Question

A uniform sphere of mass $m$ and radius $r$ rolls without sliding over a horizontal plane, rotating about a horizontal axle $OA$. In the process, the centre of the sphere moves with a velocity $v$ along a circle of radius $R$. Find the kinetic energy of the sphere.

Answer 1

Here the motion of the body is orbital and spin combined.
The sphere is spinning about a horizontal axis and orbiting about a vertical axis.
We write total kinetic energy of the sphere, i.e.,
The kinetic energy of the sphere due to its rotation about its own axis and its motion along circular path with velocity $v$.
$KE=\frac{1}{2}{I}_1{\omega }_1^2+\frac{1}{2}{I}_2{\omega }_2^2$
where $I_1=\frac{2}{5}m{r}^2$ and $I_2=(\frac{2}{5}m{r}^2+m{R}^2)$
${\omega }_1=\frac{v}{r}$ and ${\omega }_2=\frac{v}{R}$
Substituting these values in the above equation, we get
$KE=\frac{7m}{10}{v}^2[1+\frac{2r^2}{7R^2}]$.