A ballet dancer, dancing on a smooth floor is spinning about...

A ballet dancer, dancing on a smooth floor is spinning about a vertical axis with her arms folded with an angular velocity of $20 \mathrm{rad} / \mathrm{s}$. When she stretches her arms fully, the spinning speed decrease in $10 \mathrm{rad} / \mathrm{s}$. If $I$ is the initial moment of inertia of the dancer, the new moment of inertia is (a) $2 I$ (b) $3 I$ (c) $I / 2$ (d) $1 / 3$

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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 motion. It depends on how the object's mass is distributed relative to the axis of rotation. When a dancer spreads her arms, the mass distribution changes, effectively increasing the moment of inertia.

Relationship Between Angular Velocity and Moment of Inertia

For a system conserving angular momentum, the product of the moment of inertia and angular velocity remains constant. Therefore, if the moment of inertia increases, the angular velocity decreases proportionally, and vice versa. This relationship enables one to calculate the new moment of inertia when the angular velocity changes due to a redistribution of mass.

Conservation of Angular Momentum

This principle states that if no external torques are acting on a system, the total angular momentum remains constant. In the context of rotational motion, any change in the distribution of mass (and thus the moment of inertia) will lead to an inverse change in the angular velocity to maintain the constant angular momentum.

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