Two ice skaters, both of mass 68 kg, approach on parallel paths 1.6 m apart. Both are moving at 3.5 m/s with their arms outstretched. They join hands as they pass, still maintaining their 1.6-m separation, and begin rotating about one another. Treat the skaters as particles with regard to their rotational inertia. Calculate the change in kinetic energy for this process.

Verified step by step guidance

1. First, identify the initial kinetic energy of the system. Since both skaters are moving in straight lines at the same speed, their initial kinetic energy is purely translational. The formula for the translational kinetic energy of each skater is \( KE_{\text{initial}} = \frac{1}{2} m v^2 \), where \( m \) is the mass of one skater and \( v \) is their velocity. Multiply this by 2 to account for both skaters.
2. Next, consider the final state of the system. After the skaters join hands, they begin rotating about their center of mass. The system now has rotational kinetic energy. The formula for rotational kinetic energy is \( KE_{\text{rotational}} = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia of the system and \( \omega \) is the angular velocity.
3. Calculate the moment of inertia \( I \) for the system. Treat the skaters as point masses located at a distance \( r = 0.8 \ \text{m} \) (half the separation distance) from the center of mass. The moment of inertia for each skater is \( I = m r^2 \), and the total moment of inertia is \( I_{\text{total}} = 2 m r^2 \).
4. Determine the angular velocity \( \omega \) of the system. Use the principle of conservation of angular momentum, which states that the initial angular momentum equals the final angular momentum. The initial angular momentum is \( L_{\text{initial}} = 2 m v r \), and the final angular momentum is \( L_{\text{final}} = I_{\text{total}} \omega \). Solve for \( \omega \) using \( \omega = \frac{L_{\text{initial}}}{I_{\text{total}}} \).
5. Finally, calculate the change in kinetic energy. Subtract the final rotational kinetic energy from the initial translational kinetic energy: \( \Delta KE = KE_{\text{final}} - KE_{\text{initial}} \). Use the values obtained for \( KE_{\text{initial}} \), \( I_{\text{total}} \), and \( \omega \) to compute \( KE_{\text{final}} \).

Key Concepts

Conservation of Angular Momentum

In a closed system with no external torques, the total angular momentum remains constant. When the two skaters join hands and start rotating, their combined angular momentum before they join must equal their angular momentum after they start rotating. This principle is crucial for analyzing the motion and determining the final state of the system.

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by an object due to its rotation, calculated using the formula KE_rot = 1/2 I ω², where I is the moment of inertia and ω is the angular velocity. Understanding how to calculate the moment of inertia for the skaters and how it changes when they start rotating together is essential for determining the change in kinetic energy.

Kinetic Energy Change

The change in kinetic energy during a process is the difference between the initial and final kinetic energies of the system. In this scenario, it involves calculating the initial kinetic energy of the skaters before they join hands and the final kinetic energy after they start rotating together, allowing us to quantify the energy transformation that occurs during the interaction.