tain percentage of their energy is reflected by the floating particles and drops in the air, and also by the molecules of the air themselves ; and so does not reach the earth. There is also true body absorption as the waves traverse the atmos- phere. The energy that is not reflected or absorbed reaches the earth and is there almost completely absorbed and spent in producing heat effects. The earth itself is therefore also radiating energy, and this again passes through the atmos- phere and is partially absorbed. If there are clouds, this radiation from the earth itself is absorbed by them, and they radiate a certain proportion back toward the earth. Our appreciation of the temperature of the air in which we live depends largely upon the quantity of heat energy absorbed by the air, not so much upon the radiation received by us directly from the sun. Thus we see the reason why the temperature is lower upon mountain tops where the air is rare and so absorbs poorly than at sea level where the air is denser and absorbs more. CHAPTER XV THERMODYNAMICS Nature of Heat Effects. -- Throughout the previous chap- ters we have assumed that heat effects are due to work done against the molecular forces of a body and that for a definite amount of energy received the same effect is produced regardless of the source of the energy. These assumptions are justified by countless experiments, some direet and some indirect. Thus Joule, in a series of in- vestigations beginning about 1843 and lasting over forty : s, caused heat effects to be produced in many different ways ; compression of gases, friction of various kinds, con- duction of an electric current through a wire, chemical re- actions, etc. The only possible explanation of the results of these experiments is the assumption that heat effects are due to energy being given the molecules of the body and are proportional to the amount received. Thus, if a certain amount of hoat energy received from friction is used to boil some water or to melt some ice or to raise the temperature rater, we ean. by allowing this water to cool through tins temperature range, obtain an amount of energy which will melt an equal amount of ice, etc. — it being rem- that heat energy passes from high to low temperature. Again, when the energy received from any two source heat — for instance, a candle and the sun — is compared, if under any condition they raise the temperature of the same amount of water through the same range of temperature, 11 melt the same amount of ice, or boil the same water, et< •. Therefore the production of heat m 304 HEAT effects depends upon the energy received, not upon the tem- perature or condition of the source of the energy. Mechanical Equivalent of Heat. — As stated before, the practical unit of heat energy is that required to raise the temperature of one gram of water from 15° to 16° C. The value of this in ergs, or the " Mechanical Equivalent of Heat," has been determined by different observers. For many years it was thought that the specific heat of water was the same at all temperatures ; and in the early work no distinction was made between the amounts of energy required to raise the temperature of water between different degrees. Robert Mayer calculated the mechanical equivalent from experimental data secured by other observers on the heat energy required to raise the temperature of a gas at constant volume and at constant pressure. We have shown that, on the assumption of no internal forces, if J is the " mechanical equivalent," J (<7p — •#„) = R for a gas ; and hence knowing 0^ Cv, and R for a gas, J may be calculated. Joule, and afterwards Rowland, used the method of turn- ing a paddle rapidly in water, and measuring the mechanical work, the quantity of water, and the rise in temperature. The mechanical work was measured by a simple dynamom- eter method (see page 121) ; and in Rowland's work most accurate results were obtained. In fact, it was he who first measured the variations in the specific heat of water. This method was later modified by Reynolds and Moorby, who used revolving paddles to raise the temperature of a known quantity of water from 0° to 100° C. ; and so their results are independent of thermometers. The most recent work, and the best, has been done by Griffiths, Schuster and Gannon, and Callendar and Barnes, using electrical methods. It will be shown later that when an electric current passes through a conductor, heat-energy is produced ; the amount of this depends upon the electrical quantities involved, all of which can be measured with great TIlEHMnl) YNAM1CS o< I.J •tness. much more so than mechanical power. If i is the iLTth of the current; R, the resistance of the conductor; t. the time the current flows; the heat energy produced is expressed in ergs, if the electrical units are properly 11. Therefore, calling the measured heat energy IT, JH = iRt. In practice the current is passed through a wire which is in contact with water ; so H is measured directly. Thus J may be determined. (The weak point in the method is the uncertainty as to the value of the electrical units in terms of mechanical ones.) As a result of all of these experiments \ve know that to a high degree of approximation the work required to raise the temperature of one gram of water from o!6°C. is 4. 187x107 ergs. The First and Second Principles of Thermodynamics. — The •meiit. made so frequently, that the conservation of energy includes li eat effects is sometimes called the "tirst principle of thermodynamics," — the science which applies principles of mechanics to heat phenomena. The additional statement that h»-at energy of itself always flows from bodies at hi^h to those at low temperatures is called the "second principle of thermodynamics" The founder of the science of ther- modynamics was Sadi Carnot (1796-1832). He was inter- ested in the practical problem of increasing the efficiency of the steam engine. The action of this engine is compli- cated. The steam — or •• \\m -kin«^ substance" — : heat energy in the boiler and loses it while expanding, ng to conduction of the cylinder walls, and ilao while heiii'.r rnndenxrd. It dors work during part of thr process, work done on it during the rest. The HVu-i, of the process has been defined to be the ratio of the net external work done in any one stroke, H'. i.. the heat in that limr fr..m thr boiltT, //. Carnot cone ;,<) of engine, which has been called AMKS'S PHYBICf — 20 306 HEAT "Garnet's engine." In it the processes are considered as taking place as follows : the working substance expands slowty from A to B, the temperature remaining constant ; it expands from B to D in such a manner that no heat energy enters or leaves, and so its temperature falls ; i
t is compressed from D to C at a constant temperature ; and finally it is compressed from O to A under conditions such VOLUMES that no heat energy enters or FIG. i32.-ACarm>tcycie. leaves, and so its temperature rises to its initial value. The curves AB and CD are then " isothermals " ; AC and BD are "adiabatics." In this cycle the working substance in passing from A to B is sup- posed to receive heat energy from some large reservoir at the temperature indicated by the isothermal, and to lose heat energy in passing from D to C to another large reservoir at the temperature of this lower isothermal. The external work done equals the area of the curve ABDC. If H1 is the heat energy received, and Hv that lost, and W the ex- ternal work done, W= Hl — N2 by the conservation of energy (using the same units for heat energy and external H\ work), and the efficiency is * — 2. In discussing the TT __ TT action of this engine Carnot was led to several most impor- tant conclusions which Clausius and Lord Kelvin have shown to be rigid consequences of the two principles of thermo- dynamics. One of these was that the efficiency of his ideal engine was independent of the nature of the substance used to work it : steam, water, alcohol, etc. Another was that the efficiency of his engine varied directly as the difference in temperature between the two reservoirs referred to above. Absolute Thermometry. — This last fact led Lord Kelvin, then William Thomson, in the year 1848 to propose a system 307 of thermoraetry depending upon the use of Carnot's engine; this has the great merit of being independent of the sub- stance used in the thermometer (or engine). Thomson's system of " absolute " thermometry, as it is therefore called, is equivalent to defining the ratio of the temperatures of two bodies as equal to the ratio of the quantities of heat energy received and given out by a Carnot engine working between these two temperatures. Thus, if in the above description of an ideal Carnot's engine the ratio of the tem- peratures on Thomson's scale of the hot and cold reservoirs T i is written -=±t T H /2 //, 2 rI = 771' Hence the efficiency is 1 — ?. Since it is impossible to T — T have an efficiency greater than unity, there is a minimum value of trmpfi-ature, that for which T2 = 0 ; for, if T2 had a negative value, the efficiency would be greater than unity. This minimum temperature is called "absolute zero." Thomson's definition of absolute temperature does not specify the " size " of a degree, but simply the ratio of two temperatures : we can choose the degree to suit ourselves. Let us agree to use the Centigrade scale, so that if T is the temperature of melting ice, T + 100 is that of boiling water. By a series of most ingenious experiments Thomson showed that this system of temperature agrees most approximately with that which we have been using, namely, that of a con- s tai it-pressure hydrogen thermometer on the Centigrade scale, if \ve add to each temperature reading the reciprocal of the coefficient of expansion of the gas, i.e. 273 approximately. (This is what we have called on page 240 "absolute gas temperature.") Thus, if large quantities of boiling water and in» -It ing ice could be used as the reservoirs between which a Carnot engine worked, the quantities of heat energy received and given out, ITj and Hv would have such values 308 11KAT TT that their ratio ~~i almost exactly equaled . It can be x/2 21 & proved that, if the gas had no internal forces and obeyed Boyle's law exactly, the agreement between Thomson's ab- solute system of thermometry and that of a gas thermometer as above described would be perfect. Therefore, in the formula for a gas, pv = RmT, T is very nearly equal to Thomson's absolute temperature. The slight differences be- tween this system and that we have been using were deter- mined by Thomson and Joule ; and in all exact work in Heat this correction is made to the gas temperatures. There are several ways of determining the absolute tem- perature of a body and of calculating the value of absolute zero on the system of thermometry ordinarily used. The accepted value is -273°. 10 C. Historical Sketch of Heat Phenomena It was believed by the Greek philosophers that all the phenomena of a material body which we associate with the word " heat," such as expansion, change in temperature, boiling, etc., were due to the addition to the body of a sub- stance; but to Newton and his immediate predecessors and associates it seemed clear that in some way they were due to motion of the parts of the body. Thus Boyle gave a correct explanation of the heat effects observed when a hammer strikes a nail. The materialistic theory of heat, however, was again proposed, and prevailed for nearly two hundred years. Its great defenders were Gassendi (1592- 1655), Euler (1707-1783), and Black (1728-1799). Even as late as 1856, in the eighth edition of the Encyclopaedia Britannica, this theory is offered as the accepted explana- tion of heat phenomena. One great reason why this theory was so universally accepted was because it was so analogous to the accepted explanation of combustion. Stahl (1660- 1734), who was professor at Halle, advanced the theory that Til Kit Unit VXAMICS 309 •luring the process of combustion a material substance, called "phlogiston," was given off; and this idea persisted until the \\ork of Lavoisier, about 1800, and even later. Fan- tastic theories in regard to the properties of phlogiston and of the substance heat (or "caloric") were of necessity brought forward in order to account for the observed facts. Joseph Black showed that when two bodies at different temperatures were brought together he could speak of '•quantities of heat" leaving or entering the bodies; and we owe to him our ideas and methods in regard to specific heats. Black considered two kinds of effects when " heat " was added to a body : if the temperature was raised, the heat was called " sensible," and it was supposed to be free in the body; but if the temperature did not change, the heat was said to be "latent," and it was supposed to form some kind of a compound with the molecules of the body changing their state. But the experiments of Rumford, in 1798, and of Davy, in 1799, convinced nearly every one that heat effects in a body were due primarily to the transmission of motion to its minute parts. Rum ford showed that in such a process as that of boring out a brass cannon, " quantities of heat " could be produced, limited only by the amount of work done. Similarly, Davy arranged an apparatus which caused one block of ice to rub violently against another, and showed that the quantity of ice melted varied directly with the work done. Tin- first one, however, to express clearly the belief that heat effects were due entirely to the addition of energy to tin- small parts of a body was Rolx-rt Mayer, in lsj-J. He followed hy Joule, in 1848, and later by I Irlmholt/., in 1M7. i;\ the epoch-making research.- ••!' Joule, the prin- « -iplo of the conservation of energy — a phrase nf Rankine's -was soon extended so as to cover all heat phenomena. The fact that -radiation" is a phenomenon due to wave motion in thr « th« i, of exactly the same nature as that win »-h 310 UEAT produces the sensation of light, has been established by a long line of investigators. William Herschel showed, in 1800, that there were rays in the solar spectrum invisible to the eye and yet having the power of affecting a ther- mometer. Herschel speaks of these rays as subject to the laws of reflection and refraction; and this fact was fully established by Melloni some thirty years later. It was proved in the following years that these rays could be dif- fracted, be made to interfere, be polarized, etc.; and that, in short, they were due to waves in the ether. BOOKS OF REFERENCE EDSER. Heat for Advanced Students. London. 1901. PRESTON. The Theory of Heat. London. 1894. An advanced text-book, containing extended descriptions of the experi- ments which form the basis of our knowledge of heat quantities. TYNDALL. Heat a Mode of Motion. London. 1887. In this the author describes a series of experiments illustrating the fact that heat effects are due to energy changes. BRACE. The Law of Radiation and Absorption. New York. 1901. This contains the original papers of Prevost, of Balfour Stewart, and of Kirchhoff. AMES. The Free Expansion of Gases. New York. 1898. This contains the papers of Gay-Lussac and Joule on the expansion of gases through small openings. RANDALL. The Expansion of Gases by Heat. New York. 1902. TAIT. Heat. London. 1884. MAXWELL. Theory of Heat. London. 1892. YOUMANS. The Correlation and Conservation of Forces. New York. 1876. This contains reprints of papers by Helmholtz, Mayer, and others on the Conservation of Energy. STEWART. The Conservation of Energy. New York. 1874. GRIFFITHS. Thermal Measurement of Energy. Cambridge. 1901. This gives a most interesting description of various methods for the measurement of heat energy. VIBRATIONS AND WAVES CHAPTER XVI WAVE MOTION General Description. One of the most important phe- nomena in nature, and one that is of most frequent occur- rence, is the transmission of energy from one point to another by what is called " wave motion." This motion is illustrated in many ways: if a stone is dropped in a pond, waves are produced on its surface, which do work on any movable object which they meet; a vibrating bell produces waves in the air, which do work in a similar way, or by bending a suitably stretched membrane such as the drum of the human ear or the diaphragm of a telephone instrument ; if one end of a long, stretched rope is fastened to a movable object and if the other is given a sudden side wise or lentil i\\ ise motion, a disturbance will pass along the rope and will do work on the object; etc. In all these cases it is evident that there is a vibrating centre which produces motions in those portions of the surrou
nding medium immediately in contact with it; portions affect those next them, etc. The vibrations at th< centre of disturbance must be of such a nature as to produce in tin- medium a displacement or change which can be propagated l»\ it. Thus, if the hand is moved through air or \\at.-r, <.r if a pendulum vibrate* in air or water, waves ai« not pn.duerd, because the fluid flows around the obstacle as it moves through it. and is not Q pressed; in order to Loot waves, the vibration must be so rapid or the motion so sudden that the fluid ,t have time to flow, and is 312 VIBRATIONS AND WAVES therefore compressed on one side and expanded on the other. Again, any sidewise motion in a fluid of an object like a thin board, however sudden, would not produce waves, because in a fluid there is no elastic force of restitution when one layer is moved over another. In order that a medium should carry waves, there must be forces of restitution called into action when its parts are displaced ; for these forces are due to the action of the neighboring parts on each other ; and owing to the reaction of the displaced parts on those in contact with them the latter are displaced also, and so waves are produced and propagated. If these conditions as to the medium and the centre of disturbance are satisfied, and if the motion of this centre is a vibration, or a series of vibrations, the various portions of the surrounding medium will in turn be set in vibration. If the motion of the vibrating centre ceases, that in the medium will persist until forces of friction (or other causes) bring it to rest. In all these cases it is clear that the particles of the medium vibrate, but do not advance with the waves ; a certain " condition " moves out from the centre. " Water Waves " and Elastic Waves. — There are several kinds of forces of restitution that enable a medium to propagate waves. If the surface of a pond is disturbed at one point by the motion up and down of a stick dipping in the water, waves spread out, owing to the fact that the force of gravity tends to maintain a level surface. Again, if a vertical cord, like a fishing line, is moved sidewise through the surface of a pond (or, if a quietly flowing river flows past a stationary vertical cord), short waves called " ripples " may be observed on the side toward which the cord is mov- ing (or up stream in the other case). These waves are due to the force of restitution of surface tension. The waves just described occur on the surface of a liquid; but any portion of matter that is elastic can also carry waves through its interior. Thus, fluids can propagate com- pressional waves, that is, waves produced by having a centre WAVE MOTION 313 of disturbance where the fluid is compressed or expanded : as is illustrated by a bell vibrating in air or under the surface of a lake. Fluids cannot, however, propagate a distortional disturbance. An elastic solid body can carry both com- pivssiunal and distortional waves, as is illustrated by a long, stivtehed wire. If this is disturbed at some point by a transverse vibration, or if it is twisted back and forward, tin-re will be distortional waves; if it is pulled to and fro longitudinally, in the direction of its own length, at some point, there will be compressional waves. (This fact is also illustrated by the disturbances that we call earthquakes; for these are due to some great disturbance in the interior of the earth that produces both kinds of waves in the body of the earth. As we shall soon see, these two types of waves in a solid travel with different velocities ; and this fact is ob- served in all earthquakes.) Compressional waves are often called "longitudinal," and distortional ones, "transverse," •bvions reasons. Polarization. — One distinction between longitudinal and transverse waves, other than the one which gives rise to the names, is worth noting. A longitudinal wave, in whieh the particles of the medium move to and fro along the line of advance of the waves, appears the same to the eye from what- ever side this line is viewed. But, since in a transverse wave the particles are vibrating in planes that are at right angles to the direction of propagation, it will as a rule appear different when viewed from different directions. Thus, if a t raus verse t rain of waves is produced in a long, stretched rope by n loving one end up and down in a vertical line, the rope will at any instant have a sinuous form when viewed from ide, but will appear straight when viewed from above. In this case all the particles are vibrating in straight lines through which a plane can be drawn; and such a train of '1 to be " linearly " or - plane polarized." Simi- larly, if the end of the rope is moved rapidly in a circle or in 314 VIBRATIONS AND WAVES an ellipse, all the particles of the rope will in turn move in circles or ellipses ; and the waves are said to be " circularly " or " elliptically polarized." Thus only transverse waves can be polarized ; and, conversely, if in any wave motion we can detect polarization phenomena, we know that the waves must be transverse. Intensity. — In all classes of waves it is at once evident that we are dealing with the propagation of energy. For wherever there is wave motion the moving parts have kinetic energy, and the existence of the waves presupposes forces of restitution, so when the parts are displaced there is potential energy. Therefore, as the waves spread out from a centre of disturbance, the medium into which they advance gains energy. If the cause of the waves is a temporary disturb- ance, any portion of the medium gains energy when the waves reach it and loses it again when the motion ceases, owing to the waves passing on. Thus a portion of the medium simply transmits the energy. It gains none perma- nently unless there are forces of friction when the particles of the medium move relatively to each other as the wave passes. (Of course if there are foreign bodies immersed in the medium carrying the waves, they may be set in vibration and may continue to vibrate after the wave passes ; in which case this portion of space — not the medium itself — has an increased amount of energy in it afterward.) If at any point in a medium a plane having a unit area be imagined described at right angles to the direction in which the waves are prop- agated, the amount of energy transmitted through it in a unit time is called the " intensity " of the waves at that point. (If the waves are varying, the exact definition is as follows : if E is the energy transmitted in time t through an area A, the intensity is the limiting value of the ratio — -, as t and A are taken smaller and smaller.) Detection of Waves. — The effect of the waves is perceived in two ways. If the medium is limited in one direction by \\ ATE MOTION 315 e object, with which it is connected in such a mannei that the vibration of this object would produce waves in the medium, this will be set in motion by the waves unless it is restrained by mechanical means. Thus, waves in the air set in motion the drum of the ear or the diaphragm of a tele- phone receiver. If the object is surrounded by the medium, l)ii t cannot move bodily with the waves, it may be set in vibration by them, owing to " resonance," a process whieh will l)e described in detail later. Thus, waves produced in tin- air by one tuning fork may set in vibration another one, if the periods of vibration of the two are identical ; waves in the ether may set in vibration particles of matter, as described in the chapter on Radiation. Other Kinds of Waves. — So far we have spoken of mechan- ical waves only, that is, waves in matt-rial media or in the ether, in which the disturbances are displacements of par- ti, hs. lint we can have many other kinds of waves, de- pending upon the property of the medium that is varied. Tli us, if the temperature of one end of a metal rod is first d gradually, then lowered, raised again, etc., we have a vibration of temperature; and, if we observe the tempera- tun; of any point in the rod not too far away from this end, we shall find that its temperature also rises and falls. Since it takes time for the conduction of heat, the temperatures at rent points of the rod will not have their maximum values at the same time, so there is a wave of temperature in the rod. This is illustrated in the daily heating and cool- ing of the earth's surface as it is tunu-d t.. \\ard and a\\.i\ ill. sun, also by the seasonal heating and cooling owiiiLr to th«- revolution of the earth around the sun. There are temperature waves, then, going down into the earth for a short distance. The daily wave is appreciable for a depth of about 2 or 3 ft. ; the annual one for about 50 i\|Hrarup <>t th. * M it }i'-cru8t increases gradually but OOD- tn.'i..-. :\- with the depth; the rate of increase varies greatly with the VIBRATION H AND WAVES geological conditions, but is on the average about 1° C. for a depth of 28 m. This condition requires that heat energy should be continually flowing from the interior of the earth to the surface. From considera- tions based on this fact it is possible to make an estimate of the " age of the earth " ; that is, the interval of time since the earth was in a liquid condition. This is probably about 50,000,000 years. Similarly, if one end of a long electrical conductor, e.g. an ocean cable or a telephone wire, has its end suddenly joined to an electric batter}^ the effect is gradually felt along the conductor ; and, if the electric battery at the end is varied, there is an " electric wave " in the conductor. Again, if a metallic body is charged electrically, there are electric forces, so called, at points in space near by ; so, if this electric charge is varied, these forces will vary ; and, as they change, variations are produced at neighboring points. Therefore, when the electric charge on a body varies, that is, when there are "electric oscillations," waves of electric fo
