PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Torque and Rotational Dynamics · Torque and Rotational Dynamics
Problem
Two solid uniform cylinders, X and Y, are free to rotate independently about the same fixed horizontal axle with negligible friction. Cylinder X has mass M and radius 2R, while cylinder Y has mass 21M and radius R. Identical motors can be coupled one at a time so that each cylinder, while disengaged from the other, experiences the same constant torque τ0 about the common axis. Each cylinder starts from rest.
(a) Derive an expression for the ratio of the initial angular accelerations αYαX for cylinders X and Y, leaving your answer in terms of the given constants.
(b) Cylinder X is allowed to accelerate under the torque τ0 until it reaches an angular speed ω. Determine the time interval required for cylinder X to reach this speed. Express your answer in terms of M, R, τ0, and ω.
(c) Cylinder Y is then subjected to the same procedure. Without re-deriving all algebra, state whether the time required for cylinder Y to reach the same angular speed ω is greater than, less than, or equal to the time found in part (b). Support your answer with clear reasoning based on physical principles.
(d) For each cylinder, calculate the rotational kinetic energy at the moment it first reaches angular speed ω. Identify which cylinder requires more work from its motor to attain that speed, and explain why.