Motion of the Center of Mass
The definition of the center of mass of a system of particles can be rewritten as
where M is the total mass of the system. Differentiating this equation with respect to time shows
where v
cm
is the
velocity of the center of mass
and v
i
is the velocity of mass m
i
. The acceleration
of the center of mass can be obtained by once again differentiating this expression with respect to
time
where a
cm
is the
acceleration of the center of mass
and a
i
is the acceleration of mass m
i
. Using
Newton's second law we can identify m
i
a
i
with the force acting on mass m
i
. This shows that
This equation shows that
the motion of the center of mass is only determined by the external
forces
. Forces exerted by one part of the system on other parts of the system are called internal
forces. According to Newton's third law, the sum of all internal forces cancel out (for each
interaction there are two forces acting on two parts: they are equal in magnitude but pointing in
an opposite direction and cancel if we take the vector sum of all internal forces). See Figure 9.5.

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