rce are produced in the surrounding medium. The medium which carries these waves has been proved to be the same as that which carries the waves that affect our sense of sight ; namely, the "luminiferous ether." It has been proved, too, that, wherever there are variations in the electric force, there are also variations in the magnetic force ; so these waves are called " electro-magnetic." (It should be borne in mind that if we look upon matter as the fundamental concept in nature, then as soon as we are able to explain electric and magnetic forces as in some way due to the motion of matter, we shall be able to describe electro-magnetic waves in the ether as dis- placements of material portions of the ether. But, if an elec- tric charge is the fundamental concept, as soon as we can explain the properties of matter as due to the motion of charges, we shall be able to describe all waves in matter in terms of electric forces.) These electro-magnetic waves may be detected by suitable means, as is shown by the various systems of "wireless telegraphy." CHAPTER XVII HARMONIC AND COMPLEX VIBRATIONS \Vi: shall now proceed to discuss in detail the two funda- mental features of waves : liist. the properties of the centre of disturbance; second, those of the medium through which the waves pass. The effects produced when waves in the air or in the ether are perceived by our senses of hearing or of vision will be considered later in the sections devoted to Sound and to Light. The Kinematics of Vibrations Simple Harmonic Vibration. — A disturbance may be peri- odic <>r not ; that is, it may after a definite period of time called the "period" be repeated identically, and again at the end of another period, etc.; or it may be irregular. Thus, the vibrations of a pendulum, of a tuning fork, of a violin string. «-tc., are periodic; while the motion of a piece of tin as it is " crackled," of two stones when struck together, or of one's hand ; s it is moved to and fro at random, are not peri- The simplest case of periodic motion is that which is called "simple harmonic." and which is discussed on page 48. The period of this has been defined above; and the numher «.f \il.rati. .us in a unit of time is called the u fre- quency"; this is, of course, the reciprocal of the period, nplitude " has been defined aa one half tin- length of the suing, or as the value of the maximum displacement. • harmonic motions having the same period and ampli- tude may yet differ in "phase"; that is. t h<- infant s at which they pass through their on ;ins ma\ lie different. 317 318 VIBRATIONS AND WAVES Composition of Harmonic Motions. — 1. In the Same Direc- tion. If a point is subjected to two harmonic motions, the resulting motion may be found by compounding the dis- placements geometrically at consecutive instants of time, or by simple algebraic processes. Thus, if the two harmonic motions have the same period and are in the same direc- tion, they may be represented by xl = A1 cos (nt — ax) and #2 = A2 cos (nt — a%) (see page 51) ; and the resultant mo- tion is x = xl + 2r2 = Al cos (nt — ctj) -f A2 cos (nt — a2). By ordinary trigonometrical formulae this takes the form x = A cos (nt — a), where A2 = Af + A22 + 2 AlAl cos (a, - a2), A , sin a, + A „ sin a9 and tan a = -^ — • A l cos ax + A 2 cos a2 This shows that the resulting motion has the same period as that of its two components, but a different amplitude and phase. Graphical Methods. — If the two component harmonic motions are in the same direction but have different periods, the algebraic formulae are much more complicated, but their resultant may always be found by a simple graphical method. FIG. 133. — Graphical representation of harmonic motion. Any harmonic motion may be represented by a curve drawn on a diagram whose axes are intervals of time and displacements. Thus, the motion x = A cos (vit — a) will be given by a curve, a portion of which is shown in the cut — .\\i> COMPLEX \'ii;i;.\Tif)N8 319 wliirh i> ealled the "sine curve." Thus, let. as mi the harmoiiie motion be that of the point (J. the projection on the diameter of a point P, which is moving in a circle with e< distant speed. Jf, when we begin to count time, the point Pis at S, the "initial" value of the displacement is the projection of OS on the diameter. From this time on the displacement assumes different values; OA is the great- est, OB is the least ; when P is at M, the displacement is Therefore, if we erect at each point of the "axis of time " a line whose length equals the displacement of Q at that instant, and if we remember that displacements in one direction are positive, but in the other negative, we obtain a curve like that shown in the cut, which is known as a "sine curve." Sv Av Pr Mv Bv etc., indicate points corresponding to points S, A, P, Af, J5, etc., in the circular diagram. As the motion is periodic-, the curve repeats itself. This curve can be obtained practically hv fastening a wire to the bottom of a heavy pendulum, ami ilra\\in«,r under it, at right angles to Flo. 134. — Sine cnrvo drawn <>n Mn»k< .1 irlaM. the plane (,f vibration, a piece of glass which has been blackened with camphor >inok«-. in siirh a manner that the point of tin- \\ire JIM -crapes i»? black soot, thus \|.-;i\ ing a trace on the glass. The amplitude is the maximum value of ar, i.e. the length of the line OA. •ves the value of the displacement at the epoch of t hue \\ hen we begin mimtin^, i.0. at / <>. I l.-n.-e ft8\| equata Acosa. < AJ the curve is drawn, OSl is posi- . hence a is an angle lyiir_r in Hie first <»• fourth <piad- rants.) So the length of this line varies with the phase of the \il. ration, other things hein^ niichan 320 VIBRATIONS AND WAVES At the points Mr Nv Mv etc., the displacement is zero ; and if the corresponding instants of time are called tr tz, £3, etc., A cos (n^ — a) = 0 ; .•. cos (ntl — a) = 0. A cos (n<2 — a) = 0 ; .«. cos (ntz — a) = 0. A cos (n<3 — a) = 0 ; .•. cos (n/3 - a) = 0. etc. etc. So if ntl — a = -, it follows that nt2 - a = — , w?3 - « = — , etc.; and t3 — t1 = — . The interval of time taken for the 2-7T ^ /v displacement to traverse the curve from M1 to M2 is the period ; and so we have a proof of the fact that - - is the value of the period. This is the length of the line M^M^ ; and therefore, if dif- ferent harmonic motions have different periods, i.e. different values of w, they will have curves which cut the axis of time at points whose distances apart are different. The method, then, of compounding harmonic motions which are in the same direction is to draw on one diagram the curves for each one and to superimpose them. This is done by adding the values of x which correspond to each value of £, remembering that, when the curve is below the axis, the values of x are negative. Illustrations of this pro- cess are given in the accompanying cuts. This addition of harmonic motions in the same direction may be illustrated physically by suspending a series of simple pendulums one from another, as shown in the cut on page 322, and setting them in vibra- tion through small amplitudes. The motion of the lowest pendulum is the sum of the harmonic motions which each of the pendulums would have by itself. It is at once seen how this varies if the component pen- dulums have different phases, periods, or amplitudes. Complex Vibrations ; Fourier's Theorem. — It is evident that if the component vibrations are numerous, the resulting vibration is, as a rule, exceedingly complicated. The study ,l.Y/> COM PL K A I'll! RATIONS 3-21 of the converse problem, that of separating a complex vibra- tion into simpler parts, led Fourier to a most important and striking mathematical theorem: any complex motion can be Composition of three harmonic vibrations whose periods are in the ratio 1:2:8. Composition of two harmonic vibrations whose periods are la the ratio 1 : 2. Competition of 100 harmonic vibrations whose periods are proportional to 1. J. J. otc., and whose amplitudes are proportional to 1, \|, \|, f , ft, ate. 180. /rd into simple liannoni,- motions of proper amplitudes whose periods arc in the ratios of 1:2:8: Mo, Chat is, if the longest period is 7T. the others are •£, £, etc« These \ il-vat ions are called "harmoi Tin-re are other modes, however, of anal\ /ing a complex \il>rati«»n than hy I-'oiirier's theorem ; and, if the component PHYSICS — 21 T T 2, o 322 VIBRATIONS AND WAVES vibrations do not have frequencies in the above simple ratios, the one of least frequency is called the " fundamental," and the others " upper partials." The exact mode of vibration of any body may be ascertained by the smoked glass method, as described on page 319. The curve obtained is in general com- plex ; but this can be analyzed by cer- tain mechanical methods into its simple parts. Instead of using a wire attached to the body and the smoked glass, various optical methods may be used, depending upon photography. By the same methods the frequency of a vibra- tion may be determined with great exactness, a comparison being made between it and a known frequency or a known velocity. The student may consult for further details Poynting and Thomson, Sound, page 71. FIG. 136.-A complex pendulum. ^ f J , .. Composition of Harmonic Motions. — 2. At Right Angles to Each Other. When the two component simple harmonic motions have the same period, but are at right angles to each other, they may be written x = A! cos nt, y = A2cos(nt — a). The resulting motion may be repre- sented graphically by choosing two axes, one for x, the other for y, and laying off points for different values of t. This may be done as follows : Let the line of vibration of one motion be BA, and let the amplitude of the vibration be OA ; then with 0 as a centre and OA as radius describe a circle ; beginning at A, divide the how to divide a line BOA into a fc Fl°- if; ~ number of lengths which corre- motion with the amplitude p, p, HARMONIC AND COMPLEX VIBRATIONS semici
rcumference into a number of equal lengths so that there is a whole number included in each quadrant; from tin- i' i ids of these equal lengths drop perpendiculars upon the diameter BA\ the points thus marked off on the diam- eter correspond to equal intervals of time for the point making harmonic motion, as is evident from the definition of this motion as given on page 48. If the amplitudes of the two vibrations are equal, two lines AB and CD may be drawn at right angles to each other at Q, «\s, - / 7 \ \ Q, ! \ 0 / / FIG. 148. — Com position of two harmonic vibrations at right angles to each other, WHOM periods and amplitudes are equal and whose difference In phase is (1) zero, (2) quarter of a period, or =. their middle points, as shown in the cut, and the points cor- responding to equal intervals of time will l>e as indicated. through these points lines parallel to AS and CD. :he vibrations AN in the same phase, l><»th will he repre- sented by the point 0 at the same instant; then, when one vil. ration has reached /',. the nth.-r.ha8 reached Ql ; and the geometrical sum is given by Rr When the fnrmer \ il»i at inn reaches PT the latter rca< -In •- '/, : and the geometrical sum is givm l>\ /{., : etc. It is fvid.-nt that the resulting motion is a Straight line. If the \ihrati.ms dilT.-r in phase by a quarter of a period, i.e. by in angular measure, one vihratinn will be at the 324 VIBRATIONS AND WAVES end of its path, J., when the other is at 0. The geometrical sura is given by A. When the former vibration reaches P2, the latter is at Q1 ; and the geometrical sum is given by S1 ; etc. The resulting vibration is evidently a circle. FIG. 139. — Periods equal ; difference in phase 0, -,-,—, IT. If the difference of phase is one eighth of a period, i.e. j, the resulting vibration is an ellipse. The curves are shown for a number of different differences in phase. If the amplitudes of the two vibrations are not the same, the geometrical methods are exactly similar. Lay off two lines, AB and CD, perpendicular to each other at their middle points; divide them into lengths that corre- spond to equal intervals of time ; at these points draw lines parallel to AB and CD. If the phase of vibra- tion is the same, the result- ing motion is in a straight line. If the difference in phase is one eighth of a FIG. 140. — Periods equal; amplitudes unequal; period, the vibration is in difference in phase one eighth of a period. an ellipse, as shown, etc. The simplest way, however, of compounding the vibrations is to eliminate t from the equations, and plot the resulting equation. Different curves will be obtained by giving a different values. The curves of Fig. 139 may be obtained by several physical processes. One is to use, as described on page 319, a pendulum AMt <-<>Mi>LEX VIBRATIONS 826 that can trace a path on a piece of glass ; but in this case the kept stationary and the pendulum is set swinging, not in a plane through its origin, but in a cone, as a result of a sidewise push given it when it is held out at the end of its swing. Another method, due to Lissajous, is to use two large tuning forks whose frequencies are the same, and to phice them, with their vibra- tion planes perpendicular to each other, in such a position that a pencil of light from a small source incident on the end of a prong of one fork is reflected to the end of a prong of the other and thence to a screen, or into the eye of the ver. This arrangement is .shown in the cut. If only one fork is vibrating, a straight line is seen; but if this fork is quiet and the other is vibrating, another straight line at right angles to the first is seen. These lines are caused by the rapid harmonic motions of the two forks. It now both forks are set vibrating, the path of li^ht seen is an ellipse. It' the forks are started again at NT I,.: Kio. 148. - LlM«Jou.' «rrmn««.ment of two tunlnjr millC tO rest, the shape of the ellipse will be different. in general, owing to the fact that their difference in phase i> not the same .1- 1.. This is on the assumption 326 VIBRATIONS AND WAVES that the frequencies of the two forks are exactly equal ; if they are not, the shape of the ellipse will change as one looks at it, showing that the difference of phase between the vibrations has changed. The reason for this is seen at once if one considers the two equations for the forks. If their periods are not quite the same, these may be written x = y = A2 cos (n^ - a), where n1 = n + bt and b is a small quantity. Therefore sub- stituting for Wj its value, y = A2 cos [n* — (a — &£)]. Comparing this with the equation for #, the difference in phase is seen to be a — bt\ and this is different for different Periods in the ratio 1 : 2. Periods In the ratio 8 : 4. FIG. 143. — Lissajous' figures. HARMONIC AND COMPLEX VIBRATIONS values of the time. In words, if one vibration has a period slightly less than that of the other, it will gain on the latter, thus passing through various differences in phase; it will finally gain a whole vibration, when the cycle of changes will repeat itself. If the periods agree exactly, the par- ticular kind of ellipse seen will be a matter of chance, de- pending upon the manner of starting the forks, but it will persist until the motions of the forks die down owing to friction and to loss of energy caused by the production of waves. Lissajous' Figures. — Similar statements may be made in regard to compounding two harmonic vibrations of different periods, whose directions are at ri'_rht angles to each other. The curves obtained when the iiiencies bear certain simple relations to each other are given in the accompanying cuts. different curves in any Fl°- 144. -Geometrical method of com- i t- n> pounding two harmonic vibrations whoM out correspond to different ^rtot* «* in the ratio 8:4, who. »m,.n todes are unequal, and whote phiiM are .litl.r.nt. olifferences in phase; and if, using Lissajous' method, the frequencies are not exactly in the proper ratio, one curve will gradually change into another. The number of cycles of changes in a unit of time depends, of course, upon the <li\ ••]•-_;,. nee of the frequencies «>f the \|'..rkx t'rom , just- ment ; ami. if tliis numher is counted, the general si be curve noted, and the frequency of one fork km>\\n. tl-.it of the other may be deduced at once by a simple formula. NOTE. — Altli Hurli t hf se curves are generally called Lisaajous figures, were first drawn and described by Nathaniel Bowditch of Salem, 1815, forty years before Lissajous did the same. 328 VIBRATIONS AND WAVES Energy of Vibrations Damped Vibrations; Dynamics of Vibrations. — If any actual harmonic motion is observed, it is seen that its ampli- tude slowly decreases; it is said to be "damped." If this decrease is very gradual, there is no change in the period ; but if it is rapid, as for in- stance if a pendu- F.G. 145. - A damped vibration. lum has a plCCC of paper fastened to it, the period is sensibly increased. This decrease in ampli- tude is due to loss of energy by the vibrating body, generally by friction as it moves through the air or at the pivot. This is evident if we calculate the energy of the vibrating particle. If its motion is given by x = A cos (nt — a), its speed at any instant is 8 = An sin (nt — a) (see page 51) ; and so if its mass is m, its kinetic energy at this instant is £ mAW sin2 (nt — a). This varies at different instants of the vibration, but is always proportional to A2. (The mean value over one period may be proved by the infinitesimal cal- culus to be %mAW.) During the motion, as fast as kinetic energy is gained, potential energy is lost ; and so the mean total energy during' a vibration is twice the mean kinetic energy ; and this, from what has just been said, varies directly as the square of the amplitude. Therefore as the amplitude of vibration decreases, the particle loses energy. Forced Vibrations; Resonance. — Even though a body which can make vibrations like a simple pendulum has a definite period of its own, it may be given a different period, if it is attached to some other vibrating system ; thus, if the point of support of a pendulum is moved to and fro by a hand making harmonic motion, the motion of the pendulum is HARMONIC AND COMTLKX VIBRATIONS 329 i In.' resultant of two, one due to the force applied by the hand, the other its own natural motion. If this last is greatly damped by attaching a sheet of paper to the pendulum, it soon dies down ; and the iinal motion of the pendulum is that dui' to the harmonic force of the hand. This motion has the same period as that of the harmonic force; and is called a ••forced vibration," to distinguish it from the natural free vibration. The amplitude of this forced vibration depends to a great extent upon how closely the period of the force agrees with that of the natural vibration ; if they are exactly equal, the amplitude is very large. This condition is called Mmance," and is illustrated in many ways. The case of a child in a swing being set in motion by a series of pushes given at intervals agreeing exactly with the natural period of the swing has been mentioned already. In a similar manner a heavy church bell may be set swinging. If a tun- ing fork is vibrating near another one of the same frequency, tin- latter will be set in vihr.it ion. Many other simple mechanical cases of resonance are given in Rowland's Physi- cal Papers, page 28. If the ju-riod of the force varies slightly from the natural period of the vibrating body, the amplitude is not so great as when there is resonance ; and in most cases one can tell with considerable accuracy when the resonance is exact. The fad that a harmonic force produces in a system whose own vibrations are greatly damped a vibration whose period is the same as its own is of great importance. If the force is periodic, but complex, each of the component harmonic forces produces a corresponding harmonic vibration having its period
; but the phases and amplitudes of these component vibrations bear relations to each other that are not the same as for the component forces. Then-Tore, die resultant complex \ibration is different from the complex force in "form." Il Is Only a harmonic force that can - reproduce " . of OOUlVe \\ ith variations in the amplitude. CHAPTER XVIII VELOCITY OF WAVES OF DIFFERENT TYPES Wave Front. — When waves spread out from a centre of disturbance, a surface can be described that marks at any instant the points which the disturbance has reached. This is called the "wave front." Thus, if a stone is dropped in a pond, or if a raindrop falls on a pool of water, the wave front is marked by an ever-expanding circle. If waves are pro- duced in air by a vibrating tuning fork, the wave front at some distance from the fork is very approximately a sphere. These are called " spherical " waves. The waves in the ether that reach us from a distant star, or from any distant terres- trial source of light which is small, have a spherical wave front ; but this sphere has such a large radius that the portion of the wave front that affects us is practically a plane; so we call these "plane waves." Intensity of Spherical Waves. — If we consider a point source, which therefore produces spherical waves, we can easily calculate the relative intensities (see page 314) at different distances from the source. Let us describe two spherical surfaces of radii rl and r2 around the source ; their areas are 4<7rr12 and 4?rr22. So if the source emits in a unit of time an amount of energy equal to E, and if there is no absorption by the medium, the intensity at any point of the first surface Tfl TJ1 is 2 2' an^ ^at a^ any P0^ °f the other surface is 2- If the former intensity is called 7X and the latter J2, it is seen that l 1 '• 2"^V' 330 VI:L<>< ITY OF WAVES OF DIFFERENT TYPES 331 Or, in words, the intensity varies inversely as the square of the distance from the source. (If the source is of such a kind as to produce harmonic motions at all points in the surrounding medium, we see at once that the amplitude of tin- vibration at a point in the first spherical surface bears a ratio to that at a point in the second surface given by ri ri A1 : A% = — : — ; for we have shown on page 328 that the energy of a harmonic vibration is proportional to the square of the amplitude, i.e. /j : J2 = Af : A£ ; and therefore by the formula just deduced for the intensity, the one for the ampli- tude follows at once.) Velocity of Waves. — The rate at which the wave front ad van * < > is called the "velocity" of the waves. In the next article we shall deduce its value in certain simple cases in trims of the physical properties of the medium; but from general considerations it is evident that in an elastic medium the velocity will be increased if the elastic force of restitu- tion of the medium is increased, and will be decreased if the inertia of the medium is increased; and conversely. In fact, we can prove without difficulty that the velocity of compres- sional waves in a homogeneous fluid whose density is d and whose coefficient of elasticity is E^ is given by the formula Tli is formula is due to Newton, and is deduced in the Principia. WHY.- front of waves in the air is affected naturally by winds. if plane waves are advancing in a direction opposite to the \vin.l. the iij>\|"T portions of the wave front will be more retarded than the lower, because, owinir to friction, the wiml m-ar the earth has a less velocity than hi^h.-r up. So the wave front will lean backward and will proceed up in the air away from the earth. Similarly, if the wind is l.lowini: in tin- direction of the advance of the waves, the wave front will lean forward. VIBRATIONS AND WAVES If a ••point source" that is producing spherical waves has motion of translation apart from its vibration, the wave front may change. Thus, if the source is moving in a straight line with a constant velocity, it is evident that, so long as this velocity is less than that of the waves, the wave front remains spherical. If, however, the velocity of the FIG. 146. -Waves produced by a point source which source [g greater than is moving with a velocity greater than that of the waves ° in the medium. While the source moves from 0 to Ot, that of the WaVCS, the waves from O reach a distance OA. things are different. In the cut let 0, Or 02, Oy 04 be the position of the source at instants zero, £, 2 £, 3 £, 4 1 ; draw a sphere round 0 with Ji radius 4 tv, where v is the velocity of the waves ; one round Oj with a radius 3 tv ; etc. These spheres mark the distances the disturbances have spread out at the time 4 1 ; so, when the source is at 04, the wave front is a cone, whose section is shown by the cut. This fact is illustrated by the waves seen at the bow of a rapidly moving boat. A convenient model for the study of wave motion in an elastic medium is made by suspending a great number of lead balls by long strings at equal &w™r(y^ intervals in a Straight Fl°- 147- — A baU and spring model for illustrating wave horizontal line, and joining them by spiral springs. If the end ball is pushed in quickly toward the next one, a compression is propagated down the line; if it is pulled away rapidly from the next motion. VEL(>< 11 Y <>!• \VAVES OF />//•'/•'/•:/,•/•; AT TYPES 333 ball, an expansion is propagated ; if it is given any periodic motion, a corresponding periodic train of waves is produced. The velocity of these disturbances is evidently increased by using stiffer springs or lighter balls, and diminished by usin!_r heavier balls or weaker springs. Reflection of Waves. — When waves in any medium meet an obstacle, there is, as a rule, reflection, as is illustrated by water waves being thrown back by a pier wall, by the phe- nomenon of echoes, by the use of a mirror in light, etc. We can deduce the exact conditions of reflection by considering t\\o l>all-an<l-spring models like the above, which have differ- ent velocities for compressional waves, and which are con- nected so as to form a continuous "medium." Let the two sets of balls be arranged in the same straight line, the ones which carry waves with the less velocity on the left, as shown in tin- cut. We can call this set th«- --.slow" one, the other the "quick" one. When a com- pression is prop- agate* 1 from Irft •»g«nn t IOXH1 f^^ noon k..-r,-r< k-m-rr. T>,T! rn > n n"n •i to ri"ht. it trav- *'"'• 1K A """1('l illustrating two media ; in the one on the left els with con- the velocity of waves Is less than in the one on the right stant velocity unt il the second set is reached. At the bound- arv the compression is carried ort more rapidly by the quick set of balls, and therefore the spring joining the two sets is not mm pressed as much as it would he if the velocity were not increased; the last ball of the slow set is thus pulled slightly toward the right, and so produces an extension of th«- spring hark <>! it; etc. Therefore a compression in the slow set produces a compression in the quiek set, and an expansion is reflected back along the slow set from the 334 VIBRATIONS AND WAVES boundary. If an expansion is propagated from left to right, there will be an expansion produced in the fast series of balls ; but at the boundary the first ball of this set yields more easily to the force of the spring pulling it toward the left than does the last ball of the slow set. Therefore the spring at the boundary is not extended as much as the others, and this will then produce a compres- sion in the springs of the slow set. An expansion, then, propagated along the slow set produces an expansion in the fast one, but a compression is reflected back along the slow one. If a series of waves consisting of alternate compressions and expansions, such as would be produced in this model by giving harmonic motion to the first ball of the slow set, meets a set of " faster " balls and springs, the reflected waves are of the same nature ; but whenever a com- pression reaches the boundary, an expansion is reflected, and conversely. In an exactly similar manner it may be shown that, if such a series of waves in a set of fast balls and springs is incident upon a boundary beyond which there is a slow set, a similar series of waves is reflected ; but in this case a com- pression produces a compression, and an expansion an expan- sion at the boundary. There is ,a difference, then, in the reflection at the boundary between the fast and slow sets, depending upon the direction from which the incident waves come ; and this difference is equivalent to a substitution of a compression for an expansion, or vice versa. It should be particularly noted that if the velocities of the waves in the two media are the same, there is no reflection. (It is assumed that the waves are not damped, that is, that there is no absorption.) Therefore, in order to have reflec- tion of waves at a boundary separating two media, these must be such that the velocity of the waves is different in the two. (The "rolling" of thunder is an obvious illustra- tion of the reflection of air waves owing to the presence in the air of foreign bodies, namely clouds, or of regions in or M-.ir/-> Of i>in-i-:i;!-:.\T TYPES 335 which the velocity is different.) Another obvious condition for securing reflection from un obstacle is that its area should hi- large compared with the length of the waves; otherwise they will puss around it. Velocity of Transverse Waves along a Flexible Stretched Cord. — It is not difficult to calculate the velocities of certain ses of waves. This is true of the propagation of trans- verse disturbances along a stretched but perfectly flexible cord in which the tension is constant. Imagine a tube, which is straight except for a circular portion near its middle, slipped over this cord and moved rapidly along it with a con- st ant velocity v. Let us consider the motion of the particles of the cord as the curved portion of the t
