PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems
Problem
Two identical uniform solid spheres, A and B, each of mass M and radius R, are placed on the same horizontal rough floor.
• Sphere A is given an initial center-of-mass speed v0 with no initial rotation.
• Sphere B is given an initial angular speed ω0=v0/R with its center of mass initially at rest.
The coefficient of kinetic friction between each sphere and the floor is the same constant value, and it is large enough to ensure both spheres eventually achieve pure rolling.
Use qualitative reasoning and symbolic relationships to answer the parts below.
(a) For sphere A, identify which of the quantities v, ω, translational kinetic energy, and rotational kinetic energy change during the slipping interval. Explain why each quantity changes or remains constant.
(b) Answer the same question for sphere B.
(c) Determine which sphere reaches the pure-rolling condition first, or whether they do so at the same time. Justify your answer by comparing the relevant frictional forces and torques.
(d) Compare the total mechanical energy lost to thermal energy by each sphere by the time pure rolling is achieved. Provide a clear, qualitative justification.
(e) Suppose both spheres instead had the same mass but twice the radius. Discuss qualitatively whether your answers to parts (c) and (d) would change, and explain why or why not.