Question

Show that the total kinetic energy of a sphere of mass $m$ rolling along horizontal plane with velocity $v$ is $7/10m{v}^2$

Answer 1

As we know that

Kinetic energy of a rolling body ,$K=\frac{1}{2}{I}_{C.M}{\omega }^2+\frac{1}{2}m{v}^2$

Here $I_{C.M}=\frac{2}{5}m{r}^2$, which is the moment of inertia of sphere about its centre of mass

where $r$ is radius and $m$ is mass of sphere

Hence ,$K=\frac{1}{2}\times \frac{2}{5}m{r}^2{\omega }^2+\frac{1}{2}m{v}^2$

Also $v=r\omega$

$\therefore K=\frac{1}{5}m{v}^2+\frac{1}{2}m{v}^2=\frac{7}{10}m{v}^2$