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.

Intuitive explanation

This is a conservation of angular momentum problem where the system's moment of inertia changes as masses move inward. Since no external torque acts, the total angular momentum $L = I\omega$ stays constant—but when two masses move closer to the axis, the moment of inertia decreases, so the angular speed must increase to compensate. Calculate the initial and final moments of inertia carefully (accounting for which masses move), then use $I_0\omega_0 = I_f\omega_f$ to find the new spin rate.

Intuitive explanation

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