Center of Mass for Two Particles beyond One Dimension
Now that we have the position, we extend the concept of the center of mass to velocity and
acceleration, and thus give ourselves the tools to describe the motion of a system of particles.
Taking a simple time derivative of our expression for
x
cm
we see that:
v
cm
=
Thus we have a very similar expression for the velocity of the center of mass. Differentiating
again, we can generate an expression for acceleration:
a
cm
=
With this set of three equations we have generated the necessary elements of the kinematics of a
system of particles.
From our last equation, however, we can also extend to the dynamics of the center of mass.
Consider two mutually interacting particles in a system with no external forces. Let the force
exerted on
m
2
by
m
1
be
F
21
, and the force exerted on
m
1
by
m
2
by
F
12
. By applying Newton's
Second Law we can state that
F
12
=
m
1
a
1
and
F
21
=
m
2
a
2
. We can now substitute this into our
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 Fall '10
 DavidJudd
 Physics, Acceleration, Center Of Mass, Force, Mass

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