A lightweight rigid rod of length is mounted so it can rotate without friction about…
PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems
Problem
A lightweight rigid rod of length 2R is mounted so it can rotate without friction about a fixed vertical axis through its center. Two small identical masses m are originally clamped to the rod ends at distance R from the axis. The system spins about the axis with angular speed ω0.
At t=0 the clamps are released and massless springs embedded in the rod push the two masses simultaneously toward the axis. Each spring exerts a force only while it is expanding; by the time the masses reach a final distance R/2 from the axis the springs have returned to their relaxed lengths and exert no further force. Neglect any external torques during the entire process.
(a) Derive an expression for the angular speed ω1 of the rod–mass system after the masses have reached the distance R/2 and the springs are no longer exerting forces. Express ω1 in terms of ω0.
(b) While the springs are active, each mass experiences two horizontal forces: the spring force and the normal force from the rod. Compare the magnitudes of the average spring force and the average normal force on a mass during the interval in which the mass moves. Justify your comparison using appropriate conservation principles.
(c) Consider the total mechanical energy of the isolated system consisting of the rod, the two masses, and the springs. Is this energy conserved over the entire process? Provide a reasoned explanation supporting your answer.