Question

Derive an expression for kinetic energy, when a rigid body is rolling on a horizontal surface without slipping. Hence find kinetic energy for a solid sphere.

Answer 1

Total Rolling kinetic energy$=$Translational K.E$+$Rotational K.E

$=\frac{1}{2}M{V}^2+\frac{1}{2}I{\omega }^2$

But $\omega =\frac{V}{R}$

K.E$=\frac{1}{2}M{V}^2+\frac{1}{2}I{\left(\frac{V}{R}\right)}^2$

$=\frac{1}{2}M{V}^2+\frac{1}{2}I\left(\frac{V^2}{R^2}\right)$

For solid sphere $I=\frac{2}{5}M{R}^2$

K.E$=\frac{1}{2}M{V}^2+\frac{1}{2}\times \frac{2}{5}M{R}^2\times \left(\frac{V^2}{R^2}\right)$

$=\frac{1}{2}M{V}^2+\frac{1}{5}M{V}^2$

$=\frac{7}{10}M{V}^2$