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chapter we 0 be able to compute not only the motion of the oscillating mass,
but also the perturbations on the planet Ủranus produced by Jupiter and Saturnl
Galileo made a great advance in the understanding of motion when he dis-
coverecd the prznciple oƒ tmnertia: 1f an objJect is left alone, is not disturbed, 1$
continues to move with a constant velocity in a straight line if it was originally
moving, or it continues to stand still iŸ it was just standing still. Of course this
never appears to be the case in nature, for if we slide a block across a table it
stops, but that is because it is no left to itself—it is rubbing against the table.
Tt required a certain imagination to ñnd the right rule, and that imagination was
supplied by Galileo.
Of course, the next thing which is needed is a rule for fnding how an object
changes 1s speed iŸ something ?s afecting it. Thhat is, the contribution oŸ Ñewton.
Newton wrote down three laws: The First Law was a mere restatement of the
Galilean principle of inertia just described. The 5econd baw gave a specifc way
of determining how the velocity changes under diferent infuences called ƒorces.
The Third Law describes the forces to some extent, and we shall discuss that
at another time. Here we shall discuss only the Second Law, which asserts that
the motion of an object is changed by forces in this way: the time-rate-oJ-change
--- Trang 178 ---
oƒ a quantitụ called momentum is proportional to the [orce. We shall state thìs
mmathematically shortly, but let us first explain the idea.
Momentum is not the same as 0elocit. A lot of words are used in physics,
and they all have precise meanings in physics, although they may not have such
precise meanings in everyday language. Momentum is an example, and we must
defne it precisely. IÝ we exert a certain push with our arms on an object that
1s light, it moves easily; if we push Just as hard on another object that is much
heavier in the usual sense, then it moves much less rapidly. Actually, we must
change the words from “light” and “heavy” to Ïess rnass?ue and more 1ndssiue,
because there is a diference to be understood between the 0øe¿ghf of an object
and its ?nerta. (How hard ïE is to get it goïng is one thing, and how much it
weighs is something else.) Weight and inertia are proportional, and on the earth”s
surface are often taken to be numerically equal, which causes a certain confusion
to the student. Ôn Mars, weights would be diferent but the amount of force
needed to overcome inertia would be the same.
We use the term rmøss as a quantitative measure of inertia, and we may
mmeasure mass, for example, by swinging an object in a circle at a certain speed
and measuring how much force we need to keep it in the circle. In this way we
ñnd a certain quantity of mass for every object. Now the mmơmentum of an object
1s a product of bwo parts: Its zmass and its 0elocit. Thus Newton”s Second Law
may be written mathematically this way:
t= qi0n9). (9.1)
Now there are several points to be considered. In writing down any law such as
this, we use many intuitive ideas, Implications, and assumptions which are at
first combined approximately into our “law.” Later we may have to come back
and study ïn greater detail exactly what each term means, but if we try to do this
too soon we shall get confused. Thhus at the beginning we take several things Íor
granted. First, that the mass of an object is consfand; it isn't really, but we shall
start out with the NÑewtonian approximation that mass is constant, the same all
the time, and that, further, when we put two objects together, their masses ødd.
These ideas were of course Implied by NÑewton when he wrote his equation, for
otherwise 1t is meaningless. For example, suppose the mass varied inversely as
the velocity; then the momentum would ne0er chơnge in any circumstanece, so
the law means nothing unless you know how the mass changes with velocity. At
first we say, ? does not chơœngc.
--- Trang 179 ---
Then there are some implications concerning force. Âs a rough approximation
we think of force as a kind of push or pull that we make with our museles, but
we can defñne it more accurately now that we have this law of motion. The most
Iimportant thing to realize is that this relationship involves not only changes in
the rmagnitude of the momentum or of the velocity but also in their đứccfion. TẾ
the mass is constant, then Eq. (9.1) can also be written as
t'=m - = ma. (9.2)
The acceleration ø is the rate of change of the velocity, and Newton°s Second
Law says more than that the efect of a given force varies inversely as the mass;
1t says also that the đirection of the change in the velocity and the đireclion of
the force are the same. Thus we must understand that a change in a velocity, or
an acceleration, has a wider meaning than in common language: The velocity
of a moving object can change by its speeding up, slowing down (when it sÌows
down, we say it accelerates with a negative acceleration), or changing its direction
of motion. An acceleration at right angles to the velocity was discussed in
Chapter 7. There we saw that an object moving in a circle of radius with a
certain speed œ along the cirele falls away from a straightline path by a distance
equal to 2(02/R)£2 if £ is very small. Thus the formula for acceleration at right
angles to the motion is
a = 02/R, (9.3)
and a force at right angles to the velocity will cause an objecE to move in a curved
path whose radius of curvature can be found by dividing the force by the mass
to get the acceleration, and then using (9.3).
9-2 Speed and velocity
In order to make our language more precise, we shall make one further
defnition in our use of the words speed and 0elocit. Ordinarily we think of speed
and velocity as being the same, and in ordinary language they are the same. But
in physics we have taken advantage of the fact that there øre two words and have
chosen to use them to distinguish two ideas. We carefully distinguish velocity,
which has both magnitude and direction, from speed, which we choose to mean
the magnitude oŸ the velocity, but which does not include the direction. We can
formulate this more precisely by describing how the z-, -, and z-coordinates
--- Trang 180 ---
F4
DR — TT .
TƯ INNG
“~—-- |
I “ˆ.
SA AV l/
/Ax___W
Fig. 9-1. A small displacement of an object.
of an object change with time. Suppose, for example, that at a certain instant
an object is moving as shown in Fig. 9-1. In a given small interval of time Af
it will move a certain distance Az in the zø-direction, A2 in the -direction,
and Az ïn the z-direction. The total efect of these three coordinate changes is a