Problem — Rolling sphere, speed at the base

Problem

A solid sphere of radius $R$ and mass $m$ rolls without slipping down an incline, starting from rest at height $h$. Derive the speed of the centre of mass when it reaches the base.

Solution

Static friction does not dissipate energy (no slipping): mechanical energy is conserved, with the kinetic energy split between translation and rotation. With $I = \tfrac{2}{5}mR^2$ and the constraint $\omega = v/R$: $mgh = \tfrac{1}{2}mv^2 + \tfrac{1}{2}\Big(\tfrac{2}{5}mR^2\Big)\frac{v^2}{R^2} = \tfrac{1}{2}mv^2 + \tfrac{1}{5}mv^2 = \tfrac{7}{10}mv^2$ Isolating $v$: $\ev{v = \sqrt{\tfrac{10}{7}\,gh}}$ Independent of mass and radius: two solid spheres of different sizes reach the base with the same speed.

Connections

Topics: Dinamica rotazionale Concepts: Rotolamento · Conservazione dell’energia meccanica · Energia cinetica rotazionale Objects: Sfera che rotola · Piano inclinato