Question

A solid sphere and a thin spherical shell of same radius rotate about their diameter with same angular momentum but with different angular velocities. If $M _{1}$ and $M _{2}$ are the masses of solid sphere and hollow sphere and if their angular velocities are in the ratio $1: 3$, then $\left(\frac{ M _{1}}{ M _{2}}\right)$ is ______.

Answer 1

As, $L=I \omega$
$\therefore L _{1}= I _{1} \omega_{1}$ and $L _{2}= I _{2} \omega_{2}$
$\therefore \frac{L_{1}}{L_{2}}=\frac{I_{1} \omega_{1}}{I_{2} \omega_{2}}$
$\Rightarrow \frac{L}{L}=\frac{\frac{2}{5} M_{1} R^{2} \omega_{1}}{\frac{2}{3} M_{2} R^{2} \omega_{2}}$
$\cdots .\left(\because L _{1}= L _{2}= L\right.$ and $\left.R _{1}= R _{2}= R \right)$
$ 1=\frac{3 M _{1} \omega_{1}}{5 M _{2} \omega_{2}} $
$ \therefore 1=\frac{3 M _{1}}{5 M _{2}} \times \frac{1}{3} .....(\because \frac{\omega_1}{\omega_2} =\frac{1}{3})$
$ \therefore \frac{ M _{1}}{ M _{2}}=5$