·The spinning figure skater. The outstretched hands and arms...

$\cdot$ The spinning figure skater. The outstretched hands and arms of a figure skater preparing for a spin can be considered a slender rod pivoting about an axis through its center. (See Figure $10.53 .$ ) When the skater's hands and arms are brought in and wrapped around his body to execute the spin, the hands and arms can be considered a thin-walled hollow cylinder. His hands and arms have a combined mass of 8.0 $\mathrm{kg}$ . When outstretched, they span 1.8 $\mathrm{m}$ ; when wrapped, they form a cylinder of radius 25 $\mathrm{cm} .$ The moment of inertia about the axis of rotation of the remainder of his body is constant and equal to 0.40 $\mathrm{kg} \cdot \mathrm{m}^{2} .$ If the skater's original angular speed is 0.40 $\mathrm{rev} / \mathrm{s}$ what is his final angular speed?

Key Concepts

Geometric Modeling in Rotational Dynamics

In problems involving rotation, complex bodies can often be approximated by simple geometric models, such as a slender rod or a thin-walled cylinder. This simplification allows the use of standard formulas for the moment of inertia, facilitating the analysis of rotational behavior as structural configurations change.

Conservation of Angular Momentum

This principle states that in the absence of external torques, the total angular momentum of a system remains constant. In rotational dynamics, when a skater changes the configuration of their body (thus changing the moment of inertia), the angular speed adjusts to maintain the constant product of moment of inertia and angular speed.

Moment of Inertia

Moment of inertia is a measure of an object’s resistance to changes in its rotational motion, depending on the mass distribution relative to the axis of rotation. It is calculated differently for various shapes and configurations, and understanding its computation is essential in analyzing rotational motion.

Rotational Kinematics

Rotational kinematics involves the study of angular motion variables like angular speed and angular acceleration, analogous to linear kinematics in translational motion. It provides the framework for understanding how changes in configuration (and thereby moment of inertia) affect the rotational speed of an object.

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