Spinning Mass with Variable Radius

A small ball is attached to a string passing through a pipe as shown in the figure. The ball is initially spinning around in a circle of radius with tangential velocity (the instantaneous linear velocity at some point). When the string is pulled down, the velocity of the ball increases as a consequence of the conservation of angular momentum.


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We can write the conservation of angular momentum as , where is the mass, and are the initial and final radii, and and are the initial and final tangential velocities; then . For simplicity, and are chosen equal to 1.
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