Physics
Physics
Problem
Two objects of equal mass and radius—a solid disk and a thin hoop—are each given the same initial center-of-mass speed across an identical rough horizontal surface; neither object is initially rotating. When each object reaches the state of pure rolling, which one has lost the greater amount of mechanical energy to thermal energy? Briefly justify your answer.Intuitive explanation
Both objects experience friction that spins them up until rolling without slipping begins, but they end up with different final speeds—and that's the key. A rolling object's kinetic energy depends on both translational and rotational motion, with the rotational contribution determined by the moment of inertia. The hoop, having all its mass farther from the axis, stores more energy in rotation relative to translation than the disk does. Since friction does the same work on both (same force, same distance), the object that ends up storing more energy in rotation must have lost less to heat—meaning the disk dissipates more thermal energy overall.
Intuitive explanation
Both objects experience friction that spins them up until rolling without slipping begins, but they end up with different final speeds—and that's the key. A rolling object's kinetic energy depends on both translational and rotational motion, with the rotational contribution determined by the moment of inertia. The hoop, having all its mass farther from the axis, stores more energy in rotation relative to translation than the disk does. Since friction does the same work on both (same force, same distance), the object that ends up storing more energy in rotation must have lost less to heat—meaning the disk dissipates more thermal energy overall.