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. They now pull on each other’s hands, reducing their radius to half its original value. Calculate the change in kinetic energy for this process.

Verified step by step guidance

1. Step 1: Identify the initial and final states of the system. Initially, the two skaters are moving in straight lines parallel to each other, separated by a distance of 1.6 m. After joining hands, they begin rotating about one another, and later reduce their radius of rotation to half its original value (0.8 m).
2. Step 2: Use the principle of conservation of angular momentum. Since no external torques act on the system, the angular momentum of the skaters is conserved. The initial angular momentum can be calculated using the formula: $L=I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.
3. Step 3: Calculate the moment of inertia for the initial and final states. Treating the skaters as particles, the moment of inertia is given by $I=m{r}_2^{}$, where $m$ is the mass of each skater and $r$ is the radius of rotation. Initially, $r$ is 1.6 m, and finally, $r$ is 0.8 m.
4. Step 4: Relate the angular velocity before and after the radius change using conservation of angular momentum. Since $I\omega$ is conserved, the final angular velocity can be expressed as ${\omega }_{\mathrm{f}}={\omega }_{\mathrm{i}}\times \frac{I_{\mathrm{i}}}{I_{\mathrm{f}}}$, where $I_{\mathrm{i}}$ and $I_{\mathrm{f}}$ are the initial and final moments of inertia.
5. Step 5: Calculate the change in kinetic energy. Rotational kinetic energy is given by $K=\frac{1}{2}I{\omega }_2^{}$. Compute the initial and final kinetic energies using the respective moments of inertia and angular velocities, and find the change in kinetic energy by subtracting the initial kinetic energy from the final kinetic energy.

Key Concepts

Conservation of Angular Momentum

In a closed system, the total angular momentum remains constant if no external torques act on it. When the two skaters join hands and reduce their radius, they must conserve angular momentum. This means that the product of their moment of inertia and angular velocity before the change will equal the product after the change, allowing us to analyze their rotational motion.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses 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. As the skaters pull on each other and reduce their radius, their moment of inertia decreases, which affects their rotational kinetic energy, leading to a change that can be calculated.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, dependent on the mass distribution relative to the axis of rotation. For point masses, it is calculated as I = m r², where m is mass and r is the distance from the axis. In this scenario, as the skaters pull closer together, their moment of inertia decreases, which is crucial for determining the change in kinetic energy.