A solid cylinder of mass ' M ' and radius ' R ' is rotating about its geometrical axis. A solid sphere of same mass and same radius is also rotating about its diameter with an angular speed half that of the cylinder. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder will be

Kinetic energy ratio calculation

Given a solid cylinder and solid sphere with equal mass $M$ and radius $R$, where the cylinder rotates with angular speed $\omega$ and the sphere with $\omega/2$. The rotational kinetic energy ratio $\frac{K_s}{K_c}$ is determined.

The moment of inertia for the cylinder about its axis is $I_c = \frac{1}{2}MR^2$, yielding kinetic energy $K_c = \frac{1}{2}I_c\omega^2 = \frac{1}{4}MR^2\omega^2$.

For the sphere about its diameter, $I_s = \frac{2}{5}MR^2$, with kinetic energy $K_s = \frac{1}{2}I_s(\omega/2)^2 = \frac{1}{20}MR^2\omega^2$.

The ratio simplifies as follows:

$\frac{K_s}{K_c} = \frac{\frac{1}{20}MR^2\omega^2}{\frac{1}{4}MR^2\omega^2} = \frac{1}{5}$

The kinetic energy ratio of sphere to cylinder is $1:5$.