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 (Fig. E10.43). 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 thinwalled, hollow cylinder. His hands and arms have a combined mass of 8.0 kg. When outstretched, they span 1.8 m; when wrapped, they form a cylinder of radius 25 cm. The moment of inertia about the rotation axis of the remainder of his body is constant and equal to 0.40 kg m² . If his original angular speed is 0.40 rev/s, what is his final angular speed?

Verified step by step guidance

1. Identify the initial and final moments of inertia. Initially, the arms are outstretched and can be modeled as a slender rod pivoting about its center. The moment of inertia for a rod about its center is given by \( I_{rod} = \frac{1}{12} m L^2 \), where \( m \) is the mass and \( L \) is the length.
2. Calculate the initial moment of inertia of the arms: \( I_{arms, initial} = \frac{1}{12} \times 8.0 \text{ kg} \times (1.8 \text{ m})^2 \).
3. When the arms are wrapped, they form a thin-walled hollow cylinder. The moment of inertia for a hollow cylinder about its central axis is \( I_{cylinder} = m r^2 \), where \( r \) is the radius.
4. Calculate the final moment of inertia of the arms: \( I_{arms, final} = 8.0 \text{ kg} \times (0.25 \text{ m})^2 \).
5. Apply the conservation of angular momentum: \( I_{total, initial} \omega_{initial} = I_{total, final} \omega_{final} \). Solve for the final angular speed \( \omega_{final} \), where \( I_{total, initial} = I_{body} + I_{arms, initial} \) and \( I_{total, final} = I_{body} + I_{arms, final} \).

Key Concepts

Angular Momentum

Angular momentum is a measure of the rotational motion of an object and is defined as the product of the moment of inertia and the angular velocity. It is a vector quantity that depends on the axis of rotation. In a closed system, angular momentum is conserved, meaning that if no external torque acts on the system, the total angular momentum before and after an event remains constant.

Moment of Inertia

The moment of inertia quantifies how mass is distributed relative to an axis of rotation. It is calculated by summing the products of each mass element and the square of its distance from the axis. Different shapes have different formulas for calculating moment of inertia, and it plays a crucial role in determining how easily an object can be rotated. In this scenario, the skater's arms change shape, affecting the moment of inertia as they move from an outstretched position to a compact one.

Conservation of Angular Momentum

The conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of that system remains constant over time. This principle is essential in analyzing the skater's spin, as the change in the distribution of mass (from outstretched arms to arms wrapped) alters the moment of inertia, which in turn affects the angular velocity. As the skater pulls in their arms, they must spin faster to conserve angular momentum.