Question

A sphere of mass m rolls on a plane surface. Find its kinetic energy at an instant when its centre moves with speed $\nu$.

Answer 1

Let radius of the sphere be R and its angular speed be ω.
Moment of inertia of sphere, $I=\frac{2}{5}m{R}^2$
Total kinetic energy,
$$\begin{gathered} K=\frac{1}{2}I{\omega }^2+\frac{1}{2}m{v}^2 \\ K=\frac{1}{2}\times \left(\frac{2}{5}m{R}^2\right)\times \frac{v^2}{R^2}+\left(\frac{1}{2}m{v}^2\right) \\ K=\frac{2}{10}m{v}^2+\frac{1}{2}m{v}^2 \\ K=\frac{\left(2+5\right)m{v}^2}{10}=\frac{7}{10}m{v}^2 \end{gathered}$$