A circular platform that can rotate freely about its central vertical axis is modeled…
PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems
Problem
A circular platform that can rotate freely about its central vertical axis is modeled as a solid disk of mass M and radius R. Two students, each of mass m, are seated immovably at opposite ends of a diameter. The platform is spinning with angular speed ω0 when Student A, who is holding a medicine ball of mass 2m, throws the ball straight toward Student B along the diameter. The speed of the throw, relative to the platform, is v. Student B catches the ball inelastically. The ball travels entirely in the horizontal plane of the platform, and no external torques act during any part of the process.
(a) Qualitatively describe how the angular speed of the platform changes at the instant the ball leaves Student A’s hands and again at the instant it is caught by Student B. For each event, state whether the platform’s angular speed is greater than, less than, or equal to ω0, and explain your reasoning.
(b) Derive an expression for the final angular speed ωf of the combined system (platform, two students, and ball) immediately after the ball is caught, in terms of M, m, R, v, and ω0.
(c) Discuss how the linear momentum of the ball, as measured in the platform’s frame just after the throw, contributes to the total angular momentum of the system about the central axis. Explain why this contribution is essential for the conservation analysis in part (b).