2
has a discernible reason, namely, interaction with one or more
other bodies, for example, by gravitational attraction. Given the
dependence of the forces on the locations of the bodies (for
example, the inverse-square law of gravity), then the collection
of F = m a equations for the bodies is an equation of motion
(technically, a set of coupled second-order differential equa-
tions), from which the motion of the bodies can be calculated,
given their locations and velocities at one time.
The third law (Fig. 2), often referred to as "action equals reac-
tion," has the important consequence that all the many internal
forces exerted by parts of a body on one another are irrelevant
to the motion of the body because they cancel out. The effects
of quantum mechanics and special relativity on the motions of
the parts, no matter how light or rapidly moving they are, are
also irrelevant as long as the body itself is sufficiently heavy
and slowly moving. However, the third law is not always true
in Newton's sense; for example, the magnetic forces between
moving charged bodies do not obey the third law because the
electromagnetic field can carry momentum.
There also is a rotational analog of the second law (the force on
a body equals the rate of change of its momentum), namely,
that the torque on a body equals the rate of change of its
angular momentum. However, there is no analog to the first
law, except in the very special case of rotation of a rigid body
around a fixed axis.
Newton was well aware that there is no way to measure "ab-
solute" motion of a body, that is, without comparison to the
motion of some other material thing. One can measure the mo-
tion relative to oneself ("the observer"), but two observers who
are accelerating or rotating relative to one another will see the
acceleration of a body to be different, and hence they will not
both see Newton's laws obeyed. An observer who sees Newton's
laws obeyed is called "inertial." It is important to note that the ve-
locities (although not the accelerations) of such observers may
be different; this is usually called Galilean relativity. Einstein's
special relativity is the modification that allows electromagnetic
phenomena to be included in this relativity of motion.
Newton's Laws of Motion (continued)
Fig. 2: Newton's third law of motion: Two bodies (m
1
and m
2
) exert forces (F
1
and F
2
) on one another that are equal in magnitude, but opposite in direction.
(Copyright © McGraw-Hill Education)
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