Question

The angular speed of a body changes from ${\omega }_1$ to ${\omega }_2$ due to changes in its moment of inertia without applying torque. The ratio of radius of gyration in the two cases is

• A. ${\omega }_1:{\omega }_2$
• B. $\sqrt{{\omega }_2}:\sqrt{{\omega }_1}$
• C. $\sqrt{{\omega }_2^2}:\sqrt{{\omega }_1^2}$
• D. $\sqrt{{\omega }_2^3}:\sqrt{{\omega }_1^3}$

Answer 1

The correct option is B $\sqrt{{\omega }_2}:\sqrt{{\omega }_1}$

Net external torque $\left({\tau }_{\text{ext}}\right)$ is zero on the body, hence its angular momentum will remain conserved.
$\Rightarrow$Applying $P.C.A.M$ on the body undergoing change in moment of inertia:
$I_1{\omega }_1=I_2{\omega }_2....\left(i\right)$

Moment of inertia $I=m{K}^2$,where $K$ is radius of gyration.
Substututing in Eq.$\left(i\right)$ we get:
$m{K}_1^2{\omega }_1=m{K}_2^2{\omega }_2$
$\Rightarrow {\left(\frac{K_1}{K_2}\right)}^2=\frac{{\omega }_2}{{\omega }_1}$
$\therefore \frac{K_1}{K_2}=\sqrt{{\omega }_2}:\sqrt{{\omega }_1}$