The 80-kg man is holding two dumbbells while standing on a...

The $80-\mathrm{kg}$ man is holding two dumbbells while standing on a turntable of negligible mass, which turns freely about a vertical axis. When his arms are fully extended, the turntable is rotating with an angular velocity of $0.5 \mathrm{rev} / \mathrm{s}$. Determine the angular velocity of the man when he retracts his arms to the position shown. When his arms are fully extended, approximate each arm as a uniform 6 -kg rod having a length of $650 \mathrm{~mm}$, and his body as a $68-\mathrm{kg}$ solid cylinder of $400-\mathrm{mm}$ diameter. With his arms in the retracted position, assume the man is an $80-\mathrm{kg}$ solid cylinder of $450-\mathrm{mm}$ diameter. Each dumbbell consists of two $5-\mathrm{kg}$ spheres of negligible size.

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Key Concepts

Conservation of Angular Momentum

This principle states that in a closed system with no external torques, the total angular momentum remains constant. In rotational dynamics problems, if a system changes its mass distribution (and therefore its moment of inertia), its angular velocity will adjust to keep the product of moment of inertia and angular velocity constant.

Moment of Inertia

The moment of inertia is a measure of how resistant a body is to changes in its rotational motion, depending on how its mass is distributed relative to the axis of rotation. Different shapes such as rods, cylinders, or point masses have specific formulas used to calculate the moment of inertia, and when dealing with composite systems, the total moment of inertia is the sum of the individual parts.

Angular Velocity

Angular velocity describes the rate at which an object rotates around an axis. In rotational dynamics, when the moment of inertia of a system changes but angular momentum is conserved, the angular velocity must change accordingly to maintain the constant product of moment of inertia and angular velocity.

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