ube reaches it, and the forces which the tube exerts on the cord. As this curved P Q F i... 149. — A cord, over which has been slipped a bent tube, is stretched between Pand Q. portion of the tube reaches any particle of the cord, it gives the particle a motion which may be resolved into two com- ponents : uniform motion in a circle with constant speed v, and uniform motion parallel t<» that of the tube along the cord with constant speed v. The former motion will be con- si ( In -i •« I presently. As the particle enters the curved portion, it is given, therefore, a momentum along the line of the cord, whirl i it keeps until it leaves the other end of the curved portion, when it is given an e<\|iial momentum in the opposite (liivriinM and brought to rest. The two forces that pro- duce these changes in momentum are due to the tube; hut one balances the other exactly; so then- is no resultant action or reaction due to them. The only other acceleration is that occasion. <1 by the particle being made to move in a circle with constant speed. It' / is the length of any minute portion of the ronl. </ its mass per unit length, an<l r 336 VIBRATIONS AND WAVES tin- radius of the circular portion of the tube, the force which must be exerted on this portion of the cord to make 2 it move in the circle is dl - - There are three forces acting on it, the tension in the cord acting at its two ends and the reaction of the side of the tube against which it presses. Let us calculate the former. In the cut let A, B, O represent three consecutive points in the cord, drawn on an immense scale ; let the length of the arc ABO be Z, the portion of the cord which we are considering; let 0 be the centre of the circle drawn through these three points ; and T be the ten- sion in the cord. Owing to this tension B there are two forces acting on this por- tion of the cord, as shown in the cut. Calling the angle between AO and OB T (and also between OB and 00) N, the component of each of these forces along BO is T sin N, and their components perpendicular to BO balance each other. The resultant, then, of these two ten- Fio. 150. — A, B, C are i — i three consecutive points of sions is 2 T sin N", and its direction is c«rd.CUrV6d P°rti0n °f " along BO' N is an extremely small angle, and so sin N may be replaced by JV; but 2 N is the angle (^4.0(7), and this equals the length of the arc ABC, i.e. I, divided by r, the radius. So the resultant force due to the tension acting on the elementary portion of the cord considered is — . If the tube is moving Tl with such a velocity that this equals the force required to make the portion I move in a circle, there is no reaction of the tube on the moving cord; that is, the tension in the cord is itself sufficient to maintain the cord in the given curvature ; and the tube now exercises no force on the string whatever. In other words, the tube could now be removed entirely, and the " hump " in the cord would be propagated of itself VELOCITY OF IIM T/-> <>F l)irri-:i!l\l IYPES 337 along the cord with the velocity of the tube before it was removed. The mathematical condition for this velocity, as given above, is that This si 10 us that the velocity of any hump, whatever its radius, is a constant quantity for a given cord under a definite tension ; therefore, any kind of a disturbance, whatever its nature, will be propagated with this same velocity. The fundamental conditions are that there is no force in the string except its tension and that this is constant. Velocity of Compressional Waves in a Fluid. — The velocity of compressions] waves in a fluid may also be calculated; but the process is not quite so simple as in the previous case. The student will find it given in Maxwell, Heat, 4th c<lit ion, page 223, and in many other text-books. The result • prove that, if E is the coefficient of elasticity of the tin id, and d its density, the velocity of the waves is given by d Fri the formula V=\— • Nothing need be said in explanation of d\ but we have seen that the "elasticity" of a gas is an indefinite quantity, unless the condition is defined under which the change in volume occurs (see page 194). In the present case, the number of compressions and expansions in one second is so great that the change in volume of the fluid takes place adiabatically ; for there is not time for heat transfer to occur. Consequently in the above formula E is the adiabatic coefficient <>f elasticity. Its value for any gas has been given already (page 253). If c is the ratio of the two specific heats, and p the pressure of the gas, E equals the product cp. Therefore, for a gas, and this can be simplified by using the gas law p- RdT. Thus, V=^Tftft Then-fore, the velocity varies directly as the square mot of the absolute temperature; and AME' -- 22 338 VIBRATIONS AND WAVES further, if F", 72, and T are known for a gas, c may be calculated. Laplace was the first to see (1816) that the coefficient of elasticity in this formula was the adiabatic one. Newton, who was the first to derive and apply the formula, used the isothermal coefficient, whose value equals p, and thus made an error. Assuming the value of c for air to be 1.40, and substituting proper values of p and d, the velocity of air waves may be calculated at any temperature. The value of this velocity at 0° C. is thus equal to 33,170 cm. per second, if the C. G. S. system is used. The fact that the velocity varies with the temperature is illustrated by the observations of arctic travelers who have noticed that the so-called " velocity of sound " is less at low temperatures. The velocity of waves in air is also seriously aifected by the presence of moisture, because the density of the air is changed. The velocity is seen to be independent of the nature of the disturbance propagated, and also of the pressure. When there is a violent explosion in the air, there are slight variations in the velocity near the centre of disturbance, owing to the fact that the value of the elasticity of the gas, as given above, is true for small variations in the pressure only. At some distance away from the centre, however, the velocity becomes normal. a Similarly, the velocity in other gases may be calculated. The velocity of waves in liquids may also be deduced from the original formula V=\— - For water at 8° C. it is found, as stated before (page 172), that an increase in pressure of one atmosphere, i.e. of 76 cm. of mercury, decreases a unit volume by 0.000047 of its value. If the thermal effects may be neglected, „ 76 x 981 x 13.59 , 7 , therefore, V= 145,000 cm. per second. -£666617 — 'and'/ = There are also experimental methods for the determination of these velocities in gases and liquids ; the details of which VEL»< ILY or WAVES <>r Dirri:nEM TTPMS 339 are described in larger text-books such as Poynting and Thomson, Sound. These methods may be divided into two classes : direct and indirect. In the former, a disturb- ance is produced at some point and the time taken for the waves to reach a point at a measured distance away is accu- rately measured. The disturbance may be the ringing of a bell, a mild explosion, etc. ; and the instant of arrival of the waves may be determined by the mechanical motion of a diaphragm or by the perception of the sound. In the indirect methods, the gas or the liquid is set in vibration by some periodic disturbance whose frequency is known : since this is due to resonance, the frequency of the vibration of the gas or liquid is known ; and, as will be shown presently, the velocity of waves in it may be at once calculated. Velocity of Waves in a Solid. — In solids a purely compres- sional wave cannot exist, because when there is a compression produced by two opposite forces there is at the same time a distortion. The velocity of longitudinal waves in a solid that extends in all directions is given by the formula when- /r is the coefficient of elasticity for a change in volume, and // the one for a chanty in shape. A train of waves can, however, be produced where there is only distortion, as when one end of a long wire is rapidly twisted to and fro. In this case, if n is the coefficient of rigidity for the solid, V— \pj- If longitudinal waves are produced in a wire or a rod by stroking it lengthwise with a resined or damp cloth, the ve- locity of the waves is given by V=^— , where E is Young's modulus. ( S66 page I •"> I . ) Water Waves. — Tin- velocity of waves upon the surface of a liquid di-j.rmls upon many quantities, and we can do no 340 r//;/,M770ATs AMD WAVES more here than state certain facts in regard to liquids whose viscosity may be neglected. These statements involve the quantity known as the " wave length," which in the case of waves on liquids may be defined to be the distance from crest to crest or trough to trough. If this quantity has the value Z, we have the following formulae for the velocity : If the liquid is deep, V—\— , where g has its usual 2 7T meaning. If the liquid is shallow, V= Vfy/, where h is the — , surface tension and d the density. The general formula for waves on liquids which are deep is Id 2 r y o V and it is clear that, if the waves are long, the first term is negligible, while, if they are short, the second one is. It is seen by calculation that in the case of water if Z>10 cm., the first term may be neglected, and if Z<0.3 cm., the second. For intermediate values of Z, the full expression must be used. There are many most interesting applications of these formulae. The fact that, if one sailing boat has a longer water-line than another, the latter is given a " time allowance " in a race, is due to an attempt to equalize the advantage of the longer boat ; for a boat moving through the water produces waves that are comparable in length with its own ; and as the boat is helped on by these waves, the longer boat is helped the more because the velocity of the waves it produces is greater. Again, as waves approach a shelving s
hore, if they are oblique to- the shore line, they will gradually turn so as to approach parallel to it, owing to the fact that in shallow water the waves are faster in the deeper portions than in the ones less so. The motion of the individual particle of a liquid as a wave passes over its surface is in general an elliptical path ; and the effect of the waves is felt only a short distance down VELOCITY <>F \VAVES OF DIFFERENT TYPES 341 from the surface, as the amplitude of the vibration decreases rapidly \\ith the depth. This is not the place to discuss the velocity of temperature waves or of electric waves along wires. But it may be stated that in both these cases part of the energy of the waves is dissipated in heat effects throughout the media and in other ways, and as a consequence of this the waves die down and are not propagated as far as they otherwise would be. The FIG. 151. — A drawing1 of Lyman's wave model for water waves, showing the form of the wave, the motions of the indi- vidual particles, etc. waves are said to become "attenuated." It may be proved also that long waves persist for a greater distance than short ones ; and this fact is of fundamental importance in tele- phone service, as will be shown later. CHAPTER XIX HARMONIC AND COMPLEX WAVES — « STATIONARY WAVES" Trains of Waves and Pulses. — In discussing many of the properties of wave motion it is essential to distinguish two types of waves : one is produced by a sudden irregular dis- turbance, and may be called a " pulse " ; the other is pro- duced by a periodic disturbance, and is called a " train of waves." Harmonic Waves : Wave Length, Wave Number, Amplitude. — The simplest type of a train of waves is one produced by a centre of disturbance whose motion is harmonic. This is called a "train of harmonic waves," or a "harmonic train." As a consequence of this disturbance, each particle of the medium will be set in harmonic motion, but the phase of the vibration varies from point to point at any one instant. (This may be illustrated on the ball-and-spring model.) The simplest mode of representing waves graphically is to choose two axes, one giving the distance the wave front advances in any direction, the other the displacement at any instant at points along a line drawn in this direction. A curve on this diagram gives the displacements at FIG. 152. — Distances along a line in the direction of prop- . ,, ,, agation. A harmonic train of waves. any instant Ot all the particles in the medium along a line in the direction of advance of the waves. 342 HARMONIC AND COMPLEX WAVES 343 I III! II I ILL LL ILL ILL I MM II i i in INI III I II III III Hill If the waves are due to a harmonic disturbance, the curve is as shown, where the line PQ indicates the displacement of the particle whose posi- »?????• tion in the undisturbed I M l medium is P. (If the waves are transverse, this line may be the actual displacement ; if they are longitudinal, it equals, or is proportional to, the displacement of P in the direction of the axis. In this latter case a displace- ment of P toward the right may be indicated by PQ being drawn ver- tically upward, as shown; and a displacement tow a n 1 the left by the line PQ being drawn downward.) As time goes on, the dis- placements all change, the waves advance ; and successive conditions in the medium arc illust rated i i i i 1 1 1 1 nun i i i ill Jn i i 1 1 rdt i i i i Mini 1 1 i 1 1 H" i 1 1 ILL ILL M LLL I I III JJ_L ILL 11 in I i-. 1"'L where three different curves are drawn for three different instants <>f time ; and it should be note.l that in drawing these it is aSSUinrd that there is no decrease in tin' amplitude. If the IT iMMi»fiit •«<•» duplaoement of any particle P in the inr<liinii i-, i.Wrved, it is SITU that it is given in succession l>\ /'',',. /'$,, and F.,.. IM.-A eompn».Nl.>n»l train of waves id i nun MM n n m ILL U4 VIBRATIONS AND WAVES PQB; so it is making vibrations as previously described. The amplitude and the period of the vibration of any particle are called the amplitude and the period of the waves, and the frequency of vibration of the particle is called the "wave number." The distance along the line of propagation from any point to the next one where at t la- same instant the motions are identical is called the " wave length " ; e.g. from P to R. (This may also be defined as tlip distance between two consecutive points in the direction FIG. 154. — Diagram showing three successive positions of the train of wavesj_the vibration of an individual particle of the medium P\ and the wave length PR. of propagation at which the phase of vibration is the same.) In the time of one complete vibration of a particle the waves advance a distance equal to the wave length, and so in a unit of time the waves advance a distance equal to the product of the wave number and the wave length. Therefore, if V is the velocity of the waves, I the wave length, and ^V the wave number, V— Nl. Or, if T is the period of the waves, V= — . (The velocity of an individual particle depends upon the instant we consider it, for it is making harmonic vibrations ; and so the velocity varies from zero to a maximum value, then decreases to zero, etc. It is clear that there is not the faintest connection between this varying velocity and the constant velocity of the waves.) //.l/M/o.y/r AM) ro.u/'/./;.v \\\l\ 345 Doppler's Principle. — If we speak of that portion of a train of waves whieh is a wave length long as a " wave," we may say that, if the wave number is 2\T, the source emits N waves in a unit of time ; and, in general, N waves pass any point in the medium in a unit of time. This is true if the vibrating source, the point in the medium, and the medium as a whole are not moving. It is interesting, however, to consider the two cases, when the source is moving and when the point in the medium where the waves are counted is moving, the medium not being in motion in either. If the vibrating source is at rest, and the point in the me- dium is moving toward it in a straight line, let N be the frequency of the source, V the velocity of the waves in the stationary medium, I their wave length, and v the velocity of the moving point. Owing to the motion of this point it would pass - waves in a time if the waves were stationary in the medium ; but since they are moving toward the point at such a rate that N waves would pass a fixed point in a unit of time, the total number that passes the moving point in that time is the sum JV+y. (But-ZV=-; and so this number jr. \ may be written — j^-. J For similar reasons, if the point is moving away from tin- fixed source with a velocity v, the number of waves which pass it in a unit of time is N— ' / T " \ * If the point in the medium is fixed, and tin- vibrating source is approaching it in a straight line with the velocity v, the case is entirely different. If tin- source were n«>i the length «,f ;v wave would be I; but. when it is the wave fnmt advances a distance T from the in ,i unit time, and in this same time the source advances a diMam-e v; so the X \\aves that have been 346 VIBRATIONS AND WAVES emitted in this time are crowded together in the interval of space V— v, and the length of a wave is now — ~^- The new wave number, or the number of these waves that pass a fixed point in a unit time, is the velocity V divided by this wave length, or — — N. Similarly, if the vibrating source v — v is receding from the fixed point with the velocity v, the new wave number is - — N. y V+v If v is small compared with FJ these last two expressions may be written fl + iW and (l - £\flT, or N+ - and N— - ; so the formulae for this and the preceding case agree under this condition. It is thus seen that when the source of waves and the point under consideration are approaching each other, the wave number is apparently increased ; while, if they are receding from each other, it is apparently decreased. These formulae express what is called Doppler's Principle. It is illustrated in the case of a star approaching or receding from the earth, in a whistling locomotive approaching or receding from a station, etc. Attenuation of Waves. — The energy of a harmonic vibra- tion varies as the square of the amplitude, and hence the intensity of harmonic waves varies as the square of their amplitude. This amplitude decreases as the waves advance, owing to various causes. One case has been considered already on page 331, where it was proved that in spherical waves the amplitude varies inversely as the distance from the source. Further, there may be friction involved in the relative displacements of the particles of the medium, as is the case to a greater or less degree with all waves in all forms of matter ; or, motion may be given particles of foreign matter immersed in the medium (see page 311); in both of HARMONIC AND COMPLEX WAVES 347 which cast's the amplitude of the waves decreases and they lose energy. Similarly, if waves in one medium are inci- dent upon a boundary separating it from another in which the velocity is different (see page 333), waves are transmitted into the latter medium, and waves are also reflected back into the former. The combined intensity of these two trains of waves must equal that of the incident train ; and so the amplitude of the transmitted waves and that of the reflected waves must both be less than that of the incident waves. (An obvious law connects them.) Superposition of Waves; Complex Waves. — There can be two harmonic trains of waves of the same wave length and amplitude, but differing in phase at any instant, depending upon when or how their motion was begun. Similarly, we may have in any medium waves of different wave length, different amplitude, etc. ; and their combined action may be found by compounding them as was done for the vibrations of a particle, for it may be proved that this is allowable, pro- vided the individual displacements of the particles of the medium are sma
ll in comparison with the wave length. The best method of considering the superposition is a graphical one. Thus, two plane polarized transverse waves (see page 313) which are harmonic, and whose directions of vibrations are at right angles, may be compounded as shown in Lissa- jous' figures. In particular, two such waves having the same period would combine to produce an elliptically polar- ized train of waves, or a circularly polarized train if their amplitudes are equal and their difference in phase ^- Con- versely, an elliptically or circularly polarized train of waves may be resolved into two plane polarized trains of waves which are harmonic and whose vibrations are at right angles to each other. Two plane polarized transverse waves which are harmonic and whose directions of vibration an- the same, or two 348 VIBRATIONS AND WAVES tudinal harmonic trains of waves, may be compounded in the same manner as were two vibrations in the same direc- tion ; and any of the illustrations given on page 321 may be applied to the case of waves. Conversely, by Fourier's theorem, any complex train of waves may be resolved into trains of harmonic waves whose wave lengths are in the ratio of 1:2:3: etc. There are other modes of resolution, also, which often are more convenient. We see, then, that a harmonic vibration produces a har- monic train of waves ; a complex vibration, a complex train. A special case of this last is a non-periodic or a confused vibration ; it will produce a corresponding wave. If the disturbance is intense, but lasts only a short time, it produces in the medium what we have called a "pulse." Its effect when it reaches a portion of the medium containing foreign matter is naturally different from that of a long train of harmonic waves ; because owing to these last there are peri- odic forces brought into action on the foreign particles, and resonance may follow. Distortion of Complex Waves. — As we have seen in speak- ing of the attenuation of waves owing to their decrease in amplitude, there are cases in which long waves are less affected than short ones. (See page 341.) In such cases, if a complex vibration is producing a train of complex waves, its harmonic components of long wave length will persist longer than those of short wave length ; and so the char- acter of the complex vibration at different points in the medium will vary. This phenomenon of the change in the "form" of the wave is called "distortion." The further one is from the vibrating source, the more nearly does the vibration approach that of being simple harmonic. This fact is illustrated by ocean cables and by long-distance telephone wires. However complex the electrical disturbance at one end of a cable, that at the other is nearly, if not quite, har- monic. In using a telephone over a long distance the qual- 349 ity of the sound is entirely changed, only the graver notes l>cin«r heard. The great merit of Pupin's new system of constructing cables and telephone lines is that it not alone il< -creases the attenuation, but also diminishes greatly the distortion by making the attenuation of all the waves, long and short, the same. Nodes and Loops " Stationary Waves. " — A most important effect of wave motion is illustrated by a simple experiment which may be performed with a long flexible cord; e.g. a long spiral spring or a long rubber tube, one of whose ends is fastened to a fixed support and the other is held in the hand. If the cord is stretched fairly tight and the free end moved sidewise with a rapid harmonic motion, waves will be produced in the cord which will be propagated up to the fixed end, and will there suffer reflection and be propagated back to the hand, etc. Consequently, at any instant there are in the cord two trains of waves traveling in opposite directions. If the frequency of the motion of the hand is exactly right, it will be observed that the cord ceases to have the appear- ance of being traversed by waves, and vibrates transversely in one or two or more portions or " segments." That is, there are certain joints in the cord where there is very little, almost no, motion, which are called "nodes" ; and the cord in between these vibrates just like a short cord whose two ends are fastened. The points halfway het ween the nodes, where therefore the motion is greatest, are called "loops." Tli is type of vibration is sometimes, but improperly, called "stationary" or "standing" Craves, In reality, the waves have disappeared in the production of a vibration. Such vibrations as this are extremely common; and may occur with waves of all kinds. The explanation i^ evident Consider a medium through which are passing in opposite directions two harmonic- in in- 350 VIBRATIONS AND WAVES of waves which are not suffering attenuation and whose wave lengths and amplitudes are equal. At any instant they may be represented by curves as in the cut. Since the actual motion in the medium is found by superimposing the two wave motions, there will be certain points, one of which is P2 in the cut, at which the displacement is zero, owing to one wave neutralizing the effect of the other. As the waves ad- vance— in opposite directions — they continue to neutralize x...../ \ . p FIG. 155. — Formation of nodes and loops by two trains of waves advancing in opposite direc- tions. PU P2, PSt etc., are nodes. each other at this point, as is seen from the cut; therefore this is a node. The importance of the conditions that the waves should not be damped and that the amplitudes of the two trains should be the same as well as their wave lengths is evident. Further, since in a train of waves at a distance of half a wave length from any point the displacement is exactly reversed, if two trains of waves neutralize each other at a point JP, they will also do so at points distant from P by half a wave length, a whole wave length, etc. Therefore, the distance apart of two nodes equals one half the wave length of either of the component trains of waves. In the case of the cord which was first considered, the fixed end is obviously a node ; and the one held in the hand is approximately one, as is evident if one observes the motion. If the long flexible cord is held suspended vertically from a balcony so that the lower end hangs free, a vibration of the same kind can be produced ; only in this case, since reflection takes place at a " free end," this point is a loop. As the frequency of vibration of the hand is increased gradually, it is found that there are certain definite fre- " STATION ABT \\.