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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
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
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, it 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
LMir-'. 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 iOOOrd 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
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 '
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- lustrations 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
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
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 illustrated 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
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,
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,
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
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 (
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
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 around 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
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.
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 system 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
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 2*2 - 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 finally 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
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 F
v 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
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
, 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 assuming 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 — \| Trr*D, 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?////
onism 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- ing 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 numer
i- 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
.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 volume 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
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
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
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 fo
i.-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 any 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 called, 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
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- piez
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
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 case 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*.
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(
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,
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. Co
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
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
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 black
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 its 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\
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
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 another, 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
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

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