The angular velocity of a body changes from $ω_{1}$ to $ω_{2}$ without applying external torque. Then find the ratio of the initial radius of gyration to the final radius of gyration.

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Solution

Moment of inertia , $I = m k^{2}$ , Where $k$ is radius of gyration.
Angular velocity changes by without applying external torque.
$\Rightarrow I_{1} ω_{1} = I_{2} ω_{2}$
$\Rightarrow m k_{1}^{2} ω_{1} = m k_{2}^{2} ω_{2}$ $\Rightarrow \frac{k_{1}}{k_{2}} = \sqrt{\frac{ω_{2}}{ω_{1}}}$

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The angular velocity of a body changes from ω1 to ω2 without applying external torque. Then find the ratio of the initial radius of gyration to the final radius of gyration.

Moment of inertia , I=mk2 , Where k is radius of gyration.Angular velocity changes by without applying external torque.⇒I1ω1=I2ω2⇒mk21ω1=mk22ω2 ⇒k1k2=√ω2ω1

The angular velocity of a body changes from $ω_{1}$ to $ω_{2}$ without applying external torque. Then find the ratio of the initial radius of gyration to the final radius of gyration.

Moment of inertia , $I = m k^{2}$ , Where $k$ is radius of gyration.
Angular velocity changes by without applying external torque.
$\Rightarrow I_{1} ω_{1} = I_{2} ω_{2}$
$\Rightarrow m k_{1}^{2} ω_{1} = m k_{2}^{2} ω_{2}$ $\Rightarrow \frac{k_{1}}{k_{2}} = \sqrt{\frac{ω_{2}}{ω_{1}}}$