\VES" 351 quencies for which the cord separates into vibrating seg- ments ; ami the number of these segments increases, that is, the distance apart of the nodes decreases, as these critical frequencies increase. Similarly, if the tension of the cord is increased, the critical frequency must be increased also. The explanation of these facts is not difficult. Let us consider the first case, that of the cord with its end fixed ; and let the length of the cord be L, the wave length of the component waves be Z, the distance apart of two nodes be c?, the velocity of the waves in the cord be K, the frequency of the vibration be N, and the number of segments be n (necessarily a whole number). Then, we have the relations: c? = -, V •= Nl, L = nd; and therefore, on substitution, N= — . So if the tuision is kept constant, i.e. if V is constant, and if the length L of the cord is not varied, n varies directly as JV; 1 1 ml is, tin- frequency of the vibration must have a definite value, since n must be an integer, and if the number of seg- ments is increased from 1 to 2 to 3, etc., the frequency must be increased in an equal ratio. Again, if the tension in the cord is increased, the velocity V is increased ; and therefore, if // U tin- same, iV, the frequency, must be increased. Vibration of a Stretched Cord. — These vibrations in a stretched flexible cord are not always produced by the method described. If the cord is stretched between two fixed points, it may be set vibrating by using a violin bow, by plucking it with a tinker, by striking it a blow, etc. The vibrations are exactly like those just described. By lightly touching the middle point of the cord, so as to hold it nearly at rest, and bowing the cord or plucking it at a point distant from the end by quarter of the length of the cord, the string vibrates in two equal segments; if the point touched is at a distance from one end equal to one third the length of the • ••>! -d, it may be made to vibrate in three segments ; etc. The positions of the nodes and loops of any vibration of the cord 352 r //;/,-. i noNS AND WAVES may be determined experimentally by putting astride the vibrating cord, which is supposed to be horizontal, small paper saddles. They will be thrown off at the loops, but not at the nodes. (This method was used as early as 1677 by two Oxford scholars, William Noble and Thomas Pigott.) If the cord is bowed or plucked at random, the vibration is a complex one, which can be resolved into the simple compo- nents just described. (Naturally, the point of the cord which is plucked or bowed must be a loop ; and so only those component vibrations are present in the complex one which have a loop at this point.) We shall consider each of these simple modes of vibration n V more in detail. The general formula is N= — - ; and so, if V there is one segment, N— •— ; if there are two segments, O pr *i g TT 2 L ^JL ^V"=_; if there are three segments, N=— — ; etc. The frequencies are therefore in the ratio 1:2:3: etc. These ___ vibrations are, as we have ^5> said in speaking of Fou- rier's theorem (page 321), called "harmonics." The first one, in which n = 1, is also called the "funda- mental"; and the others, "upper partials." (It is seen that the frequency when there are two seg- ments is the same as if the cord were of half the length and had but one segment.) A few AFio.l56.-Vibrationsof stretched cords: (1) one <>UtS
are given of these vibrating segment; (2) two vibrating segments ; tranSVCrSC vibratlOUS of (SUhree vibrating segments ; (4) superposition of •• STATIONARY MM r/>" 353 Tin- velocity of transverse waves in such a stretched flex- ible cord has been proved to be given by the formula, * d IT r=\' — , where T is the tension in the cord and d is the mass per unit length. Substituting this in the general for- mula, we have N= ^'^\—', which shows that if the tension of cord is increased, the frequency is increased ; if the length of the cord is increased, the frequency is decreased; etc. All of these facts are illustrated in various musical instru- 2.L * a ments, as will be noted later. Vibrations of a Column of Gas. — The same type of vibra- tion can be produced in a column of gas, such as we have in the case of an organ pipe, a flute, a horn, etc. There are many ways in which the vibrations may be produced: by blowing a blast of air over the sharp edge of an orifice open- ing into the column ; by making some solid body in contact with the gas at one end vibrate harmonically with a suitable frequency; etc. (The first is illustrated when one blows an ordinary whistle or when one blows over the end of a hollow key; the latter when one blows a horn by means of the vibrations of the lips or holds a tuning fork over the mouth of a bottle in which water is poured until there is resonance.) If a particle in a column of gas is at a node, that is, if its motion is a minimum, it must be in contact with the end wall or it must be held stationary by symmetrical conditions on its two sides, up and down the column. Thus, as the particles in the gas vibrate, there are the greatest fluctua- tions in pressure and density at the nodes. Similarly, at a loop there is the greatest motion, but tl <- least change in pressure and density. Tin- vibration ..f a column of gas is illustrated in t he accompany MILT cut, in which the transverse lines indicate the positions of 1 1 -a us verse layers of gas. If the column <>f gas is closed by a solid partition, this point is a node; while if the en. I ix opm to thr air, so that AMES'S PHYSICS — 28 354 VIBRATIONS AM) WAVES the pressure there cannot change greatly, this point is approx- imately a loop. (Actually, the loop is a short distance beyond the open end in the air outside. If the tube contain- I 1 FIG. 157. — Stationary vibration in a column of gas. Vertical lines represent positions of layers of gas. Curves represent by their vertical displacements the horizontal displacements of the layers of gas from their positions of equilibrium. Arrows represent the directions of motion of the layers of gas. ing the gas is a circular cylinder with a radius R, the loop is at a point beyond the end at a distance given approxi- mately by 0.57J2.) As in the case of a stretched cord, the column of air can vibrate in different ways, depending upon the number of seg- ments. We shall consider several special cases : "STATION A It Y II .1 I'A'.s 855 1. A column of gas closed at both ends. — There is there- fore a node at each end. The simplest mode of vibration is wht'ii there is only one loop, which must then be at the mid- dle point : in this case the distance from node to node is the length of the column L. This must equal half the wave length of the component waves, J ; and, if 2 is the fre- quency of the vibration and V the velocity of the compres- sional waves in the gas, ^ = — . Hence, L = A y i JVj = ^-y. The next simplest mode of vibration is when the column of gas is divided into two segments by a node at its middle point. In this case the distance between two nodes V or is half the length of the column, — . Hence, f Tin- analogy with the transverse vibrations of a cord stretched between two fixed points is complete. The possible vibrations make a complete harmonic series. 2. A column of gas open at both ends. — There is then a loop at each rnd. The simplest moil,- ut' vibration is wlu-n tin -re is only one node, which must be at the middle point. We saw in our general discus- sion that a loop came halfv 1..-I vreen two nodes; so the distance from loop to loop equals tl ii'Mlu to node. In this rasr. thr L N L N L N L '•. — Vibration* of A column of RM open »t both en.U : (1) ftimhuncnul ; (2) flrtt prtUl ; (.1) Mcond partial. 356 \'li;ilATlONS AND WAVES distance from loop to loop is the length of the column, neglect- 2 2JV, ing the slight correction for the ends ; and so L = -1 = -— , The next simplest case is when there is a loop at the middle point and a node at each of the points halfway from it to the ends. The distance from loop to loop is one half the length of the column. Hence, So this case is the same as the previous one ; the vibrations form a complete harmonic series. This kind of a column of gas is called an " open " one ; and, as in previous cases, if it is set vibrating by some random or indefinite means, the resulting motion is a complex vibra- tion equivalent to the addition of these simple ones. 3. A column of gas open at one end and closed at the other. — There is thus a loop at one end and a node at the other. The simplest mode of vibration is when there are no other nodes or loops. The distance from node to loop is one half the distance from node to node ; so in this case 2 L= J = — — , v 2 2JV] orai-Jl The next simplest mode of vibration is when there is a node at a distance — from the open end, and a loop at a dis- 2 L 2 L tance — • The distance from node to node is then — ; and we have the relation =£ = = -^-, or N2 = 2±- Simi- 9 T 7 V ^ V o 22 J\/ 4 Jj larly, the next mode of vibration will give a frequency JV3 = — r ; etc. So in this case the vibrations have fre- quencies in the ratios 1:3:5: etc. ; a series of the odd har- monics only. It should be noted that the fundamental in "8TAT10\Ai;r 357 this column of gas has a frequency one half that of the fun- damental in the other two cases, when their lengths are the same. This kind of a column is called a "stopped" one. If it is set vibrating at random, its complex vibration is com- pounded of these simpler ones just described. Manometric Flame. — A simple method of studying the vibrations of the column of gas in a tube is to pierce openings at intervals along the tube and close them with flexible rubber diaphragms ; these are covered on the outside with small hemispherical caps into which there are two <>i -fil- ings: one permits the introduction of illuminating gas, the other is joined Fio. 169. —Apparatus fur the comparison of UM vioraUng movement* of two sonorous tube. to a pas burner with a fine opening. If the gas is ignited at this opening, it v. ill tlirk-T if tin- .li.ij.lir;i-.;iii '-nine. at a node in the vibrating column of gas, because the fluctuations of pressure there will cause tli<- <lia\|>hragm to vibrate and so affect the pressure on th<> illuminating gas. These tlui-t nations have the frequency of the vibration <>f th> column of gas, : ore, as a rule, very rapid. So the flickering of the 358 VIBRATIONS A\D WAVES must be observed in a revolving mirror, which separates the images when one looks at the reflection of the flame in the mirror. This appa- ratus is called a manometric flame. In its present form it is due to Rudolph Konig. FIG. 160. — Attachment to organ pipe in order to obtain inanonietric flames. Longitudinal Vibrations of a Wire or Rod. — Similarly, a wire or a rod can be set in longitudinal vibrations by stroking or rubbing it lengthwise. If the wire is stretched between two fixed points at a distance L apart, there are nodes at these points ; and if V is the velocity of compressional waves in the wire, we have the following relation for the frequency of the fundamental vibration, FIG. 159 a. — Manometric dames. If a rod is clamped in the middle, this point is a node and the ends are loops. If L is the length of the rod and V the velocity of compressional waves in it, the frequency of the compressional vibrations is given as before by the formula » STATION AB¥ \\AVEs" 359 It may be proved by methods of the infinitesimal calculus, as has been already stated, that the velocity of these com- pressional waves in a wire or rod is given by the formula * d V = \- , where E is Young's modulus of the wire and d is its densiu -. (This modulus E is found, however, by experi- ment not to have exactly the same value as the one found for the same material by the statical method as described on page 154.) Other longitudinal vibrations than the fundamentals can be produced in wires and rods; but their frequencies do not have simple ratios with that of the fundamental; so they are not harmonics, but upper partial*. Transverse Vibrations of a Rod. — A rod can also be set in Averse vibration. If it is not clamped at any point, but is supported in such a manner as to be perfectly free to vibrate, its fundamental mode is shown in the cut ; and its next X-x:-''' - --JiS: simplest mode of vibration is as '' ^ ^X shown. The ratio of the fre- quency of this vibration to that ... N of the fundamental is 2.92; and ,.,-J^V,...,'.>'C _____ '-jg^.., ill a similar manner the Other ^.,61.- Transverse vibrations of possible Vibrations do not have • rod: (I) fundamental; (2) first up- frequencies which hear simple relations to that of the fundamental or to each other; so are upper partials, but not harmonics. It the rod, instead of IMMULT straight, is bent into the shape of a long, narrow U- a\|«l has a projection attached at its middle point, as shown in the cut, it makes a "tuning fork." In all actual forks the rod is not uniform, but is thickened at its base; so that the two nodes, which in the uniform rod are near i \\hen vibrating in its fundamental mode, are brought closer together. The point \\here the projeetin:.: ittached is still a loop, as may be shown l.y 360 VIBRATIONS AND WAVES the fork vibrating and then pressing the stem against a board; the latter will be set in vibration. A tuning fork is, in fact, generally
attached to a hollow box, open at one end, and of such a length that the column of air inside has the same fre- quency as the fundamental of the fork. This is called a " resonance box." An- other effect of giving the fork its peculiar shape is so to enfeeble the vibrations other than the fundamental FIG. 162. — Set of tuning forks on resonance that the latter is the Only one present when the fork is bowed or struck so as to be set in vibration. The frequency of a fork may be measured with great accuracy by comparing its period with that of a standard clock ; but for details of the methods of comparison, reference should be made to some larger text-book, such as Poynting and Thomson, Sound, Chapter III, or to some laboratory manual for advanced students. (The frequencies of two forks may be compared by Lissajous' figures, or by " beats," if the frequencies are close together. See pages 327 and 412.) .Vibrations of Metal Plates. — Thin metal plates cut in squares or circles can be made to vibrate transversely by clamping them at some point — generally the centre — and drawing a violin bow across their edges. When this is done, it is easy to show that there are certain lines along the plates where there is comparatively no motion ; these are called "nodal lines." The simplest method of proving their existence is to scatter fine dry sand over the plates before they are set vibrating; when the vibrations begin, the sand collects along the nodal lines, being thrown there by the other vibrating parts of the plates. There are thus formed most beautiful, regular geometrical figures, which are "STATIONARY WAVM&" 361 called "Chladni's figures," after the physicist who first sys- tematically investigated them. Their shape and complica- tions depend upon the point of support of the plate, the point where it is touched with the finger so as to make a node, the point of bowing, and the manner of bowing, which. to a certain degree, determines the number of vibrations, besides the fundamental, which are present. Similar nodal lines may be observed on stretched mem- branes, such as drumheads, when they are set in vibration in a proper manner. If some extremely light powder is used instead of the sand, with which to cover the plate, it is observed that it gathers, not at the nodal lines, but over those portions of the plate which are moving with the greatest amplitude. The reason for this was discovered by Faraday, who showed that small whirls were formed in the air near the vibrating portions of the plate, and that these carry the light powder off the nodal lines on to those portions. Vibrations of Bells. — Metal bells or bell-shaped objects, like glass tumblers or bowls, may be set in either transverse or longitudinal vibrations. The simplest mode of transverse vibration is shown in the cut. The position of the loops is fixed by the I MI i 1 1 1 where the clapper strikes. Many other vibrations than the fundamental are always present, but there is no simple relation between their frequen- It there ll an irregularity in the thickness of the rim at some point, this will produce comparatively little FIG. 168. - Ono mode of vibr. effect if it comes at a node; but if it is at a loop, it will result in a change of the frequency. So a hell like this .-an vibrate in two ways, giving frequencies that are n..t very different : and if the bell is Struck a random l»lo\\, both these vibrations will occur. This is the 362 V1BRATH).\ AM) }\'A\rES cause of the "beating" of large church bells, as will be explained in a later chapter. (See page 412.) Other Illustrations of Stationary Waves. — The illustrations so far given of these vibrations due to the superposition of two trains of waves in opposite directions in a medium have been purely mechanical ones, the medium being either a solid or a fluid. But waves in the ether can produce these vibra- tions also, as is shown by the experiments of Wiener. A node in this case corresponds to a point where there is no motion in the ether (if we retain our mechanical concept of the ether) ; and a loop, to a point where there is the greatest motion. Rapid vibrations in the ether produce in the sur- rounding matter various effects, such as chemical changes which may be shown by photographic processes, fluores- cence, etc. ; and by all of these the existence of nodes and loops in the ether has been proved, when waves in it fall upon a mirror and suffer reflection. The distance apart of the nodes equals half the wave length of the incident waves. Electrical waves along wires and electromagnetic waves, produced by ordinary electric oscillations, cannot produce these vibrations with nodes and loops, because these waves are all "damped"; and owing to this fact the amplitudes of the reflected and incident waves are not equal. Measurement of the Velocity of Waves. — These facts in regard to the vibrations of stretched cords, of wires and rods, and of columns of gas (or, in fact, liquids also), lead at once to obvious methods of comparing the velocities of waves in different media. Thus, suppose two stretched cords have the same frequency of vibration when vibrating transversely in their fundamental modes ; then if L^ and L^ are their lengths, and Vl and F^ are the velocities of transverse waves along them, •• BTA /vo.v.i/;)' HMFJ&S" 363 Similarly, if the fundamental longitudinal vibrations of two \viivs «'i rods of different material are the same, and if Zj and LI are their lengths, we have the same relation, I \ : ] 2 = Ll : Ly for the velocities of compressional waves. Or. it t\ 11 pipes contain different gases, and if their fundamental vibrations have the same frequency, the same formula applies for the velocities of waves in these gases. There an- two general methods for determining when two vibrations have the same frequency if the medium is a ma- il-rial one. As we shall see later, when two vibrating bodies that are not moving bodily are producing sounds, they have the same frequency if the " pitches " of the two sounds are the same; and this can be told with great exactness by a trained ear; or, if they differ slightly, the exact difference in frequency can be determined by the beats. Again, the vibrations may both be produced by resonance from a third vibration. Thus, if a vibrating tuning fork is held at the opening of a column of a gas whose length can be varied, resonance may be secured by noticing for what length the sound i> most reenforced ; and similarly with the column of a second gas. (A column of gas can vibrate, as has been shown, in many ways, breaking up into a different number of segments. So with a Lfiven tuning fork, resonance may be secured for several different lengths of the column of gas. The distance from node to node is the same, however, for all these different nodes of vibration ; and it is the quant it I. in the previous formula. If the frequency of the fork Nis known, and this length L is m.-asm-.-d, the velocity of tin- waves in the gas is obviously *2LN.) Again, stretched • •ords or wires may have their lengths .-han^i-d until they are in resonance with a given tuning fork by the following mechanical method : stretch the cord or wire from two pegs or over two knife edges which are fastened to a large box open at its ends ; place loosely on the cord or wire a light saddle of paper ; set the stem of the vibrating tuning fork 364 VIBRATIONS AND WAVES on the box ; if there is resonance, the wire or cord will be set in vibration and the paper saddle will be thrown off. Kundt's Method. — We can also compare the velocity of congressional waves in a solid rod with their velocity in a gas or liquid by a method devised by Kundt. The gas is contained in a long tube, which is closed at one end by a tightly fitting piston and at the other by a very light one, which can move to and fro easily, but which nearly fits the tube. This last piston is attached rigidly to one end of the rod which is along the axis of the tube but projects beyond it, and which is clamped at its middle point. This rod is set in longitudinal vibration by stroking its free end with a damp cloth, and so the piston attached to its other end FIG. 164. — Kundt's apparatus for measuring the velocity of waves. vibrates and produces waves in the gas contained in the tube. The length of this column of gas is altered by means of the piston at its further end until it is vibrating with nodes and loops. The method of determining this condition is to sprinkle through the tube some light powder, such as is obtained from the finest cork dust ; when nodes and loops are formed, the powder collects in ridges across the bottom of the tube, leaving, however, the nodes perfectly bare. The frequency of the vibration of the rod is the same as that of the column of air, because the latter is " forced " by the former ; so, if L^ is the distance from loop to loop in the rod, that is, its length if we neglect the mass of the vibrating piston ; and if Lz is the distance from node to node in the column of air, the velocity of compressional waves in the rod is to the velocity of waves in the gas as Ll : Z2. (This method may also be used for a liquid instead of a gas by using suitable powder to mark the nodes.) 365 It should be observed that the piston on the end of the rod is at a loop for the rod, but a node for the gas; exactly as when the long elastic cord is set in vibration by the hand, as d---n il»ed on page 350, — the motion at a loop in the solid rod is extremely small compared with that at a loop in the gas. The explanation of the formation of the transverse ridges of the pow- der depends upon the fact that, as the particles of gas away from the nodes vibrate back and forward between the particles of powder, their mean velocity depends upon the arrangement of the dust particles, and varies at different points, thus producing pressures in certain directions. (See page 169.) The full description of the process is long, and will not be gi
