The angular velocity of the body changes from ω1 to ω2 without applying torque but by changing m

step 1 In this question we have given that angular velocity of the body changes from omega 1 to omega 2

The angular velocity of the body changes from ω1 to ω2...

The angular velocity of the body changes from $\omega_{1}$ to $\omega_{2}$ without applying torque but by changing moment of inertia. The initial radius of gyration to the final radius of gyration is (a) $\omega_{2}: \omega_{1}$ (b) $\omega_{2}^{2}: \omega_{1}^{2}$ (c) $\sqrt{\omega_{2}}: \sqrt{\omega_{1}}$ (d) $\frac{1}{\omega_{2}}: \frac{1}{\omega_{1}}$

The angular velocity of the body changes from $\omega_{1}$ to $\omega_{2}$ without applying torque but by changing moment of inertia. The initial radius of gyration to the final radius of gyration is
(a) $\omega_{2}: \omega_{1}$
(b) $\omega_{2}^{2}: \omega_{1}^{2}$
(c) $\sqrt{\omega_{2}}: \sqrt{\omega_{1}}$
(d) $\frac{1}{\omega_{2}}: \frac{1}{\omega_{1}}$

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Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational state. It depends on both the mass of the object and its distribution relative to the axis of rotation. In systems where the geometry changes without applying external torques, the moment of inertia plays a key role in determining the resulting angular velocity.

Radius of Gyration

The radius of gyration provides an alternative way to express the moment of inertia. It is defined as the distance from the axis of rotation at which the total mass of the body could be assumed to be concentrated to yield the same moment of inertia. This concept is particularly useful when analyzing how changes in the distribution of mass affect rotational motion.

Conservation of Angular Momentum

This principle states that if no external torque is acting on a system, the total angular momentum remains constant. It is a fundamental conservation law in rotational dynamics, allowing us to relate changes in a body's moment of inertia to corresponding changes in its angular velocity.

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