(Solved): A frictionless turntable with radius R and mass m1 uniformly distributed over its area is spinni ...

A frictionless turntable with radius $R$ and mass $m_1$ uniformly distributed over its area is spinning at an angular velocity ${\omega }_0$ about its center. A man and a woman are standing on the spinning turntable. They are initially located on opposite sides of the turntable at the outer edge, but start walking toward each other at the same rate until they meet at the center. You may approximate the people as objects, each with mass $m_2$, and negligible size. (a) (2 pt) Compute the moment of the inertia of the turntable together with the people on it, as a function of $r$, the distance of the man and woman from the center. (b) (4 pt) Compute the angular velocity $\omega (r)$ of the turntable (which is the same as that of the people) as a function of $x$. If $m_1=2m_2$, what is the angular velocity when the people meet at the center? (c) (2 pt) In the case that $m_1=2m_2$, what is the initial rotational kinetic energy? What is the final rotational kinetic energy when the people meet at the center? Express your answers as a numerical factor multiplying $m_2{\omega }_0^2{R}^2$. (d) (2 pt) Is rotational kinetic energy conserved? Explain why or why not in a sentence or two.