ven here. It may be found in Rayleigh's Theory of Sound, Vol. II, page 46. Therefore, if we know by direct experiment, or otherwise, the velocity of waves in air, we may by Kundt's method de- termine the velocity of compressional waves in any solid mate- rial out of which a rod can be made ; and then, by replacing the air in the tube by some other gas (or by a liquid), we may determine the velocity of waves in it. (This is the .standard method for all ^ascs which can be secured in a small quantity only, such as helium, argon, and the other new gas« The values of the velocity of compressional waves in a few substances is given in the following table : Air • • • . 0° C. . . . 33,140 cm. per second Hydrogen . . . 0° C. ... 128,600 cm. per second I Humiliating gas . . 0° C. ... 49,040 cm. per second Oxygen . 0° C. ... 31,7i>0 cm. per second ...\| (abx.lnt.-) . 8°.4C. . . . 126,400 em. IMT. second :-nm . . . 73.4C. . i Water . . 8° C. ... 1 1.V)<H> em. i>er second Braes 350.000 cm. per second < "pper ... 20° C. ... 356,000 cm. per second 600,000 to 600,000 cm. per second . 20° C. . . . f> 13.000 cm. IMT second tfin ... 16° C. ... 130,400 cm. per second CHAPTER XX HUYGENS'S PRINCIPLE. REFLECTION AND REFRACTION Huygens's Principle. — One of the most important theorems in regard to wave motion is in part due to Huygens, and it is called by his name. In its most general form it is com- plicated, and can be demonstrated only by the aid of the infinitesimal calculus. We shall give certain special applica- tions of it ; and, although the statements to follow are not rigorous, they are sufficiently so for all present purposes. We cannot do better than to use Huygens's own language, as it appears in the translation of his TraitS de la Lumiere by Crew in The Wave Theory of Light (New York, 1900). Huygens's treatise was written in 1678, but was not pub- lished until 1690. " In considering the propagation of waves, we must remem- ber that each particle of the medium through which the wave spreads does not communicate its motion only to that neigh- bor which lies in the straight line drawn from the luminous point, but shares it with all the particles which touch it and resist its motion. Each particle is thus to be considered as the centre of a wave. Thus, if DCF is a wave whose centre and origin is the luminous point A, a particle at B, inside the sphere DCF, will give rise to its own individual [secondary] wave, KCL, which will touch the wave DCF in the point (7, at the same instant in which the principal wave, origi- nating at A, reaches the position DCF. And it is clear that there will be only one point of the wave KCL which will touch the wave DCF, viz., the point which lies in the straight line from A drawn through B. In like manner, each of the 366 REFLECTION AND REFRACTION 867 other particles, bbbb, etc., lying within the sphere DCF, gives \ n wave. The intensity of each of these waves may, however, be infinitesimal com- pared with that of I) OF, which is the resultant of all those parts of the other waves which are at a maxi- mum distance from the centre A. " We see, moreover, that the wave DCF is determined by the extreme limit to which the motion has traveled from the point A within a certain interval of time. For there is no motion beyond this wave, whatever may have been produce (I inside by those parts of the secondary waves FIG. 165. which do not touch the sphere DCF." We shall also quote Huygens in his explanation of reflec- tion and n-fractioii. 1. Reflection of Plane Waves by a Plane Mirror. — " Having explained the effects produced by light waves in a homogene- ous medium, we shall next consider what happens when they impinge upon other bodies. First of all we shall see how reflection is explained by these waves, and how the equality of angles fol- lows as a consequence. l.«-t AB represent a plane polished surface of some metal, glass, or other sub- stance, which, for the \| Ft... 1M. ent, we shall consider as perfectly smooth (concern- in^ iiT("_nil;int i> - which are unavoidable, \\.- shall have some- thing to sav at the close of this demonstration); and let the liii. .I/..', inclined to AB, represent a part of a light 368 VIBRATIONS AND \YAVES wave whose centre is so far away that this part, AC, may be considered as a straight line. It may be mentioned here, once for all, that we shall limit our consideration to a single plane, viz., the plane of the figure, which passes through the centre of the spherical wave and cuts the plane AB at right angles. " The region immediately about 0 on the wave A 0 will, after a certain interval of time, reach the point B in the plane AB, traveling along the straight line CB, which we may think of as drawn from the source of light, and hence drawn perpendicular to AC. Now, in this same interval of time, the region about A on the same wave is unable to transmit its entire motion beyond the plane AB ; it must, therefore, continue its motion on this side of the plane to a distance equal to CB, sending out a secondary spherical wave in the manner described above. This secondary wave is here rep- resented by the circle SNR, drawn with its centre at A and with its radius AN equal to CB. " So, also, if we consider in turn the remaining parts, H, of the wave AC^ it will be seen that they not only reach the surface AB along the straight lines HK parallel to CB, but they will produce, at the centres, K, their own spherical waves in the transparent medium. These secondary waves are here represented by circles whose radii are equal to KM, that is, equal to the prolongations of HK to the straight line BGr, which is drawn parallel to AC. But, as is easily seen, all these circles have a common tangent in the straight line BN, viz., the same line which passes through B and is tangent to the first circle having A as centre and AN, equal to BC, as radius. " This line BN (lying between B and the point N, the foot of the perpendicular let fall from A) is the envelope of all these circles, and marks the limit of the motion produced by the reflection of the wave AC. It is here that the motion is more intense than at any other point, because, as has been RE I '!.!•:< T/o.Y l.\/> UEFRACTION 369 explained. HX is tlie new position which tin? wave AC has assumed at the instant when the point C has reached B. For there is no other line which, like BN, is a common tan- gent to these circles . . . " It is now evident that the angle of reflection is equal to the angle of incidence. For the right-angled triangles ABC and BXA have the side AB in common, and the side CB equal to the side NA, whence it follows that the angles oppo- site th.-se sides are equal, and hence also the angles CBA and NAB. But CB, perpendicular to CA, is the direction of the incident ray, while AN, perpendicular to the wave BN, has the direction of the reflected ray. These rays are, therefore, equally inclined to the plane AB. " I remark, then, that the wave A C, so long as it is con- sidered merely a line, can produce no light. For a ray of light, however slender, must have a finite thickness in order to be visible. In order, therefore, to represent a wave whose path is along this ray, it is necessary to replace the straight line AC by a plane area . . . where the luminous point is supposed to be infinitely distant. From the preceding proof it is easily seen that each element of area on the wave [front], having reached the plane AB, will there give rise to its own secondary wave ; and when C reaches the point B, these will all have acommon tangent plane, vi/.., [a plane through BN~\. This [plane] \\ill he cut ... at right angles hy the same plane which thus cuts the [wave front at A C at right angles, i.e. the plane of incidence]. "It is thus seen that the spherical second a r\ waves can have no common tangent plane other than BN. In this plane will he located more of the reflected motion than in any other, and it will therefore receive the light transmitted from the \\.ive CH. ******* VM! -'« I II •, - i 370 VIBRATIONS AND WAVES "We must emphasize the fact that in our demonstration there is no need that the reflecting surface be considered a perfectly smooth plane, as has been assumed by all those who have attempted to explain the phenomena of reflection. All that is called for is a degree of smoothness such as would be produced by the particles of the reflecting medium being placed one near another. These particles are much larger than those of the ether, as will be shown later when we come to treat of the transparency and opacity of bodies. Since, now, the surface consists thus of particles assembled together, the ether particles being above and smaller, it is evident that one cannot demonstrate the equality of the angles of inci- dence and reflection from the time-worn analogy with that which happens when a ball is thrown against a wall. By our method, on the other hand, the fact is explained without difficulty. " Take particles of mercury, for instance, for they are so small that we must think of the least visible portion of sur- face as containing millions, arranged like the grains in a heap of sand which one has smoothed out as much as possible ; this surface for our purpose is equal to polished glass. And, though such a surface may be always rough compared with ether particles, it is evident that the centres of all the second- ary waves of reflection which we have described above lie practically in one plane. Accordingly, a single tangent comes as near touching them all as is necessary for the pro- duction of light. And this is all that is required in our demonstration to explain the equality of angles without allowing the rest of the motion, reflected in various direc- tions, to produce any disturbing effect." The law of reflection in regard to the equality of the angles of incidence and reflection was known to the ancients, and was made use of by Euclid as early as 300 B.C. He also deduced some of the properties of concave
mirrors. The law stating that the normals to the two waves and the surface lie Hl-:i-'LK< TIOH AND REFRACTION 371 in a plane was first given by the Arabian scholar Al Hazen, about 1000 A.I). 2. Refraction of Plane Waves at a Plane Surface. — Again quoting from Huygens : u In order to explain these phe- nomena on our theory, let the straight line AB, Fig. 10, represent the plane surface bounding a transparent body extending in a direction between 0 and N. " By the use of the word plane we do not mean to imply a perfectly smooth surface, but merely such a one as was employed in treating of reflection, and for the same reason. Let the line AC represent a part of a light wave, whose source is so distant that this part may be treated as a straight line. The region (7, on the wave AQ will, after a certain interval of time, arrive at the plane AB, ;il<niLT the straight line CBi which we must think of as drawn from the source of light, and which will, therefore, intersect AC at right angles. But during this same interval of time, the region about A would have arrived at Q-, along the straight line AG. equal and parallel to CB\ and, indeed, the whole of the wave AC would have reached the position Q-B* provided the transparent body were capable of transmitting waves as rapidly as the ether, lint suppose the rate of transmission is less rapid, say one third less. Thru the motion from the point A will extend into the transparent body to a distance which is only two thirds of CB. while producing its secondary spherical wave as described above. This wave is repr. ^. nted by the circle vV// mitre is at A and whose radius is equal to } CB. Fio. 167. It' \\e consider in like manner the. other points /STof the wave 372 VIBRATIONS AND WAVES AC, it will be seen that, during the same time which 0 cm- ploys in going to B, these points will not only have reached the surface AB, along the straight lines HK, parallel to OB, but they will have started secondary waves into the trans- parent body from the points Kas centres. These secondary waves are represented by circles, whose radii are respectively equal to two thirds of the distances KM ; that is, two thirds of the prolongations of HK to the straight line B G-. If the two transparent media had the same ability to transmit light, these radii would equal the whole lengths of the various lines KM. " But all these circles have a common tangent in the line BN\ viz., the same line which we drew from the point B tangent to the circle SNR first considered. For it is easy to see that all the other circles from B up to the point of contact N touch, in the same manner, the line BN, where N is also the foot of the perpendicular let fall from A upon BN. " We may, therefore, say that BN is made up of small arcs of these circles, and that it marks the limits which the motion from the wave AC has reached in the transparent medium, and the region where this motion is much greater than anywhere else. And, furthermore, that this line, as already indicated, is the position assumed by the wave AC at the instant when the region C has reached the point B. For there is no other line below the plane AB, which, like BN, is a common tangent to all these secondary waves. . . . " If, now, using the same figure, we draw EAF normal to the plane AB at the point A, and draw DA at right angles to the wave AC, the incident ray of light will then be repre- sented by DA ; and AN, which is drawn perpendicular to BN, will be the refracted ray; for these rays are merely the straight lines along which the parts of the waves travel. "From the foregoing, it is easy to deduce the principal law of refraction; viz., that the sine of the angle DAE always bears a constant ratio to the sine of the angle NAF, ui:rii:moN AND i;i-:n;.i< TION 373 whatever may be the direction of the incident ray, and that tin- ratio is the same as that which the speed of the waves in the medium on the side AE bears to their speed on the side AF. " For, if we consider AB as the radius of a circle, the sine of the angle BAC is BQ and the sine of the angle ABN is AN. But the angles BAC and DAE are equal, for each is the complement of CAE. And the angle ABN is equal to NAF, since each is the complement of BAN. Hence the sine of the angle DAE is to the sine NAF as BO is to AN. But the ratio of BCto AN is the same as that of the speeds of light in the media on the side toward AE and the side inward AF, respectively; hence, also, the sine of the angle DAE bears to the sine of the angle NAF the same ratio as these two speeds of light." Since these speeds are properties of the media and not of the direction of the propagation of the waves, we have at once the law that the ratio of these sines is independ- ent of the angle of incidence. It is evidently different for different media, and will be shown to be different for waves of different wave length in the case of ether waves in a material medium. This law of refraction was first discovered experimentally by Snell (1591-1626). I! --fraction is a much more common phenomenon with ether waves than with air waves, and it. will be disenss.-d more fully in the section devoted t<> Light. CHAPTER XXI INTERFERENCE AND DIFFRACTION Interference Young's Experiments. — A most important phenomenon of wave motion, and one of particular interest historically because by means of its discovery Thomas Young in 1801 proved that light was due to waves, is wli.it is called inter- ference. In Young's own words: "When two undulations, from different origins, coincide either perfectly or very nearly in direction, their joint effect is a combination of the motions belonging to each. " Since every particle of the medium is affected by each undulation, wherever the directions coincide, the undulations can proceed no otherwise than by uniting their mo- tions, so that the joint motion may be the sum or difference of the separate motions, accordingly as similar or dissimilar parts of the undulations are ;. 16S. — Interference of waves on the surface of a liquid, which are sent out by two point sources. coincident." (This prin- ciple is illustrated in the cut, which represents the " interference " of two trains of water waves.) Two experiments may be described ; both are due to Young, and both may be performed easily with home-made apparatus. We shall describe them as if the waves to be 374 7JV TK /,•/•/•:/.' i:\rf-: .i.v/> inrn;.ii IION 375 studied were light waves; but the same apparatus, suitably enlarged, would do equally well for air waves. Let there be trains of waves sent out by having some source placed near a long Marrow slit in an opaque screen. If the slit is sufficiently narrow, the disturbances will proceed out from the slit in all directions, making a train of waves with a cylindrical wave front. A second opaque screen with two X' if row slits, which are close together and both of which are parallel to the slit in the lirst screen, and at equal distances from it, is placed parallel to the latter. As the cylindrical waves reach these two slits, two cylindrical trains of waves are produced beyond the second screen. The importance of this arrangement lies in the fact that the two trains of waves thus produced are identical; that is, they have the same amplitude, the same wave length, and the same phase, because they arc produced by disturbances in the same wave front at the same distance from tin- lirst slit; so, if the original source of the waves changes its character in any way, the two cylindrical waves from the two slits both change in the same manner at the saint- instant. Then, if we consider the effect at any point in the space which is traversed l>y tin- two trains of waves, it is receiving disturb- ances from both waves, and the effect produced is the sum of two, one due to each train. Another mode of producing this result is to remove the screen with the two slits, and to place parallel to the slit in the first screen a narrow opaque object like a line wire or small needle. As will be shown in speaking of diffraetion (see page 880). disturbances are produced in the low . \ l( tly as if there were a source of waves along the edge of the obstacle ; so, in the case of the wire or needle. the points in the shadow are receiving distnrhanees from two parallel line sources ,,f waves along the two ed: These two disturban then, in this case also due to 1 \\ o '{•'iff trains of Vfl 376 VIBRATIONS AND WAVES If these waves are light waves of a definite color, and if, from a point in the medium traversed by the two trains of waves, one looks in the direction of the two slits (or of the wire), or, better still, if a magnifying glass is used in front of the eye, a series of black and colored lines parallel to the slits (or wire) are seen. Similarly, if short sound waves are used, it is not difficult to prove by means of a sensitive flame (see page 192) that there are corresponding " bands of silence and sound," meaning that at points along a line parallel to the slits there are disturbances in the air, while along a neighboring line there are not. B C A C FIG. 169. — Diagram of Young's Interference experiment. 0t and Ot are two sources of waves and ACi& a screen on which the two trains of waves are received. U The explanation is not difficult. Let us consider the effect at various points on a screen parallel to the plane of the two slits. Let Ol and 02 be the traces of the slits on the paper, and A C that of the last screen. Let B be a point halfway between the slits ; draw a line BA perpendicular to the screens, and let 0 be any other point on the screen. This point receives disturbances due to two trains of waves ; but the lengths of the paths from C to the two sources 01 and 02 are not the same. This is shown on a large scale in part of the diagram. i.\TKi:ri-:nENCE AND DIFFRACTK>\ 877 FKU n 0.2 draw O.J perpendicular to the line 0-^C. Since OL a jid 0.2 are iii reality extremely close together compared with the other distances
in the apparatus, O^P is the differ- ence in path from 01 and 02 to C. If it amounts to a wave length exactly, or to any integral number of wave lengths, tin- disturbances reach O in the same phase, and so the effect is great ; but, if this difference in path is exactly half a wave length, or any odd number of half wave lengths, the disturb- ances arrive at O in exactly opposite phases, and so there is no effect. At the point A the two paths are of equal length, so the effect is great ; and as points near it are considered, constantly receding from A in either direction, the effect decreases, becomes zero when the difference in path is half a wave length, increases to a maximum when this difference is a wave length, decreases to zero again; etc. The effect at all points on a line through C parallel to the two sources is evidently the same ; and so the screen is covered with a pat- tern of bands, or. as they are called, "interference fringes," alternately dark and bright. The condition for a fringe where there is a maximum effect is, in terms of the figure, that where n is any whole number 0, 1, 2, 3, etc. ; and I is, as usual, the wave length. Similarly, the condition 1m a fringe of zero intensity is that O^P = (2 n+ 1 > _>- 2 These conditions may be expressed in t< -rms of the distance of ih«- friii'_r'- from tin- mitral one at A ; i.e. in terms of the distance AC. Call this distance x: the distance apart of th«- two screens, i.e. AB, a \ thr di>taiir«- apart of the two sources 010a, 6; and tin- angle (ABC), N. It is to be re- mrmlirivd !l,;it this angle is always small, and that the two soiin-.-s are so close tOgethrr that the lim-s drawn from O to tin- points Or /f, and 03 make practically the same angle 378 VIBRATIONS AND WAVES with AB. Then, referring to the diagram, the angles (ABC) and (OjOjP) are equal, A0= AB tan (ABC), and O^P = 0X02 sin (^i^2-P); or, in symbols, or and hence 0P = _ Therefore, if £ is small com- Va2 + z2 fo pared with a, 0XP = — . \i x gives the position of a fringe of maximum intensity, nl = — , or x = — — . The position of i i a al al the next similar fringe is given by xl = (n 4- 1) — - ; so the dis- tance apart of the two, or xl — x, is — . This shows that all the bright fringes are equidistant. Similarly, if x gives the distance of a fringe where there is a zero effect, The distance of the next similar fringe is given by so their distance apart, or x1 — x, is — , the same as for the fringes of maximum intensity. It is thus seen that the greater the wave length, the farther apart are the fringes ; and that, if their distance apart, that of the screens, and that of the sources are known, the wave length may be determined. This matter will be referred to later. It should be noted that in the formation of these interfer- ence fringes there is no destruction of energy ; it is simply distributed differently from what it would be if the screen t: A.\h Dll-'Fl; ACTION 379 were receiving waves from two sources which had no perma- nent phase relation. < rfertno* frtagM irtrttlntil by Young's method. 380 VIBRATIONS AND WAVES Other cases of interference will be described in the section devoted to Light, but all interference phenomena dealing with light can be reproduced with waves in the air. Diffraction FresnePs Principle. — It is a well-known fact that, if an opaque obstacle is interposed in a beam of light, a shadow will be cast on any suitably placed screen, which is more or less sharply defined, depending upon the smallness of the source of the light. This is sometimes expressed by saying that "light travels in straight lines." In the case of sound, however, such an obstacle would not prevent a noise being heard behind it ; in other words, there is no sound shadow with such an obstacle. The explanation of the difference in the two cases was given by Fresnel, making use of Huygens's principle. It may be well to quote Fresnel's own words in Crew's translation. Fresnel's great memoir on Diffraction, from which these quotations are made, appeared in 1810. "I shall now show how by the aid of these interference formulae and by the principle of Huygens alone it is possible to explain, and even to compute, all the phenomena of dif- fraction. This principle, which I consider as a rigorous deduction from the basal hypothesis, may be expressed thus: The vibrations at each point in the wave front may be considered as the sum of the elementary motions which at any one instant are sent to that point from all parts of this wave in any one of its previous * positions, each of these parts acting inde- pendently the one of the other. It follows from the principle of the superposition of small motions that the vibrations pro- duced at any point in an elastic fluid by several disturbances * I am here discussing only an infinite train of waves, or the most general vibration of a fluid. It is only in this sense that one can speak of two light waves annulling one another when they are half a wave length apart. The formulae of interference just given do not apply to the case of a single wave, not to mention the fact that such waves do not occur in nature. /A'/-/-;/,- /••/•;/,• /-;.v< B .i.v/> inrni ACTION 381 are equal to the resultant of all the disturbances reaching this point at the same instant from different centres of vibra- tion, whatever be their number, their respective positions, their nature, or the epoch of the different disturbances. This general principle must apply to all particular cases. I shall suppose that all of these disturbances, infinite in num- ber, are of the same kind, that they take place simulta- neously, that they are contiguous, and occur in the single plane or on a single spherieal surface. ... I have thus reconstructed a primary wave out of partial [secondary] dis- turbances. We may, therefore, say that the vibrations at each point in the wave front can be looked upon as the resultant of all the secondary displacements which reach it at the same instant from all parts of this same wave in some previous position, each of these parts acting independently one of the other. "If the intensity of the primary wave is uniform, it follows from theoretical as well as from all other considerations that this uniformity will be maintained throughout its path, pro- vided only that no part of the wave is intercepted or retarded with respect to its neighboring parts, because the resultant of the secondary displacements mentioned above will be the same at every point. But if a portion of the wave be stopped by the interposition of an opaque body, then the intensity of each point varies with its distance from the edge of tin- shadow, and these variations will be especially marked near the edge of the geometrical shadow." Rectilinear Propagation. — As a simple case, consider a train of plane waves advancing from left to right; let the paper be at ri.rht angles to them and let the trace on the the wave front at any instant be given in part by AB. The etl'eet at any later time at a point P in advance of the waves is determined, as already Stated, by deducing the effects there owing to the secondary waves from ea< -h point of the wave fn.nt. and adding these geometrically VIBRATIONS AND WAVES It is evident that the effect at P of the secondary waves from any point Q on the wave front depends upon the length of the line QP for two rea- sons : the decrease in amplitude of spherical waves varies inversely as the distance, and the phase of the disturbance as it reaches P varies with it. There- fore, if 0 is the foot of the perpendicular let fall upon the FIG. 171. — AB is a section of a plane wave advancing toward P. B wave front from P, i.e. its "pole," as it is called, and if a circle with a radius equal to OQ be drawn on the wave front around 0, the secondary waves from each of the points on this circle will reach P with the same amplitude and in the same phase, because they start out with the same amplitude and in the same phase, and travel the same distance to reach P. But the directions of the displacements due to the sepa- rate secondary waves are not the same, and they must be added geometrically. (Since the phases are all the same, we have simply a case of vector addition.) Let us assume that the waves are longitudinal (the proof is similar, if they are transverse). Let Q1 and Q2 be two points at the end of a diameter of the circle round 0\ and let the displacement at P due to the secondary waves from Q1 be represented by FIG. 172. — Diagram to represent the resultant action at P of all the secondary waves from points <?,, Qt, etc., on a circle around 0. i\TKi;rEi;i-:\f'E A.\D DIFFRACTION 383 then that due to the secondary waves from Q2 is repre- sented by PA2. Their resultant is PA^ a displacement perpendicular to the plane wave front. Similarly, the resultant displacement due to all the secondary waves coming from points in the circle around 0, through Qr is proportional to PA. But calling the angle (OPQ^) the " inclination " of Qv and representing it by jY", it is evident tliat PA = 2 PAl cos N; and therefore, as the inclination increases, the resultant displacement decreases. The task of compounding the effects at P, due to the secondary waves from all the points in the wave front, is not at first sight a simple one ; but Fresnel invented a most brilliant method. His plan is as follows : Calling the length <>f the line OP a, and the wave length of the waves /, describe around P as a centre a series of spherical surfaces of radii a, a -f- -, a -M, a H- — , etc. ; these spheres will cut the wave 7 07 2 2 front in a series of concentric circles around 0, thus dividing the plane into a series of concentric zones or circular ri and the effect at P due to the secondary waves from each zone is considered as a whole. Fresnel's purpose in thus dividing the wave front into these particular zones was to secure the following condi- tion : If we consider the disturbance at P from any point in any /.
