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27. Problem 12.62P (HRW) A cockroach of mass m runs counterclockwise around a circular
disk mounted on a vertical axis of radius R and rotational inertia I and having
frictionless bearings. The cockroach’s speed (relative to Earth) is v, whereas
the disk turns clockwise with angular speed
(a) What is the angular speed of the disk after the cockroach stops? (b) Is mechanical energy conserved? |
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The cockroach finds a breadcrumb on the rim and of course stops.|
Solution: Click For PDF Version We will apply the conservation of angular momentum for solving this problem.
The angular momentum of the cockroach-disk system when the cockroach is moving
with speed v in counterclockwise direction and the disk is turning with
angular speed
After the cockroach stops the system of disk-cockroach will rotate together
as a rigid body. Let
(b) The initial kinetic energy is
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will be
.
be the
changed angular speed of the system. As the moment of inertia of the
disk-cockroach system is
, the
conservation of angular momentum implies
and the final kinetic energy is
.
Therefore, the kinetic energy is not conserved.