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34. Problem 12.49P (HRW) Two cylinders having radii
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and
and
rotational inertias
and
about
the central axis are supported by axes perpendicular to the plane of the figure
as shown. The large cylinder is initially rotating with angular velocity
. The
small cylinder is moved to the right until it touches the large cylinder and is
caused to rotate by the frictional force between the two. Eventually, slipping
ceases, and the two cylinders rotate at constant rates in opposite directions.
We have to find the angular velocity
of the
small cylinder in terms of
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Solution: Click For PDF Version In this problem neither angular momentum nor kinetic energy are constants of motion. We will solve this problem by calculating changes in angular momentum arising out of angular impulse. Let us assume that the two cylinders are in contact with each other for time
Equations of angular impulse are
Solving these three algebraic equations, we find
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and let
F be the magnitude of the force of friction exerted by each cylinder on
the other when in contact and that at the end of this time interval both the
cylinders begin to rotate without slipping. Let v be the speed of the
rims of each cylinder after slipping has ceased. Let
be the angular speed of the smaller cylinder and
be the angular speed of the cylinder of radius