one whose distance from P is 6, this will be exactly opposite in phase to disturbances reaching P at the same instant from certain points in the two emit i^mns /<>nes, tin- distances from which to P are b + - and ft — -; for. if two 2 2 waves differ by half a wave length, one produces an effect tly opposite in phase to the other: therefore, if the whole effect at P due to all the secondary frarefl fn»m tin- points in one zone is in a direction which is railed M pnsit i that due to the waves fnmi the two oontlgaoua /ones will be in a negative din-el i«»n. Consequently, calling >//r m,, my 384 VIBRATIONS AND WAVES etc., the magnitudes of the effects at P due to the first, second, third, etc., zones, the total effect at P due to the whole wave front may be written The numerical value of any m depends upon three quanti- P ties : the area of the zone, its mean distance from P, and its "inclination" — as just defined. The area of a zone is easily calculated. Let the two radii of the spheres around P which the opposite edges of the (n+i)st zone determine it be defined by the around O. iw. n3. — <>n and Qn+i are points on nl equations : Then \+l = PQ?n+l - OP = (n Therefore the area of the circle through Qn is and that of the circle through Qn+l is so the area of the zone between is the difference between these'or This may be expressed a 4 IXTERFERENCE AM) DIFFRACTION so if I is extremely small in comparison with a, this area is 7T/</. a i i.l is therefore the same for all the zones. (In the case of ether waves which affect our sense of sight I is about Ffftaff n"» an^ so this assumption is justified; for waves in tir which affect our sense of hearing the wave length varies from about half an inch to about 20 ft., and this approximation cannot be used.) If the wave length is so small that - is a small quantity, a the mean distances of two contiguous zones from P are also very approximately equal. In the general case, the area of each zone is slightly greater than that of the one inside it. and this fact would make its effect greater than that of the inner /.one; but this is counterbalanced by the fact that the mean distance of the inner zone from P is less. In any case, th«- inclination of each zone is less than that of the contigu- ous one outside it, but by a very small amount ; and then the value of m for any zone is greater than that for the outer zone. In symbols, this may be expressed mm-i>mm>mm+ii where mn is the effect of the nth zone. Then-fore, the series /= 7»j — m., -f- w8 — m4 + m6— etc., is one of terms which de- crease in numerical value by extremely small amounts. If we rewrite this series in the form f = i "M + (I ml - m, + i ros) + Q m, - 4 + J ma) + we see that each of tin- terms with the exception of the first is extremely small. The last term in the series is one hall the in for the la- and. if the wave front is not lim its value may be assumed to be vanishinglv small. Therefore, -urn of all the series except the first term is inlinitesir and we may write J= \ mr This means that the effect ,v /' due to the secondary \\a\es from any zone except the first utrali/' • • half the «•:• ntigUOUS zones; and so the total effe< t at /' due to th. \\hole wave front is one half tliat d n. iliime. Tim area of this 386 VIBRATIONS AND II zone is wla very approximately, if - is ;i small quantity. a (For visible ether waves I is not far from 0.00005 cm., and so if a = 10 cm., the area Trla equals 0.0016 sq. cm.) Spherical Waves. — The case .of spherical or cylindrical waves may be treated in the same manner ; and in some cases other modes of describing " Fresnel zones " are prefer- able. The result in them all is, however, the same. If from a point in advance of the wave front a perpendicular line is let fall upon it, the effect at this point, due to the whole wave front, is one half of that due to the first central zone around the point where this perpendicular meets the wave front, i.e. the "pole." (It is assumed, of course, that the velocity of a disturbance in the medium is independent of the direction of propagation ; otherwise, referring to the previous cuts, the effects at P from points in a circle around 0 would not reach it in the same time, and so they would be in different phases. This case of non-isotropic media will be discussed later in speaking of Double Refraction.) Let us, then, consider the propagation of a train of waves having a spherical wave front, spreading out from the point S. Let AB be a portion of the wave front at any insta n t : the effect at P at a later time will be one half that pro- duced there by the central zone around 0, where 0 is the pole of P. If the wave length is ex- tremely small so F». 174. -Case of spherical waves advancing toward P. ^ ^ ^ Q£ ^ central zone is small, we may say that the effect at P is due to the disturbance at the point 0; and so in turn, when the wave front reaches P, we may say that the effect at a point AM* birn;A< T/o.v 887 I\. farther cut on the line SOPV is due to the effect at the point 1\ etc. In this sense, the disturbance due to a train of waves having a small wave length is propagated in a ight line. 1 his line, SOPPr is called a " ray." (In an •tly similar sense, the disturbance of an exceedingly thin ••{mlse" is propagated in a straight line.) This obviously explains the general phenomenon of the casting of shadows by opaque obstacles when a small source of light is used. Diffraction past an Edge. — If the waves have a long wave length, the area of the central zone is not small, and the •t is not propagated from point to point. So witli waves in the air, which produce sounds, there are no sharp shadows; neither are there in the case of light, if one Ige of the shadow with care. To see exactly what occurs. we shall consider one simple case, following -Mel's treatment. -pherical waves from a point source S meet an opaque oh>tacle the section of whose edge is represented at A ; we shall study the etlr.-ts at various point of the ob- stacle. Straight linrs drawn from 8 through the points on the edge of the obstacle to • 11 mark uhat is called the "geomci .shadow." Thus drawing the line > 1 //. // i t he edge of the shadow. At any point /'. i his. the etl.-ct due to a wave front that U • acrwn. B b U* limit of UM fWMMtttotl fthftdow. Diffraction put • »hrp «!«• of an opaqa* obrtaeto. n.. IT-. obstacle 1 by con I6t dia\\ ii 388 VIBRATIONS AXD WAVES around 0, where this is the intersection of the wave front by the straight line SP. It is seen that only a limited number of complete zones can be drawn, that have an effect ; for the others will be cut off more or less by the opaque obstacle. If this number of complete zones is even, e.g. four, their effect may be written m^— m^-}- m^— m^ and is therefore almost zero ; while, if the number is odd, e.g. three, their effect may be written ml— m2+ ra3 = \| mv nearly, and so is great. Con- sequently, as the point P is taken farther and farther out from B, the intensity passes through a series of maxima and minima ; and, when it is some distance out, the effect becomes uniform and is uninfluenced by the obstacle. FIG. 176. — Diffraction bands seen near the geometrical shadow cast by an edge. At any point Pv inside the geometrical shadow, the effect may be considered as due to the zones around Ov where the straight line SPl meets the wave front. A number of these zones are behind the obstacle ; but a certain one, say the nth, will produce an effect, because it emerges sensibly above the edge at A. The total effect at P may then be written mn— ™n+i+ ™n+2>-., which equals ^ approximately ; so there is an effect at Px which becomes less and less continuously as the point recedes into the shadow, corresponding to the 2 tNTMRTMBMNCM AM> />//•'/•'/,•. KT/otf 389 increase in n and the consequent gradual decrease in mn. Therefore, the disturbance from impenetrates into the geo- metrical shadow. The effect may be represented by a curve so drawn that its height above a horizontal line at any point indicates the intensity of the disturbance at that point. Thus, in Fig. 176, the points P1 and B are the - represented by the same symbols in the previous cut. It is seen that the effect in the geometrical shadow is as if there were a source of waves at the point A, the edge '->!' the opaque obstacle. Fi... 177. — Photograph of diffraction bands near the geometrical shadow of a sharp edge. Naturally, everything depends upon the magnitude of the wave length of the waves. If this is small, as in light, the waves penetrate a very short distance into the ir.-omrtrieal shadow; or, better, tin- intensity of the waves decreases so rapidly inside the shadow that they can be percei\<-d for only a short distance; and the region «»utsidi- the shadou in \\ Inch there are variations in the intensity, i.e. where there are "bands," ii extremely limited. P.oth ()f these phenomena may be easily observed in the case of a shadow cast on a screen by any opaque obstacle if the source of light is small. 390 VIBRATIONS AND WAVES If we are dealing with air waves about 1 cm. long, all of the above phenomena, as described for light waves, are easily observed by the use of a sensitive flame ; and, if the waves are several metres long, " sound shadows " may be observed if the obstacle is sufficiently large, e.g. a mountain. But we see that, if the waves are long and the obstacle is of ordinary size, the former will penetrate a great distance in the shadow. This phenomenon of the peculiarities of a shadow produced by trains of waves incident upon an obstacle is called " dif- fraction." It was first described in the case of light by the Italian priest Grimaldi in 1666 ; but its explanation was given by Fresnel. Diffraction through a Small Opening. — An important illus- tration of diffraction is afforded when a train of waves falls upon an opaque obstacle which has in it a single small open- ing. The general features of the phenomena may be deduced
easily. Let the waves come from such a direction that their wave front is tangent to the plane of the opening. Draw the line OB perpendicular to this plane, and OP oblique to it ; we shall consider the effect at various points on a screen perpendicular to OB. The effect at B depends upon the number of zones that can be drawn in the small opening at 0; if it is even, the effect is zero; if it is odd, the effect is practically as great as if the whole obstacle were removed, viz., £ mv The effect at P in a similar manner depends upon the number of zones that can be drawn for it in the opening. The centre of the zones is the pole of P on the wave front, Fio. 178. —Diffraction through a small opening 0. / \ / /• /; /••/;/; /•; \ i /•; . i \ i> DIFFRA GT/OJV* 391 and is therefore far from the opening ; and, since the edges of the zones get closer and closer as one recedes farther from the centre of the zones, P will have more zones — or rather portions of them — included by the opening than does B. So, if B has three zones in the opening, it will receive a maxi- mum effect; and, if P is sufficiently far away from B, it will have portions of four zones included, and so will receive a minimum effect ; farther out still, there will be a point for which there will be portions of five zones in the opening, and which accordingly receives a maximum effect; this is, however, much less than that at BI owing to the intimation of the zones and t<> the fact that only n:irr.-\v -lit. • I a Mi, ».'!•• 'ton* of zones are intercepted by the opening; etc. Con- s' the opening 18 circular, the "diffraction pattern " mi tin- sen-en will cMiisist of concentric circular "hands" in which the intensity is alternately a maximum and minimum, hut which rapidlv fade in intensity as one recedes from their centre at B. It is also <-\ ident from the same re, that 392 VIBRATIONS AND WAVES the smaller the opening, the farther apart are the bands; and. if the opening is sufficiently small, B will receive a maximum effect, and the first ring of minimum intensity recedes to an infinite distance ; so all the points on the screen receive dis- turbances from the opening just as if it were a minute source of waves (see page 375). This is a familiar fact, because if a small "pin hole" is made in a card- board or in any thin opaque screen, and a light of any kind is placed behind it, it appears bright when viewed from any direction on the far- ther side. (This is, of course, entirely apart from any light reflected from the edges or sides of the opening.) Similarly, if a room has a small open- ing in it, and if a tuning fork is vibrating in the room, it may be heard outside in all directions from the opening. . The treatment just given of diffraction past an edge and through a small opening must not be regarded as rigorous. It is based upon many assumptions, which have been pur- posely omitted, and which are not all justified. On the whole, however, the results are sufficiently accurate for our present purposes. Criteria of Wave Motion. — The existence of any of the phenomena that have been described in the foregoing arti- cles— vibrations with nodes and loops, diffraction, inter- ference — is proof positive of the presence of wave motion. This fact will be made use of immediately in describing and explaining the phenomena of Sound and Light. SOUND CHAPTER XXII ANALYSIS OF SOUND Fundamental Facts. — To one who has the sense of hearing tin- word --sound" has a definite meaning, as describii .dn sensation, but it is impossible to define it so as to convey an idea to one who is born drat'. The cause of this Cation can always be traced to some body that is vibrat- ing rapidly. Thus, if we hear u sound from a tuning fork, a piano string, a metal bell, etc., it is a simple matter to prove that they are vibrating, and that, when tin- motion ceases, the sound does also. Again, it is a familiar fact that some time elapses between the instants when tin- vibration begins and the sound is heard, and between those when the former stops and the latter ceases; for there is a considerable inter- val of time between the instants when a distant gun is seen to be fired and when the sound of the report is heard, or when a mt steam whistle is seen to blow and when the noise is heard, etc. This proves that \\ a uses the sensation time for its transmission through space. The fact that the product ion of the sensation depends upon the pres- ence of a vmtt' rf'il medium between the vibrating bnd\ and the ear may be proved by suspending the vibrating body in a space from which the air may be more an* 1 more exhausted ; as this is removed, leaving only the ether, the sound becomes less and less intense. Vibrating bodies will produce com- pre»ioiiul waves in a surrounding fluid, provided the fre- quency of vibration is sufficiently great; and the fact that SOUND the sensation of sound is due to these waves may be proved most simply by allowing them to produce vibrations with nodes and loops, and showing that they cause sounds. Methods of doing this will be described in a few pages. We shall discuss first the characteristics of different sounds and the physical cause of these differences, then describe a few typical musical instruments and some acoustic phe- nomena, and finally give the physical explanation of harmony in musical compositions, with a brief description of musical scales. Noise and Musical Notes. — If we analyze our sensations of sounds, we are led at once to recognize two great classes which we call in ordinary language noises and musical notes. The latter have all the characteristics of periodic motion ; they are continuous and uniform in character, and are pleas- ant to the ear. The former are discontinuous, with abrupt changes, and are often extremely disagreeable to -the ear. Thus, the sounds due to a tuning fork, to a piano string, to the column of air in an organ pipe, etc., are musical notes. But the sounds heard when a piece of paper is torn, when a wagon rolls over cobblestones, when a slate pencil is sharpened, etc., are noises. We can study the nature of the vibrations of a body, as has been explained on page 319, and it is found that a musical note is always due to a periodic vibration ; a noise, to an extremely complex motion, consisting of differ- ent vibrations which differ slightly in frequency and which are rapidly damped. A confused vibration which causes a noise will produce a musical note, if the vibrating body is near a flight of steps ; for, when the pulse reaches the first step, a reflected pulse is produced; and in a similar manner others are produced when it reaches in turn the other steps. Therefore there will be in the air a series of reflected pulses at exactly equal inter- vals apart; and, as they reach the ear, a musical sound is heard. Thus, if one claps one's hands near a staircase, the noise is first heard, but it is followed immediately by a musical note. The same phenomenon occurs if u. noise is produced near a picket fence. ANALYSIS "1 SOUND ' 395 Simple and Complex Notes; Quality. — If we analyze our sensations of different musical notes. \\c (.ID >eparate them into two classes: one we call "pure" or "simple"; then: "complex." Thus, the note produced by vibrating metal plates like cymbals is complex, while that due to a tuning fork <>r to a stopped organ pipe is pure. In fact, all notes, with the exception of these last, are more or less complex. If we examine the corresponding vibrations, it is found that a pure note is always due to a simple harmonic vibration, and a complex note to a complex vibration. Complex vibrations can be analyzed, in accordance with Fourier's theorem, into simple harmonic components who>e frequencies are in the ratio 1:2:3: etc., or into other com- ponents not so related. In a similar manner, if one listens attentively to a complex sound, various simple pure notes may be distinguished. (This statement that the human ear analy/.es mechanically a complex wa\c into simpler com- ponents and hears the corresponding simple notes separately is known as Ohm's Law for Sound. ) If two complex notes differ, it is found that the corre- sponding complex vibrations differ; but t he converse state- ment that two different complex vibrations produce t\\o different complex notes is true only with one limitation. If the component parts of the two complex vibrations differ in their frequencies or in their amplitudes, the corresponding notes are different ; but the ditVerenees in phase between the coni\| -nts may be different in the two complex vibrations or may be the same; they have no influence on the note. (This is seen to be in ace. n.l with ohm's law, as stated above.) Two different complex notes are said in "quality." Thus, the complex notes produced by the vibrations of the column of air in an organ pipe, of a violin string, of a piano siring, of the column of air in a horn, etc., all differ in qualit) ; and it is by this property that we recoglii/e the 396 SOUND nature of the source of the sound. This quality depends upon the number and amplitudes of the other component vibrations besides the fundamental present in the complex vibrations ; it does not depend, however, as has been already said, upon the relative phases of these component vibra- tions. Pure notes are never used in music, because -they lack what may be called "character," or individuality. Notes that are useful for musical compositions must be complex to a certain extent, seven or more components often being present. Analysis of Notes. — This process of analyzing a complex sound or a complex vibration into its harmonic components is greatly helped by the use of resonators. Helmholtz in his epoch-making work used those of the form shown in the cut. (One of their advan- tages is that when the inclosed air is set in vibration, the motion is simple harmonic.) He constructed a set of them, and accurately determined the frequencies of the vibrations of their inc
