NEWTON’S LAWS CONTINUED
Newton’s Third Law – Action/Reaction
Surely, one of the greatest and most important insights
in classical physics is reflected in Newton’s Third Law,
which effectively states: whenever one body exerts a force
on a second body, the second body exerts a force of equal
magnitude and opposite direction on the first body,
or alternatively,
forces in nature always occur in pairs which are equal in
magnitude, opposite in direction, and act on different
bodies.
Students often seem to have difficulty grasping this
concept, and determining the “reaction force” associated
with any “action force”. But it is really very simple if
you keep the following in mind:
1. when the original (action) force is a “pull”, the
reaction force is also a “pull”, and when the action
force is a “push”, the reaction is also a “push”; and
2. to identify the reaction force, simply reverse the
roles of the two bodies, i.e. if body 1 is pushing
against body 2, the reaction will be body 2 “pushing
back” (i.e. in the opposite direction) on body 1.
Newton’s Third Law introduces a critically important
idea: that bodies (whether gigantic like galaxies and
stars or tiny like subatomic particles) are continually
interacting with one another. Some of these interactions
are subtle and could easily be overlooked (e.g. the desk
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pushes up on the bottom of a book, so the book pushes
back down on the top of the desk), while others are
significant and very obvious (e.g. the very large equal
and opposite forces which exist for a tiny fraction of a
second when a baseball bat collides with a baseball).
Incidentally, the word “collision” has an everyday
meaning that we tend to associate with violent events
(ball/bat collisions, vehicle collisions, an asteroid
colliding with the Earth, etc.). But, in physics, the word
“collision” is used much more widely. For example,
when a space probe passes near Jupiter (say) and as a
result, alters its course before heading toward Saturn,
this “gravitational interaction” is considered a
“collision”, even though no “contact” actually occurs.
Conservation of Linear Momentum
It turns out that the three concepts of Newton’s Third
Law, “collisions” and “linear momentum” are all
related. Imagine two blocks of identical mass m on a
horizontal frictionless surface, with Block 1 being given a
“push” (say to the right) so that it collides with Block 2.
In the process, Block 1 will stop altogether and Block 2
will proceed to the right with the same speed that Block
1 had just prior to the collision. During the actual
collision itself (which lasts perhaps only a few
hundredths of a second), Newton’s Third Law tells us
that the force exerted by Block 1 on Block 2 (to the right)
will result in a reaction force by Block 2 back against
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Block 1 (to the left). If we let F12 and F21 represent these
two forces, respectively it follows that
F21 = –F12 .
But, according to Newton’s Second Law (First Version),
this means
p1
p2

p1  p 2 .
t
t , or simply
In other words, the momentum “lost” by Body 1 during
the collision (when it stops) is exactly accounted for in
the momentum “gained” by Body 2 (as it begins to
move). This can be further rearranged to give
p1  p2  0 or  p1  p2   0 .
Therefore, the “total linear momentum” of both bodies
has no change as a result of the collision, which means
that the total linear momentum “remains constant” or
“is conserved”. We call this the principle of
conservation of linear momentum.
Conservation of Linear Momentum:
In the absence of a net external force, the linear
momentum of a body, or a system of (two or more) bodies
interacting with one another according to Newton’s Third
Law, is conserved, i.e. the total linear momentum remains
constant.
Actually, there are different types of “collisions”
between bodies. The example above is a so-called elastic
collision, about which more will be said in the next
chapter (on Energy). Another example of an elastic
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collision would be when something like a “superball” is
dropped from a certain height, collides with the floor
and rebounds all the way back to its original height.
A second type of collision is called an inelastic collision.
An example of this would be dropping a blob of oatmeal
porridge from a certain height, and observing it land
with a splat on the floor and remain there! In other
words, in an inelastic collision, the two bodies (in this
case the porridge and the Earth) stick together.
In fact, most collisions fall somewhere between these
extremes, and are referred to as either partially elastic or
partially inelastic, such as when a tennis ball is dropped
but rebounds to only perhaps half of its original height.
Conservation of Angular Momentum
We have briefly mentioned the concept of angular
momentum, which is the momentum associated with
objects which remain in one place while rotating or
spinning, rather than objects which “translate” or move
from place to place.
The theory and mathematical formulas required for a
full understanding of angular momentum are somewhat
more complex than those for linear momentum, and
won’t be discussed in detail. It can be shown (though not
here) that, for each individual particle of mass m
undergoing rotational motion with speed v around a
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circle of radius r, the angular momentum l of that
particle will be given by l  mvr .
In any case, it turns out that, under the appropriate
conditions, just as for linear momentum, the total
angular momentum of a body or a system of bodies is also
conserved.
However brief, this discussion is still important, because
there is very little matter that isn’t spinning or rotating
in some fashion. Particles as small as electrons and
protons naturally spin. The Earth rotates on its axis
every 24 hours. Our Sun also rotates. And even
enormous, disc-shaped galaxies are spinning.
Here is one everyday example that you can easily relate
to. Imagine a figure skater spinning slowly with one leg
and both arms extended. When the skater pulls her/his
leg and arms inward, the rate of spin dramatically
increases. The above simple equation, l  mvr , applied
to each “particle” of the skater, easily shows why. As a
limb is pulled in, each particle’s radius of rotation r gets
smaller, while the particle’s mass m remains the same.
But, since angular momentum l is conserved, this means
that the speed of rotation v must increase!
It follows that any spinning object will rotate faster as it
contracts and more slowly as it expands. There are
many applications of this fact, including the surprisingly
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high rotation speeds of “collapsed stars” (more later)
and the high wind speeds found near the centre of
hurricanes or (even more dramatically) at the centre of
tornados.