Question

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

Answer 1

A sphere having mass $m$ rolls on a plane surface. Let its radius $R$. Its centre moves with a velocity $v$

Therefore kinetic energy

$=\left(1/2\right)l{\omega }^2+\left(1/2\right)m{v}^2$

$=\frac{1}{2}\times \frac{2}{5}m{R}^2\times \frac{v^2}{R^2}+\frac{1}{2}m{v}^2$

$=\frac{2}{10}m{v}^2+\frac{1}{2}m{v}^2$

$=\frac{2+5}{10}m{v}^2$

$=\frac{7}{10}m{v}^2$