losed volumes of air. Then by bringing them in turn near a vibrating body, he could tell by holding one of the ends of the resonator near the ear whether the par- ticular vibration that corresponded to that of the air in the resonator was present in the complex vibration of the body ; for, if it were, the corresponding sound would be intensified. In this manner it is a simple matter to detect the components. In the best and most recent work on analysis of sounds, phonographs are used, and the traces are magnified. FIG. 180. — A Helmholtz resonator. A\AL}'SIS OF SOUND 397 Several other illustrations of resonance are worth mentioning. When a large seashell or a vase is lu-M near the ear, the roaring that is heard is due to the resonance of the inclosed air produced by certain sounds in tin- room. The sound may be varied by partly closing the opening of the sht'll or vase. The passage leading from the outside of the head into the eardrum forms a small resonator, and its action is often noticed when one is li>t«Miing to an orchestra, by the strong resonance of certain very shrill sounds like the buzzing of insects. Helmholtz performed with his set of resonators the con- verse of the analysis of a complex sound; he produced one by HUM MS of the superposition of simple harmonic vibrations or of pure notes. He arranged in front of each resonator a tuning fork whose frequency of vibration was the same as that of the air in the resonator, and adjusted electro-magnets to these forks in such a manner that he could set them vibrating and maintain them in motion. He could also alter their amplitude. Then by making different forks vibrate he able to produce different complex sounds, and in fact to imitate the sounds characteristic of different instruments. In a complex sound, the component note corresponding to the fundamental vibration is called the "fundamental"; and other notes "overtones." If the component vibrations form a harmonic series, the component notes are also called k* harmonics." Pitch and Loudness. — If we compare two simple notes, we recogni/e the fact that they may differ in two ways, in shrillness and in loudness. Thus, the notes of a piccolo are shriller than those from an nr-^an pipe; and any note of an in may keep the same shrillness and yet may vary in loudness. If we compare the < : ions, we tind that in every case if one note is shrill another, the frequency of its vibration, or rather the number • reaching the ear in a unit of time, is the greater. (This -st observed by G Kurt her, we find that ne note deoreaaei in Umd increases in amplitude, other things remaining unchanged. 398 We cannot measure the shrillness of a note, because we cannot imagine a unit of shrillness nor the idea of shrillness being made up of parts which can be compared. We can, however, give a number to the shrillness of a note by assign- ing it one equal to the frequency of the vibration, if the vibrating body and the observer are at rest relatively; or, more generally, we assign a number equal to the number of waves reaching the ear in one second. (See Doppler's Prin- ciple, page 345.) This number is called the "pitch" of the sound. Similarly, the pitch of a complex note is denned to be the pitch of its fundamental. The loudness of a sound, either pure or complex, varies as the intensity of the waves producing it. It is thus seen why, when a sounding body approaches the ear, the loudness of the sound heard increases. We can measure .the intensity of the waves, but we have no method of measuring the loudness, for this is a sensation, and not a physical quantity. Audibility of Waves. — In order to produce waves in the air, the frequency of the vibration must exceed a certain limit, as has been explained ; otherwise the air flows, but is not com- pressed. But all waves in the air do not affect our sense of hearing ; for this sense depends upon disturbances being conveyed to the nerve endings from the external air by a mechanism whose parts are set in motion by waves of certain wave lengths, and not by others. Thus, it is found that waves whose wave number is greater than 20,000 per second or less than 30 do not in general produce sounds ; but, of course, these limits are only approximate, and vary greatly with different individuals. In musical compositions as played by orchestras the maximum range of pitch is about from 40 to 4000. The Human Ear. — For a full description of the human ear reference should be made to some treatise on Physiology ; it is necessary to mention only a few details here. The ear consists of three parts : the external ear, which ends at the ear- .1 YALY818 OP SOtTJTD (Inini: the- middle car, which is connected with the throat and mouth I iy a tube, and in which there arc three little hones witli Hex il)lc connect ions, tli us making a mechanism joining the drum to a membrane which closes one opening into the third por- tion of ear; the inner ear, which is entirely inclosed in the hone of the skull, and which consists of several cavities tilled with a liquid. In one of these cavities there is a minute fibres of regularly decreasing length, with which the nerve endings of some of the auditory nerves are connected, and which are thought to play the part of reso- nators for musical notes. Other branches of the auditory nerve end in another cavity under conditions which have led several scientists to believe that their function was to respond to noises. In any case it is easy to trace a mechanical con- nection between the waves in the air and the nerve endings through the eardrum, the three bones, and the liquid in the inner ear. The student will find in the work of Helmholtz a full discussion of these various steps. CHAPTER XXIII MUSICAL INSTRUMENTS WE shall discuss only two types of musical instruments: stringed and wind instruments. The commonest stringed instruments are the piano, the violin, the violoncello, and the harp; and the commonest wind instruments are the organ, the flute, and the horn. Stringed Instruments. — For present purposes we may regard the vibrations of the strings in any stringed instru- ment as being identical with those of a perfectly flexible cord, although in reality musical strings are far from being per- fectly flexible, and their elasticity plays a part in addition to their tension. Only transverse vibrations are ever used. We saw on page 353 that the frequency of the fundamental 1 \T vibration of a cord was given by the relation N=—-\—'> 2 L * a where L is the length of the cord, T its tension, and d its mass per unit length; and that the parti als had frequencies 2 JV, 3 JV, etc. This formula explains how a string may be "tuned " by altering its tension ; how its frequency may be altered by shortening its length, as is done in violins, etc., by means of the fingers ; and why the different strings of any one instru- ment are made of different densities. When a string is struck at random or is plucked, the vibra- tion is complex ; but those components are absent which would have a node at the point struck or plucked ; thus the quality of the note depends largely upon where this point is, as is shown in the use of violins. The difference between pianos as made by different makers lies to a great extent in 400 MUSICAL 1. \STlirMK\TS 401 the point of the strings which is struck, in the size and hard- - of the fc' hummer," and in the duration of the blow; for these alU influence the quality ofrthe notes heard. All stringed instruments, with a few exceptions, have the strings stretched between pegs which are fastened to a wooden board or box. Owing to the vibrations of the strings, and the resulting motion of the pegs, this board is made to vibrate; and, since these vibrations are "forced," they imitate more or less closely in character those of the strings. But, of course, there are differences depending upon the thickness, area, stiffness, etc., of the boards. Simi- larly, if there is a box or cavity, the inclosed air may be set vibrating. The vibrations of this "sounding" board or box affect the surrounding air much more than do those of the tine string; and so the sound we hear depends to a great extent upon the former vibrations. We see, therefore, the reasons why the violins of certain makers have such great value, owing to their skill in the construction of the wooden parts. Wind Instruments. — We have given the theory of the vibrations of a enlnmn of air on pages 353-358, and have shown that in the case of a column open at both ends the frequency of the fundamental is N= — -, where V is the velocity of air waves, and L is approximately the length of the column: and that the partial vibrations have frequencies equal to 2 JV, 3 JV, etc.; whereas, a column which is closed 2 /. at one end — a "stopped" pipe — has for the frequency of its fundamental N= . and for those of its partials 8iV, 4 L ate. Wh'-n a column is set vibrating, as a rule both the fundamental and partials are present; but the funda- mental i> in general mm -h more intense than the others, and the partials decrease in intensity as their fr. .pinn ies i in- reuse. So, when an organ pipe is blown gently, only t he fundamental AMES'S PHYSICS — S6 402 SOUND • is lu-anl and the note is almost pure; this is specially true of 'a stopped pipe because the vibration whose frequency is "2 N is absent ; and for this reason these pipes are not generally used in orchestras. With but little practice, one may learn to blow a pipe with such pressure as to make the fundamental or a particular partial the most intense. We see from the formula that the frequency does not depend upon the material of the solid inclosing the column of air nor upon its cross section. (Of course the "correction" for the end by which L is increased by 0.57 R, as explained on page 354, is affected slightly.) If the length of the pipe is changed, the pitch of the corresponding sound is altered ; and so pipes of different lengths are used in organs, and by m
aking small changes a pipe may be " tuned" accurately. If the pipe is not of uniform cross section, but conical, there are marked differences produced — the position of the nodes and loops is affected ; as is shown by the difference between an organ note and one produced by a horn. The action of "stops" or "pistons" in horns is to vary the length of the vibrating column of air. If an opening is made in the side of the tube containing the column of gas, the vibrations must adjust themselves to this point being a loop ; and if two openings are made, their distance apart determines the length of a vibrating segment and therefore the frequency of the vibration. This explains the effect of making and closing openings in a tube as is done in flutes and similar instruments. Organ Pipes. — The column of air in a pipe or tube is set vibrating in several ways. In the ordinary organ pipe, a section of which is shown in the cut, there is a narrow pas- sage leading to the bellows or wind chest, through which a blast of air is directed against a sharp lip forming the upper edge of a narrow opening at the bottom of the pipe. This blast at the beginning of operations sends a disturbance up the tube, which is reflected at the upper end and returns. J/r>/'-.i/, INSTRUMENTS 403 When it reaches the bottom, its effect is to deflect the blast out of the opening in the tube. When this effect ceases, the' blast returns : and so there is an oscillation of the blast, which has a period determined by that of the stationary vibration of the column of air produced by the two trains of waves, the ••11 direct and the reflected. This bottom of the tube is approximately a loop, because it is open to the air. The vibration of the air would soon cease owing to loss of energy by friction and by the production of waves, 8 it not for the sup- ply furnished periodic- ally by the blast. The MUM method of produc- Fio. 181. —Section of an ordinary organ pipe. ing vibrations is used in flutes and whistles, the lungs or mouth In-ill^ the wind chest. Reed Pipe. —In another form of organ pipe, known as tin- reed pipe, the wind chest is connected directly wit h an elongated l)o\ ; and into this is inserted an inner tube, in which there is a rectangular opening closed by a strip of brass fastened at one end, called the "reed." When pressure in the wind chest is sufficient, this spring door is pushed open, a 1.1 r passes, and the pressure being ':! 1\ . the Nprin^ returns and. o\\ in^ to its inertia. continues to vibrate in its own natural period. In this manner a series of puffs «.i delivered at intervals, 404 SOUND determined by the frequency of the metal spring. There is always attached to the pipe a resonance tube of some kind, the air in which is set in vibration owing to the intermittent puffs. Without this box the sound is most complex, but with it the note becomes fairly simple. The instrument is " tuned " by altering the length of the reed by a clamp. Horns. — In the case of horns, the vibration of the column of air is produced by means of the vibrations of the lips of the player. The column in a horn of fixed length can vibrate in only a limited number of ways ; and the lips must be stretched to exactly the right degree so that, when they are set vibrating by air from the mouth being forced through them, their frequency is one of those to which the horn responds. If one is playing a horn of variable length, like a trombone, and a definite note is being produced, a change in the length changes the frequency of the vibra- tion, and therefore requires a change in the frequency of the lips ; but this change is produced almost auto- matically, owing to the reaction of the column of air itself. The Siren. — There is another acoustic instru- ment which, although not, strictly speaking, a musical one, should be described. It is called the " siren," because its action continues under water as well as above. In principle it is not unlike a reed pipe, inasmuch as it is designed PIG. 182. — Helinholtz's double siren. to deliver a number of M I > 1C A L INSTR UMENTS 4o;> pull's <>f air at iv^ular intervals, only with it this number can be varied at will and can be easily counted. As shown in ,the cut, there is a wind chest, which is closed on its upper side by a thick circular plate perforated with a definite number of holes, at regular intervals, around a circle concentric with the plate itself. These passages in the form of instrument pictured are not perpendicular to the plate, but are inclined slightly, so that the axis of the passage con- Fio. 192 a. -The sidered as a vector has a component parallel to tlu of the circle through the holes at that point. Immediately over tins fixed plate is a movable on* . \\hi< -h can rotate on MI. .m«l which is identically like the fixed plate, except that its passages slope the opposite way. Therefore, if the movable plate is in su.-h a position that its openings are over those in the fixed plate, the air rushing out from the wind < -hest will have a momentum against the sloping sides of the passages in the former plate, and will set n m 406 SOUND rotation on its .axis. Each time the openings in the two plates coincide, a puff of air escapes ; and if there are n openings in each plate, there will be n puffs during each rotation. (This would be true, also, if there were n holes in one plate only and but one in the other ; if there are n holes in each, the intensity of the puffs is increased.) So, if the rate of rotation is ra turns per second, the number of puffs in a second is mn ; and this is therefore the pitch of the resulting sound. The speed of rotation may be altered at will by regulating the pressure of the air in the wind chest. The number of openings in the plates is easily observed; and the number of revolutions in any interval of time is determined by using a mechanical counter, such as are seen on steam engines. (A screw thread is cut on the shaft of the rotating plate, so that a worm wheel is turned ; this drives a train of cogwheels, which moves a hand over an indexed dial, like the face of a watch.) In other forms of this instrument, the passages in the plates are not slant- ing, and the movable plate is made to rotate by means of a mechanical or electric motor. The simple siren as just described has been modified in two ways. One is to make in the plates several concentric rows of openings, which contain different numbers, e.g. 8, 10, 12; and thus, if all these are opened at one time, a com- plex sound is heard whose component simple vibrations have frequencies in the ratio 8 : 10 : 12. Again, two sirens con- nected with the same bellows may be arranged one over the other with their movable plates on the same shaft and facing each other; this enables one to produce two sounds whose pitches have a known ratio, that is, are at a known "inter- val" apart. This instrument was invented by the German physicist Seebeck, and was improved by Cagniard de la Tour and more recently by Helmholtz. It is not, however, as much used now as formerly. MUS1LAL IX^ilil'MENTS 407 Phonograph. — Another acoustic instrument is the phono ;>h, which consists essentially of a hardened wax cylinder ;nst whose surface pre»« •> a sharp point connected with a flexible membrane forming part of a mouthpiece. The cylinder is turned and advanced by clockwork; and, as >ounds are produced near the mouthpiece, the point makes •nding indentations in the wax, which are faithful reproductions of the displacements of the vibrating body. :i. if another point attached to a membrane is made' by mechanical means to pass over these traces on a cylinder at the same rate as that at which the cylinder was turned originally, the membrane will be set in vibration, and its motion will therefore be very nearly the same as that of the one in the mouthpiece that caused the trace. These vibrations will then produce waves in the air which will affect the ear; and so the original sound is reproduced. The Human Voice. — The human voice is due to the vibra- tion of the vocal cords of the larynx and of the various mov- able parts of the mouth. Consonant sounds such as ft, e, etc., are produced by vibration of the lips, the tongue, etc.; while vowel sounds owe their origin directly to the larvnx. This consists of two stretched membranes which have free edges along ;i nearly straight line. Thoe , an beset in vibration •lie air pressure in the Inn^s: and their frequency can be altered by voluntary changes in their tension. Owing to vibrations, which are, however, very "damped." the air in the cavities of the mouth and throat is also set vibrating. When a definite vowel sound, such as <i/t. is produced, no matter what the pitch of its most prominent component, it urn! by analysis that there are present one or more com ponents of definite pitch. These are not partials of the fun- damental vibration of the larynx reenforced by resona: but are independent vibrations; and the vowel character of '•mid depends upon them ; that is. different vowel sounds have different pi-rmain-nt component. 408 SOUND Reflection and Refraction as Phenomena of Sound. — Since sounds are due directly to waves in the air, all the proper- ties of wave motion may be studied and illustrated by sound I 2 3 Parabolic mirror ; centre of disturbance at its focus. Spherical mirror ; centre of disturbance at its focus. (Notice the effect at the edges of the wave-front.) FIG. 188. phenomena. Thus the reflection of waves is shown by echoes, by the use of concave mirrors, by sounding boards, etc. If a disturbance is produced near a curved wall, waves will spread out and will in part be reflected. Those waves fall- Ml' Sir A L 7.V.S 7-/,' fM/A'ATN 409 ing very obliquely upon the wall may by reflection meet the wall again, be reflected again, etc. Thus a certain amount of the energy of the waves will follow around the of A single pulse ft-o pulses. A noise ma BaftMdOH of A pal by • lens containing a heavier ga
t. Fio. 188 a. wall. This is the explanation of "whispering galleri Similarly, in speaking tubes" ;m«l " spr.ikin<_f trumpets" the energy of the waves is «>min, .1 within certain bounds, instead of spreading out in all directions. 410 80UM) An interesting case of irregular reflection is seen in the case of air. waves passing through a non-homogeneous atmos- phere occasioned by air currents of different density or by the presence of clouds. The waves may be reflected or scat- tered, as in the "rolling" of thunder; and many similar phenomena have often been observed with signals from fog horns, as shown by Henry and Tyndall. Sometimes the path of the waves is bent so as to rise in the air and then descend, causing regions or islands of silence. (The pres- ence of fog as such does not give rise to these phenomena; for it does not make the atmosphere non-homogeneous.) One illustration of refraction has been given on page 331, where it was shown how winds may change the direction of a plane wave front. This, however, is what may be called " mechanical" refraction. True refraction, similar to that observed in Optics, may, however, be produced. Tyndall made a lens out of a soap bubble filled with nitrous oxide gas, which had all the properties with air waves that an ordinary glass lens has for ether waves. A few photographs of pulses in the air, illustrating these and other properties of wave motion are added. They were obtained by Professor R. W. Wood, using a method devised by Toepler. These pulses are produced by the explosive action of an electric spark. The black knobs seen in the photographs are the centres of the disturbances ; and the wave fronts may be distinguished clearly. The Acoustic Properties of Halls. — Something should be said also in regard to the acoustic properties of halls that are used either for public speaking or for concerts. Great care must be exercised to avoid what is called " reverbera- tion." This, if excessive, is a great objection. It is due, of course, to the echoing and reechoing of the sounds, occa- sioned by the repeated reflection of the waves from the floor, walls, ceiling, seats, auditors, etc. The problem of inves- tigating the exact conditions that determine or prevent 411 reverberation was first undertaken by 1'mfessor Sabine of Harvard I niversity in tin- year 1895. His met hod ,,f stndv- ing it was to arrange an organ pipe at one point of a hall so that it could be blown and then stopped at any instant; sta- tioning himself in turn at different positions in the hall, he would note how many seconds he could hear a sound after the organ pipe had ceased acting. It was found that the rU ration was the same practically at all points in the hall and that it was independent of the position of the pipe. For halls of the same volume the reverberation is the same; but, as the size of tin; hall increases, the reverberation in- creases also, other things remaining unchanged, ll is de- creased greatly by putting soft coverings on the floors, walls or seats, by making the walls less rigid, and by the presence of an audience. It was found that for practical purposes the reverberation should not be decreased below 2.3 second >. otherwise the music was not fully appreciated by the audi- ence. Professor Sabine was able to deduce a general formula which can be used to predict with great exactness the dura- tion of reverberation in a hall when its dimensions and the materials used in its construction are known. The acoustic property of a hall depends upon other things than reverberation, for its shape may be such as to focus the iraYee at {.articular points, etc. "Sounding boards" which reflect the waves down upon the audience are often used. Another effect whirh must be taken into account is that due to ascending and descending currents of air: for wherever there are changes in the homogeneou of tho air, there are reflections of the waves. CHAPTER XXIV MUSICAL COMPOSITIONS Combinational Notes. — When two instruments are sound- ing at the same instant, there are several interesting phe- nomena besides the production of the two sounds. If the two instruments are setting in vibration directly the same portion of air, as when a double siren is used, or if two wind instruments are blown by the same wind chest, other sounds are heard than the two corresponding to the instruments. Thus, if n^ and n% are the frequencies of the two vibrations, other vibrations of frequencies, n^ -f- n2 and n^ — nv are pro- duced in the air. (The mathematical theory of this was given by Helmholtz.) The corresponding sounds are called "com- binational," or " summational " and " differential " notes. Beats. — If the two vibrations have frequencies which are quite close together, a different phenomenon is observed. Thus, suppose the frequencies are n and n + m, where m is a small number — not necessarily an integer. Then, when the instruments are sounded at the same time, the loudness of the sound heard fluctuates; it rises and falls at regular in- tervals. If m = 4, these intervals are a quarter of a second ; or, in general, this is — th of a second. There are then said m to be 4 "beats" (or, in general, m "beats") per second. This " beating " is due obviously to the fact that as the two trains of waves traverse the same medium before they reach the ear, there will be points at regular intervals apart where the compression of one train will neutralize the expansion of the other, and, at points halfway between these, the com- 412 MUSICAL COMPOSITIONS 413 pressions of the two will coincide. Thus, if the waves have a velocity V, and if the difference in the wave numbers is an integer p, in a distance V in the direction of propagation of the waves, there are p points where the disturbances are almost, if not quite, neutralized and p points where the dis- turbance is abnormally great. This is shown in the figure, v/ \v/ \XAAx~ Fio. 184. — Diagram showing origin of beats. The two vibrations have frequencies whose ratio is 7 : 8. where n:n+p = 1 :8. Therefore, in the course of one sec- ond, as the waves enter the ear, it will happen p times that the sound almost vanishes and p times that it is abnormally loud. Thus, if we count the number of beats in a second, we can tell exactly the difference in the frequencies of the two vil (rations. (If m is not a whole number, a distance greater than V must be chosen in the above treatment. If 7» = 4.5, a distance 2 Fmay be chosen, in which there are then 9 points where the disturbances neutralize each other, etc.) If the vibrations are complex, beats may occur owing to the prox- imity of the frequencies of any two of the component vibra- tions, or to that of the frequencies of any of these and those of the combinational ones. (The explanation of beats was first given by Sauveur, about 1700.) It is recognized by every one thai beats produce a disagree- able sensation, and for the same reason that a tickling feather or a rapidly flashing li^ht do: namely, ouiiiLT t«» fatigue of the nerve* \Vhen the b.-.us heroine so rapid that they cannot be individually re.-,.'_nii/ed, they cause a "roughness" in the sound whieh is unpleasant : hut the degree of di-.i- greeableness depends up<,n both the pitch «.f the sound and the number ,,f. befttfl p--r second. Tins matter was fully investigated by Ilelmholt/.. 414 SOUND Harmony or Consonance. — It has been known to men of all nations since the earliest ages that there are certain com- binations of vibrating bodies which produce sounds pleasant to the ear, such as that heard when two or three stretched strings of definite lengths are vibrated at the same time. This fact has nothing to do with the state of civilization or of musical cultivation ; it is a property of the human ear. It was recognized by the Greeks before the days of Aristotle (probably as long ago as 500 B.C.) that, if two stretched strings of the same size and material and under the same tension, but of lengths in the ratio 1 : 2, were sounded together, the sound heard was agreeable ; and also that, if there were three similar strings whose lengths were in the ratios 4:5:6, the same was true. Mersenne, in 1636, showed that the frequencies of the vibrations of stretched strings varied, other things being equal, inversely as the lengths of the strings. So the problem of explaining the consonance of the sounds produced by the two or three strings as just described became this : why should two notes produced simultaneously by strings making vibrations of frequencies n and 2n, or three notes produced simultane- ously by strings making vibrations of frequencies 4 n, 5 n, 6 n, cause a pleasant sensation ? Helmholtz's Explanation of Consonance and Discord. — The answer to these questions was first given by Helmlioltz. He showed that in every case of consonance when two or more notes are produced simultaneously, that is, a " chord " con- sisting of two or more notes, beats were nearly, if not en- tirely, absent ; and that in any other case, when two or more notes were sounded together, the degree of the discord could be predicted from calculations of the number of beats present and from a knowledge of the degree of their disagreeable- ness to the ear. Thus, when a note of pitch n is produced by a vibrating string, notes of pitch 2w, 3 ft, etc., may be heard in the complex sound ; and, when a note of pitch 2 n COJCPO&lTfO 415 is produced, others of pitch I //. b'n, etc., may be heard ; fur- ther, tin- combinational notes all have pitches n, 2n, etc.; so t lie re are no beats, and the two complex notes of pitches n ami 'In are therefore in harmony when produced by wind or stringed instruments. Hut, if the lower note had the pitch n + 5, its partials would have the pitches 2 n + 10, 3 n + 15, etc., and these would beat with the other note of pitch 2n and with its partials. Therefore, it t\vo complex notes of pitch n + 5 and 2n are produced, they are discordant. It is evident that the same explanati
