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

Answer: 5

As, $L=I\omega$
$\therefore L_1=I_1{\omega }_1$ and $L_2=I_2{\omega }_2$
$\therefore \frac{L_1}{L_2}=\frac{I_1{\omega }_1}{I_2{\omega }_2}$
$\Rightarrow \frac{L}{L}=\frac{\frac{2}{5}{M}_1{R}^2{\omega }_1}{\frac{2}{3}{M}_2{R}^2{\omega }_2}$
$\cdots .(\because L_1=L_2=L$ and $R_1=R_2=R)$
$1=\frac{3M_1{\omega }_1}{5M_2{\omega }_2}$
$\therefore 1=\frac{3M_1}{5M_2}\times \frac{1}{3}.....(\because \frac{{\omega }_1}{{\omega }_2}=\frac{1}{3})$
$\therefore \frac{M_1}{M_2}=5$