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## Homework Statement

A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string, whose other end passes through a hole in the table, where I am holding it. Initially the particle is moving in a circle of radius ##r_0## with angular velocity ##w_0##, but I now pull the string down through the hole until a length r remains between the hole and the particle.

a) What is the particle's angular velocity now?

b) Assuming that I pull the string so slowly that we can approximate the particle's path by a circle of slowly shrinking radius, calculate the work I did pulling the string.

c)Compare your answer to part b with the particle's gain in kinetic energy.

## Homework Equations

Rotational energy: ##E = \frac{1}{2}I\omega^2##

## The Attempt at a Solution

The main assumption I made seems correct to me, but the question seems to imply it's wrong. I thought about the rotation and decided that because we assume the path is a circle of slowly decreasing radius, there is no work done as the force of the string is always tangential to the direction of motion. If we didn't assume this, then there would be a translational component to energy when we move and maybe that would lead to some work gain. If the assumption is true, the solution is the following:

a) Because no work is done, the energy is conservative and ##T_{in} = T_{fin}##. ##T_{in} = \frac{1}{2}MR_0^2\omega_0^2## and ## T_{fin} = \frac{1}{2}Mr^2\omega^2## after a bit of math, ##\omega = \frac{R_0\omega_0}{r}##

b)As argued above, the work Is zero.

c)My gain is zero