on applies to the har- mony of the chord consisting of the three notes of pitch 4n, 5n, and 6n. In general, two or more notes are consonant, or approximate to it. it' their pitches bear simple numerical • uch other, such as 1 : 2, 1 : 3, 2 : 3, 1 : \. - If these relations can he expressed, however, only in terms of large numbers, the notes are discordant. On this fact is based the construction of all musical "scales, that is, series of notes of different pitch, which are played in chords in musical compositions. In all music, however, another <\|iiestion enters, namely, are two or n tes or chords played in rapid succession pleasant to the ear or n<>t / If they are, they are said to form a -melody." These questions belong to the Theory of Music and cannot he discussed here; but, in general, if :iant. they will also form a melod\. Musical Scales. — Many musical scales have been devised and HN.-.I ; hut only two will be discussed here. All modem mii-i.-.il compositions are based upon the idea of sele< ; •• note as a " keynote," and using this as the starting point «.t" the scale. I A i the pitch <>f the keynote, a note ;tch 'J X is . Billed its "OOtftYQ ": and the inl«T\;d between tli. -in is called "an octave." I "interval" l>etween notes is d< -lined to be equal to tin- ratio of their pit. In the • ,y scale, a definite numi- -'lave, whose,' im- ot diff. 416 SOUND widely, and which when played in chords do not produce too discordant sounds. Let their pitches be JV, 0, P, (?,•••, 2 N. Then, in the octave above this, that is, in the interval from '2Nto 4 N, the notes selected for the scale have the pitches '2N, 2 0, 2 P, 2 #,..., 4 JV; those in the octave above this, 4^, 4 0, 4P, 4 #, -, 8N; etc. The notes selected for the octave below the original one have the pitches J JV, JO, J P, J (?, •••, JVi etc. So, in defining a scale, it is necessary to choose only the pitches of the keynote and the notes in its octave. The Diatonic Scale. — The " diatonic " or " natural " scale consists of a series of notes whose pitches may be expressed as follows: If 24 n is the pitch of the keynote, the notes in its oc- tave have the pitches 24 n, 27 n, 30 n, 32 n, 36 n, 40 n, 45 n, 48 n. Thus it is seen that in the interval of an octave there are seven notes, counting only one of the end notes of the octave. If an instrument with strings of fixed lengths, like a piano, is to be constructed to play music written on this scale, some definite keynote must be chosen, and a string must be selected of such a length and size that under suitable tension it will give this note ; then other strings must be selected to pro- duce the other notes of the scale. But suppose a musical composer did not wish to use the same keynote for all his pieces ; suppose, for instance, that he wished to have as the key tone one whose pitch is 20 n. The diatonic scale of this is 20 n, gj x 20 n, fj x 20 n, fj x 20 n, Jf x 20 n, ..., 40 n, or, 20 n, 22Jn, 25 n, 26 § n, 30 n, •••, 40 n; and it is seen that, if the instrument described above is pro- vided with strings giving notes in the diatonic scale having 24 n as the keynote, many of the notes required for this composition cannot be produced ; for instance, 25 w, 26£ w, etc. For this reason, and also to make the intervals between two consecutive notes more nearly equal, the diatonic scale was altered by the introduction of five new notes in each octave : between 24 n and 27 n, 27 n and 30 n, 32 n and 36 n, MUSICAL COMPOSIT10 117 36 n and 40//. and 40 H and l~> n : and corresponding st i were introduced. But still the scale was unsatisfactory when musical compositions were written in different keys ; and so a final change was made, which solved this difficulty. hut introduced another less important one. The obvious advantage of the diatonic scale is that chords played on it are as nearly in harmony as is possible for a scale having seven notes in an octave, since the ratios defin- ing the scale are as simple as possible, vi/. - 1 : oil = 2: 3; 30:40 = 3:4; etc. But slight variations in the pitch of a note may occur in a chord without noticeable discord; and, in any case, people grow accustomed to a chord or to a piece of music and are not sensitive to its perfect harmony. The Equally Tempered Scale. — This fact was taken advan- tage of in the construction of the new scale. It is called the "equally tempered scale," because the intervals between two consecutive notes are the same throughout the scale. Twelve notes are introduced in an octave; and calling tlie pitch of the keynote n and the constant interval <i. these notes and the octave have the pitches : //. .in% a4n, •••, a^n. I Jut since d^n is the octave of n, a12 = 2 and a =^2, or 1.0595. The notes in the octave above and below this are formed as before by multiplying and dividing by 2; etc. It is evident, then, that if a piano or organ is made with strings or pipes corresponding to. the notes of this scale using any definite keynote, musical completion written in any key whose note is any where in the scale iua\ d on it. Standard Pitch. — The keynote adopted for musical scales • Stuttgart Congress in 1834 and later by the "Society \ : ts " in Kn^Iand is one whose pitch is 2<U. In Midland. however, at the present time "concert pitch " is based upon a keynote whose pitch is 273. I i scientific purposes, tuning forks are used as standards of frequency or pitch; and they are generally made in sets following the diatonic scale. The most accurate forks in general use are those AMES'S MI THICK — 27 418 SOUND made by the late Rudolph Koenig of Paris ; and he adopted as the fundamental frequency, or key tone, 256. As will be explained presently, suitable names and sym- bols have been given the notes in the various octaves of dif- ferent scales. For instance, " Ut3" is sometimes used as the name of the key tone above described. So it is seen that the pitch corresponding to a certain name or symbol in musical notation is not definite. Thus the note called Ut4 had a pitch less than 500 in the early part of the eighteenth cen- tury; in the days of Handel (1750) the note which had this same name and symbol in musical notation was one whose pitch was between 500 and 512 ; and in our days this same symbol is given a note whose frequency varies from 512 to 546, as we have seen above, for Ut4 = 2 Ut3. These facts are expressed by saying that there is a tendency as years go by for the pitch corresponding to a given symbol to rise. The octave of the equally tempered scale starting" from the keynote 264 is made up, then, of the notes whose pitches are as follows: 264, 279.6, 296.3, 314.0, 332.6, 352.4, 373.3, 395.5, 419.0, 443.9, 470.3, 498.3, 528. If this note is used as the keynote of the diatonic or natural scale, the notes in the corresponding octave have the following pitches : 264, 297, 330, 352, 396, 440, 495, 528. It is thus seen how far apart the scales are at certain points. Violins, violoncellos, etc., do not have strings of invariable lengths, because the fingers of the player can alter them at will by " stopping " at any point ; and so one can play with them on the natural scale music written in any key, if the five additional notes are introduced, as previously explained. Musical Notation. — The seven notes in the octave of the diatonic scale are called <?, d, e, f, g,a,b\ and different octaves are assigned different types or marks. Thus, the octave from 132 to 264 is written as above ; the one from 264 to 528, <?', d', etc.; the next one c", d", etc.; the one from 66 to 132, (7, D, etc. ; that from 33 to 66, Cv Dv etc. J/r>/< .\L ' Using the tempered scale and :M I as the keynote, the notes nearest these to whirh names ha\c been given are railed by the Corresponding names; thus 'J'>1 ifl oalln .:!, d ; 332.6, e ; etr. Hie note between c and d, i.e. 279.6, iharp" oru<f flat"; that between <i and «,d •harp or e flat : that between/ and g, f sharp or g flat; that be- tween y and a, g sharp or a flat; and that between a and 6, a sharp or 6 flat. <i sharp is written a$; a flat, at?; Strictly speaking, if a note is "sharpened," its pitch is in- creased in the ratio 25: 24 ; thus, if the note has a pitch 264, its "sharp" i- _7.">. Similarly, if it is flattened, its pitch is decreased in the ratio 24:25; thus the "flat" of 2(54 is \\. These in some cases differ appreciably from tin- notes which have received the names in the tempered s« Another system of notation is to call the notes making up an octave on the diatonic scale Ut, KY. Mi, Fa, Sol, La, Si, starting from the keynote, or a note which is a number of octaves above or below it ; and to distinguish the diflVrent octaves 1>\ >u!»eri\|.ts. Thus, adopting J»',J as the key tone, the octave from :'>3 to 66 is written Utj, Rer etc.; that from 66 to 132, Ut,,. I : etc. (In place of Ut, Do is now more often used. - BOOKS OF l:i i i \M- THOMSON. S.MII.I. I. <.iid, ,M. i is the best modem text-book and is a storehouse of facto. \i l . On .S..MII.I. \.-\v Y,,,-k. 182. \Mt li all the phenomena of S- ••resting description of numerous ezperimeuts dealing HMMH..II/. Sensations of Tone. (Translation by Ellw.) London. 1 966, IH contains a description of all the experiment* on \\ln.-h Hrlmholtx established his theory of harmony, and also a complete explanation of musical lustrum* Mt- ;iti<l scales. RAY i I vols. London. 1886. i * is a mathematical tren • 1 1 contains • \ i > < 1 1 > experimental knowledge of vibrations, waves, and musical notes. LIGHT CHAPTER XXV GENERAL PHENOMENA OF LIGHT To any one who has the sense of sight, the word " light " conveys a definite meaning, which, however, cannot be put into words ; but to a person who was born blind the word is unintelligible. The attention of all who have this sense of sight is attracted to many phenomena in nature, such as the colors of objects, the action of mirrors, the refraction produced by water, etc. ; and the study of these forms that branch of Physics called "Light.
" It will be seen, as we go on, that we can subdivide this subject in certain definite ways. Fundamental Facts Light is Due to Ether Waves. — The statement has been made before several times that light is a sensation due to waves in a medium called the ether ; but a brief summary of the facts on which this is based may be given again. Thomas Young showed as early as 1802 that interference phenomena, such as described in Chapter XXI, could be observed with light ; Fresnel soon after performed numerous diffraction ex- periments ; and Wiener and others have obtained evidences of "stationary waves." The experiments of Young and Fresnel may be repeated easily by any student. Thus it is established that light is a wave phenomenon. Again, Fresnel showed conclusively that these waves are transverse, because they admit of polarization (see page 313). The existence of a 420 GE.\Ki;.lL I'l/l-:\n.MENA OF LIGHT 1'Jl medium is proved by the fact of tin- wave motion; and, since we can see objects through spaces which are void of ordinary matter and through glass, water, etc., it is shown that this medium is one which fills all space known to us, even inside ordinary material bodies (see page 19). We often use the expressions " light passes " or travels, etc., meaning that the ether waves which produce light in our eyes pass or travel ; similarly, we speak of "red light," etc., meaning those ether waves which produce in our eyes the sensation that we call "red." We also speak of "waves in air" or in water, etc., meaning that the waves are in the ether, but that this medium is modified by the presence of air or water. "Velocity of Light."— The fact that light travels with a measurable velocity was first shown by Roemer, a Danish astronomer, in 1676, from observations on the satellites of Jupiter; mid in later years experimental methods have been <levise«l t<> measure this <juantity accurately. These will be discussed in a later chapter. The value of this velocity in the pure ether is not far from 3x 1(>10 cm. per second, or about 186,600 mi. per second. It was shown 1>\ Foucault by direct experiment that the " velocity in water " is less than in air, • ing that the presence of minute particles oi water influ- ence the ether more than does that of similar "particle* of air." Sources of Light. — As "sources of light " \\e make use in general of the sun, or of the sky; of electric discharges through rarefied gases; of solid bodies raised to a high tem- perature, such as the carbon rods in an electric "arc li^ht." or the filament inside an ordinary electric "glow lamp, gas-, lamp-, and candle-flames, for. in fact, the luminosity of any flame is due to the pretaDM in it of minute solid par- ticles which are raised to a hiurh temperature l»y the buHtion of the gas, etc. These are all, in general, sources vhat we call "white" light. \V owever, light of different colors, — red, green, yellow, etc., — by •undiiiL: ilf ordinary source by a colored screen. Mich \-2'2 LIGHT as a piece of colored glass; or we can in certain cases put salts in a flame and so color it — thus we can secure a brill- iant yellow light by putting common salt (NaCl) in a Bunsen flame, thus making what is called a "sodium flame." Types of Waves. — If the source of light is very small, we have approximately spherical waves; while, if they come from a distant source like a star, they are plane. We can have two kinds of spherical waves, those which are expand- ing away from a source, and those which are contracting toward a point. It is a familiar fact that an ordinary reading lens, or magnifier, may produce an image of the sun upon some opaque screen — thus acting as a " burn- ing glass " ; this means that the plane waves reaching the lens from any point of the sun are changed 011 passing through the lens into spherical waves which converge to a point on the screen. This process of converging waves is exactly the reverse of that of waves diverging from a point source. Thus, if a spherical wave front is concave when considered from portions in the medium toward which it is advancing, it will -converge to a point ; if it is convex toward that side, it will diverge more and more. Fio. 185. — Dinprram illustrating stellar scintillation. This fact is illustrated in the familiar phenomenon of stellar scintillation. The waves coming from a star are naturally plane ; but if the atmosphere is disturbed by ascending currents of hot air, the wave front is no longer plane, owing to the fact that the velocity of light in cold air is different QKNSRAL PHSNOUBNA o/-' /./<,/// from that in hot. Thus the wave front at any instant may have a "cor- rugated" form, a> indicated in the cut. Therefore the light will be concentrated in certain points Av A* etc., the centres of the concave portions of the wave front ; and, as the heated portions of tin* air change their j these points move; so if the ry«- is at a ]><.iiit.-lj atone instant. th«> n<*xt it may be between two points, .1, ami . lo; etc. So the iiitrn^ity of the light will increase and decrease intermittently. The same phenomenon is observed in the "shadow bands " seen at times of total eclipses of the sun. Homogeneous Light and White Light. — In order to deter- mine tin- wave lengths of these "light waves," it is simply necessary to use the interference method described on page 375. It is found that, if the source of light is white, the interference fringes or bands are all colored, with the excep- tion of the central one, which is white. If, however, the source is colored, the bands are alternately black and colored ; that is, in certain lines disturbances in the ether are entirely absent. In general, it will be observed that there are apparently two or more sets of fringes superimposed, each set having a definite color. It is possible, however, to secure such a colored source, that the bands are all of one color, separated by the dark fringes. (This is approximately the case with a sodium flame.) Under these conditions the light is said to be "pure," or "homogeneous." When the sourer is white, or when any ordinary colored source is used, we can analyze the complex interference pattern into series of fringes, each series having its own color and its own spacing. It was shown on page 378 that, if the waves have a definite wave length, the fringes are evenly spaced at a distance t which is proportional to the wave length. Therefore, when a source is hom- . it is emitting waves of a •'.«• definite wave Imgih: and, in general, an ordinary source of light is emitting t rains of waves of different wave length-, 'fins, th«- interference apparatus "disperses" t lie complex from the MHirO6 into simpler trains, ,-aeh ing a deiimi^ \\.i\r Length, 424 LIGHT Connection between Wave Length and Color. — We can measure the wave length of the waves emitted by any homogeneous source, by using the formula deduced on page 378, viz., distance apart of fringes =— , where I is the wave 6 length, a is the distance from the two sources to the screen, and b is the distance apart of the sources. In this way it is found that pure red light has a wave length greater than that of pure green ; and this in turn is greater than that of pure blue. The longest waves that affect our sense of sight produce the sensation of red and have a length of about 0.000077 cm. (i.e. 770/xft); while the shortest produce the sensation of violet and have a length of about 0.000039 cm. (i.e. 390 pp'). In between these limits there are all possible wave lengths, corresponding to which are all shades of color, ranging from the deepest red, through orange, yellow, green, blue, to the darkest violet. Waves shorter than 390 /i/i can be observed by photography ; and they have been obtained by Schumann as short as 100 /u/x. Waves longer than 770 pp may be studied, as described on page 292, by various means; and they have been observed as long as 25,000 pp, i.e. 0.025 cm. (Electrical oscillations produce waves in the ether, which are as a rule very long ; but, using minute conductors, waves as short as 0.6 cm. have been obtained. There is thus a gap between 0.6 cm. and 0.025 cm. which has not yet been investigated.) The fundamental facts are, then, that light is due to transverse waves in the ether ; that waves of different wave length produce different colors; that white light is, in general, due to a mixture of waves of all wave lengths which affect our sense of sight ; that the velocity of these waves is less in ether inclosed in matter than in the pure ether. (It will be shown on page 433 that, in the former case, waves of different wave length have different velocities, which is not the case in the pure ether.) mi: i OF LIGHT 425 General Properties of Light as Due to Wave Motion. — \\V described in Chapters XX and XXI certain general properties of wave motion which apply directly to light; viz., reflection, refraction, rectilinear propagation, interfer- ence, and diffraction. Each of these will be discussed more in detail later; but one or two points should be referred to here. (It should be remembered specially that we are now considering extremely short waves, viz., those whose wave length is not far from 500 pp.) Rays, Shadows. — It was shown that, if the waves are short, the disturbance at any point in an isotropic medium due to a train of waves depends directly upon the dis- turbances at previous instants along a straight line drawn backwards perpendicular to the wave front; this line is called a "ray." If the rays are all parallel, we have plane waves, or a " beam " of light. If the waves are spherical, tin- rays are radii ; and, by considering only those rays which are close together, we have a "cone" or a "pencil" of light. (Although this theorem in regard to rays was proved only for a train of waves, it may be shown to hold true for a -pulse" also.) It the waves meet an opaque obstacle, that is. on.- \\ hich does not transmit li^ht, a shadow is cast. If the source of lig

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