A solid sphere of mass ' $M$ ' and radius ' $R$ ' is rotating about its diameter. A solid cylinder of same…
A solid sphere of mass ' $M$ ' and radius ' $R$ ' is rotating about its diameter. A solid cylinder of same mass and radius is also rotating about its geometrical $1: 3$ axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation ( $\mathrm{K}_{\text {sphere }}$ to $\mathrm{K}_{\text {cylinder }}$ ) will be
$1: 8$
$1: 6$
$1: 3$
$1: 5$
Solution
Kinetic energy of sphere is given by:
$E_{\text {sphere }}=\frac{1}{2} I_{\text {sphere }} \omega_{\text {sphere }}^2$
$E_{\text {sphere }}=\frac{1}{2} \times \frac{2}{5} m R^2 \omega_{\text {sphere }}^2$
Kinetic energy of cylinder is given by:
$\begin{aligned} & E_{\text {cylinder }}=\frac{1}{2} I_{\text {cylinder }} \omega_{\text {cylinder }}^2 \\ & E_{\text {cylinder }}=\frac{1}{2} \times \frac{1}{2} m R^2 \omega_{\text {cylinder }}^2 \\ & \text { Given, } \omega_{\text {cylinder }}=2 \omega_{\text {sphere }} \\ & \frac{E_{\text {sphere }}}{E_{\text {cylinder }}}=\frac{1}{5}\end{aligned}$
.