Four identical point masses are attached to the rim of a lightweight circular ring of…

PHYSICS · Multi-part · A Level · Physics · Engineering physics · Engineering physics

Problem

Four identical point masses

m

are attached to the rim of a lightweight circular ring of radius

R

. The system spins freely about a vertical axis through its center with angular speed

\omega_0

. Two opposite masses are then pulled radially inward along frictionless spokes until they reside at radius

\dfrac{R}{2}

, while the other two remain fixed at

R

. No external torque acts during this process. (a) Derive an expression for the new angular speed

\omega

of the system in terms of

\omega_0

. (b) Determine the fractional change in the system’s rotational kinetic energy,

\dfrac{K_{\text{final}}-K_{\text{initial}}}{K_{\text{initial}}}

, in symbolic form. (c) Let

m=0.20\,\text{kg}

R=0.50\,\text{m}

, and

\omega_0=5.0\,\text{rad\,s}^{-1}

. Calculate the numerical value of

\omega

. (d) Using the numerical values in part (c), compute the total work done by the forces that pull the two masses inward.