Now let . That is, we apply an impulse of vertical momentum to the mass on the right at time 0 .

In this case, the unit of vertical momentum is transferred entirely to the mass on the right, so that

which is twice as fast as before. Just after time zero, we have , , and, because the massless rod remains rigid, .

Note that the velocity of the center-of-mass
is the
*same* as it was when we hit the midpoint of the rod. This is an
important general equivalence: The sum of all external force vectors
acting on a rigid body can be applied as a single resultant force
vector to the total mass concentrated at the center of mass to find
the linear (translational) motion produced. (Recall from §B.4.1
that such a sum is the same as the sum of all radially acting external
force components, since the tangential components contribute only to
rotation and not to translation.)

All of the kinetic energy is in the mass on the right just after time zero:

However, after time zero, things get more complicated, because the mass on the left gets dragged into a rotation about the center of mass.

To simplify ongoing analysis, we can define a *body-fixed frame
of reference*^{B.16} having its origin at the center of mass. Let
denote a velocity in this frame. Since the velocity of the center of
mass is
, we can convert any velocity
in the
body-fixed frame to a velocity
in the original frame by adding
to it, *viz.*,

The mass velocities in the body-fixed frame are now

and of course .

In the body-fixed frame, all kinetic energy is *rotational* about
the origin. Recall (Eq.
(B.9)) that the moment of inertia for this
system, with respect to the center of mass at
, is

Thus, the

This is

Adding this translational kinetic energy to the rotational kinetic energy in the body-fixed frame yields the total kinetic energy, as it must.

In summary, we defined a moving body-fixed frame having its origin at the center-of-mass, and the total kinetic energy was computed to be

in agreement with the more complicated (after time zero) space-fixed analysis in Eq. (B.13).

It is important to note that, after time zero, both the linear
momentum of the center-of-mass (
), and the angular momentum in the body-fixed frame
(
) remain
*constant* over time.^{B.17} In the original space-fixed
frame, on the other hand, there is a complex transfer of momentum back
and forth between the masses after time zero.

Similarly, the translational kinetic energy of the total mass, treated as being concentrated at its center-of-mass, and the rotational kinetic energy in the body-fixed frame, are both constant after time zero, while in the space-fixed frame, kinetic energy transfers back and forth between the two masses. At all times, however, the total kinetic energy is the same in both formulations.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University