A horizontal platform in the shape of a circular disk...

A horizontal platform in the shape of a circular disk rotates on a frictionless bearing about a vertical axle through the center of the disk. The platform has a mass of $150 \mathrm{~kg}$, a radius of $2.0 \mathrm{~m}$, and a rotational inertia of $300 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about the axis of rotation. A $60 \mathrm{~kg}$ student walks slowly from the rim of the platform toward the center. If the angular speed of the system is $1.5 \mathrm{rad} / \mathrm{s}$ when the student starts at the rim, what is the angular speed when she is $0.50 \mathrm{~m}$ from the center?

Key Concepts

Conservation of Angular Momentum

In a frictionless, isolated system where no external torques act, the total angular momentum remains constant. This principle implies that any change in the distribution of mass (and hence moment of inertia) must be accompanied by an inverse change in angular speed to maintain a constant value of angular momentum.

Moment of Inertia

The moment of inertia quantifies how mass is distributed relative to the axis of rotation, affecting how much torque is needed for a desired angular acceleration. For composite systems, like a rotating disk with an additional mass, individual moments of inertia are summed to obtain the total moment of inertia, which plays a crucial role in determining the rotational dynamics.

Relationship Between Angular Momentum and Angular Speed

Angular momentum is defined as the product of moment of inertia and angular speed. When the moment of inertia changes due to mass redistribution, the angular speed adjusts in order to conserve angular momentum. This interdependency is central to analyzing rotational motion in systems with variable mass distribution.

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