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of gravitation and this principle, but no other details.
'This principle is that acfon eguals reaction.
--- Trang 197 ---
'What is meant is something of this kind: Suppose we have ©wo small bodies,
say particles, and suppose that the first one exerts a force on the second one,
pushing it with a certain force. 'Then, simultaneously, according to Newton's
Thind Law, the second particle will push on the frst with an equal force, In
the opposite direction; furthermore, these forces efectively act in the same line.
This is the hypothesis, or law, that Newton proposed, and it seems to be quite
accurate, though not exact (we shall discuss the errors later). For the moment
we shall take it to be true that action equals reaction. Of course, If there is a
third particle, not on the same line as the other ©wo, the law does no mean that
the total force on the first one is equal to the total force on the second, since
the third particle, for instance, exerts its own push on each of the other two.
'The result is that the total efect on the first bwo is in some other direction, and
the forces on the first two particles are, in general, neither equal nor opposite.
However, the forces on each particle can be resolved into parts, there beïing one
contribution or part due to each other interacting particle. Then each pœ¿r of
particles has corresponding components of mutual interaction that are equal in
magnitude and opposite in direction.
10-2 Conservation of momentum
Now what are the interesting consequences of the above relationship? Suppose,
for simplicity, that we have just two interacting particles, possibly of diferent
mass, and numbered 1 and 2. “The forces between them are equal and opposite;
what are the consequences? According to Newton's Second Law, force is the
time rate of change of the momentum, so we conclude that the rate of change of
mmomentum ?Ø¡ of particle 1 is equal to minus the rate of change of momentum Øøs
of particle 2, or
Now lf the raf#e oƒ chønge is always equal and opposite, it follows that the £otal
chơngec In the momentum of particle 1 is equal and opposite to the #o‡øÏ change
in the momentum of particle 2; this means that if we add the momentum of
particle 1 to the momentum of particle 2, the rate of change of the sum of these,
due to the mutual forces (called internal forces) bebween particles, is zero; that is
đứm + p›)/dt = 0. (10.2)
There is assumed to be no other force in the problem. lỶ the rate of change of this
sum is always zero, that is Just another way of saying that the quantity (0 + Øa)
--- Trang 198 ---
does not change. (This quantity is also writben ?n10ị + mạøa, and is called the
total mormentum of the two particles.) We have now obtained the result that the
total momentum of the two particles does not change because of any mutual
interactions between them. This statement expresses the law of conservation of
mmomentum in that particular example. We conclude that if there is any kind
of force, no matter how complicated, between two particles, and we measure or
calculate m0 + ma0a, that is, the sum of the two momenta, both before and
after the forces act, the results should be equal, I.e., the total momentum is a
constant.
T we extend the argument to three or more interacting particles in more
complicated circumstances, it is evident that so far as internal forces are concerned,
the total momentum of all the particles stays constant, sỉince an increase in
mmomentum of one, due to another, is exactly compensated by the decrease of
the second, due to the first. That ¡s, all the internal forces will balance out, and
therefore cannot change the total momentum of the particles. Then If there are
no forces rom the outside (external forces), there are no forces that can change
the total momentum; hence the total momentum is a constant.
lt is worth describing what happens if there are forces that do nø£# come from
the mutual actions of the particles in question: suppose we isolate the interacting
particles. If there are only mutual forces, then, as before, the total momentum
of the particles does not change, no matter how complicated the forces. Ôn the
other hand, suppose there are also forces coming from the particles outside the
isolated group. Any force exerted by outside bodies on inside bodies, we call an
czternal force. We shall later demonstrate that the sum of all external forces
equals the rate of change of the total momentum of all the particles inside, a
very useful theorem.
'The conservation of the total momentum of a number of interacting particles
can be expressed as
THỊĐ1 + ThaUa + Tn303 + - - : = a constant, (10.3)
1f there are no net external forces. Here the masses and corresponding velocities
of the particles are numbered 1, 2, 3, 4,... The general statement of Ñewton”s
Second Law for each particle,
t—= qiữn9); (10.4)
is true specifically for the cømponen‡s of force and momentum in any given
--- Trang 199 ---
direction; thus the zø-component of the force on a particle is equal to the z-
component of the rate of change of momentum of that particle, or
= qi_n9z), (10.5)
and similarly for the - and z-directions. Therefore Eq. (10.3) is really three
equations, one for each direction.
In addition to the law of conservation of momentum, there is another interesf-
ing consequence of NÑewton”s Second Law, to be proved later, but merely stated
now. 'Phis principle is that the laws of physics will look the same whether we are
standing still or moving with a uniform speed ïn a straight line. For example,
a child bouncing a ball in an airplane ñnds that the ball bounces the same as
though he were bouncing it on the ground. Even though the airplane is moving
with a very high velocity, unless it changes its velocity, the laws look the same
to the child as they do when the airplane is standing still. This is the so-called
rclatiuilụ priứnciple. As we use it here we shall call it “Galilean relativity” to
distinguish it rom the more careful analysis made by Binstein, which we shall
study later.
W© have just derived the law of conservation of momentum from Newton”s
laws, and we could go on from here to find the special laws that describe impacts
and collisions. But for the sake of variety, and also as an illustration of a kind of
reasoning that can be used in physics in other cireumstances where, for example,
one might not know Newton”s laws and might take a different approach, we shall
discuss the laws of impacts and collisions from a completely diferent point of
view. WWe shall base our discussion on the principle of Galilean relativity, stated
above, and shall end up with the law of conservation of momentum.
We shall start by assuming that nature would look the same if we run along at
a certain speed and watch it as it would iƒ we were standing still. Before discussing
collisions in which bwo bodies collide and stick together, or come together and
bounce apart, we shall ñrst consider 6wo bodies that are held together by a spring