Physics
Physics
Problem
Three rigid objects each have the same mass and radius and roll without slipping across a horizontal surface, all with identical center-of-mass speeds. The objects are • a solid sphere, • a solid cylinder, and • a thin hoop. (a) Rank the three objects from greatest to least according to the fraction of their total kinetic energy that is rotational. (b) Clearly explain the reasoning that leads to your ranking, making explicit reference to how the distribution of mass within each object influences the result.Intuitive explanation
The key insight is that rolling motion splits the kinetic energy between translational and rotational parts, and different shapes store energy differently in rotation. Even though all three objects move with the same center-of-mass speed, they have different moments of inertia, so they spin at different rates and carry different total kinetic energies. To find which rolls fastest, moves farthest, or requires the most energy, you need to use the no-slip condition and compare their rotational inertias—the hoop concentrates mass at the rim (largest ), the sphere distributes it throughout (smallest ), and the cylinder sits between them.
Intuitive explanation
The key insight is that rolling motion splits the kinetic energy between translational and rotational parts, and different shapes store energy differently in rotation. Even though all three objects move with the same center-of-mass speed, they have different moments of inertia, so they spin at different rates and carry different total kinetic energies. To find which rolls fastest, moves farthest, or requires the most energy, you need to use the no-slip condition and compare their rotational inertias—the hoop concentrates mass at the rim (largest ), the sphere distributes it throughout (smallest ), and the cylinder sits between them.