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31. Problem 13.21(HR) The axis of the cylinder shown in the figure is fixed. The
cylinder is initially at rest. The block of mass M is initially moving to the
right without friction with speed
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. It
passes over the cylinder to the dotted position. When it first makes contact
with the cylinder, it slips on the cylinder, but the friction is large enough so
that slipping ceases before M loses contact with the cylinder. The cylinder has
a radius R and a rotational inertia I. We have to find the final speed
in
terms of
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Solution: Click For PDF Version As there is no external torque acting on the system of the block and the cylinder, angular momentum about the axis of rotation of the cylinder is a constant of motion. We will, therefore, use conservation of angular momentum for answering this problem. We note that as initially the cylinder is at rest its angular momentum is
zero. We will, therefore, first find the angular momentum of the block, moving
to the right with velocity
As the cylinder begins to rotate because of force of friction when the moving
block comes into contact with it and the force of friction is so strong that
slipping ceases when the block passes over the cylinder. Rotational speed
Angular momentum of the block and the cylinder after the block has crossed over to the other side, as shown in the figure, will be
Equating
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of the cylinder and linear speed
, we get