Question

The angular speed of a body changes from $\omega_{1}$ to $\omega_{2}$ without applying a torque but due to change in its moment of inertia. The ratio of radii of gyration in the two cases is :

• A. $\omega_{2}-\omega_{1}$
• B. $\omega_{1}: \omega_{2}$
• C. $\sqrt{\omega_{1}}: \sqrt{\omega_{2}}$
• D. $\sqrt{\omega_{2}}: \sqrt{\omega_{1}}$

Answer 1

Answer: (D)

In the absence of external torque angular momentum is conserved.
The radius of gyration $k$ of a body about a given line is defined as
$I=M k^{2}$
where, $I$ is its moment of inertia and $M$ is its total mass.
Also from law of conservation of angular momentum
$J =I \omega=$ constant
$\left(M k_{1}^{2}\right) \omega_{1} =\left(M k_{2}^{2}\right) \omega_{2}$
$\Rightarrow \frac{k_{1}^{2}}{k_{2}^{2}}=\frac{\omega_{2}}{\omega_{1}} $
$ \Rightarrow \frac{k_{1}}{k_{2}}=\frac{\sqrt{\omega_{2}}}{\sqrt{\omega_{1}}}$