A solid sphere of mass and radius is released from rest at a height above the lowest…
PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems
Problem
A solid sphere of mass M and radius R is released from rest at a height h above the lowest point of a track that includes a vertical loop of radius 2.0m. Static friction is sufficient for the sphere to roll without slipping everywhere on the track. Neglect air resistance.
The goal is to find the minimum release height hmin that allows the sphere to maintain contact with the track at the very top of the loop.
(a) Derive an expression, in terms of h, for the linear speed of the sphere’s center of mass at the bottom of the loop just before it starts upward.
(b) State the mechanical condition that must be satisfied for the sphere to remain in contact with the track at the very top of the loop, and express that condition as an inequality involving the required speed at the top.
(c) Combine your results from parts (a) and (b) to determine hmin in terms of the loop radius Rloop=2.0m. Give a numerical value using g=9.8ms−2.
(d) If the solid sphere were replaced by a thin spherical shell of the same mass and radius, explain qualitatively whether the required release height would be larger, smaller, or unchanged. Support your answer with physical reasoning.