Rolling Disk and Sphere Comparison · PhysicsBowl 2024 #23

When an object rolls without slipping down a ramp, gravitational potential energy converts into both translational and rotational kinetic energy. The key insight is that different shapes have different moments of inertia, so they partition energy differently—a sphere "wastes" more energy spinning than a disk does. To reach the same final speed $v$, the sphere must start from a greater height to compensate for the extra rotational energy it gains. Use energy conservation and the moment of inertia formulas ($I_{\text{disk}} = \frac{1}{2}mR^2$ and $I_{\text{sphere}} = \frac{2}{5}mR^2$) to find how much higher the sphere must be released.

When an object rolls without slipping down a ramp, gravitational potential energy converts into both translational and rotational kinetic energy. The key insight is that different shapes have different moments of inertia, so they partition energy differently—a sphere "wastes" more energy spinning than a disk does. To reach the same final speed $v$, the sphere must start from a greater height to compensate for the extra rotational energy it gains. Use energy conservation and the moment of inertia formulas ($I_{\text{disk}} = \frac{1}{2}mR^2$ and $I_{\text{sphere}} = \frac{2}{5}mR^2$) to find how much higher the sphere must be released.