A turntable of radius \(25 \mathrm{cm}\) and rotational inertia \(0.0154 \mathrm{kg} \cdot \mathrm{m}^{2}\) is spinning freely at 22.0 rpm about its central axis, with a \(19.5-\mathrm{g}\) mouse on its outer edge. The mouse walks from the edge to the center. Find (a) the new rotation speed and (b) the work done by the mouse.

When we discuss the dynamics of rotational motion, the moment of inertia is a key concept, akin to the role of mass in linear motion. It quantifies an object's resistance to changes in its rotational speed, and knowing this resistance value is crucial for solving many physics problems involving rotation.

Moment of inertia is heavily influenced by two factors: the mass of the objects and how this mass is distributed relative to the axis of rotation. It's described mathematically as the sum of the products of the mass elements and the square of their distance from the rotation axis: \( I = \sum m_i r_i^2 \). In our textbook example, the turntable and mouse system initially has a larger moment of inertia because the mouse, a mass element, is located at the edge, at a distance from the axis. Once the mouse moves to the center, the overall moment of inertia decreases, as this redistribution of mass decreases the resistance to rotational acceleration.

Demonstrating the Dependency

The exercise improvement advice here is to visualize this concept. For instance, if you hold a spinning bicycle wheel with its axle horizontal and then tilt the axle, you will feel resistance. This is because of the moment of inertia, which depends on both the mass and the distribution of the mass.

Angular momentum conservation is a principle that helps us predict the behavior of rotating systems. It's vital in situations where no external torques are acting on a system, meaning the total angular momentum remains constant over time. This principle can be stated as \( L_1 = L_2 \) or \( I_1 \omega_1 = I_2 \omega_2 \), where \(I_1\) and \(\omega_1\) are the initial moment of inertia and angular velocity, respectively, and \(I_2\) and \(\omega_2\) are the final values.

In our turntable example, as the mouse moves towards the center, there are no external torques, thereby the angular momentum of the system is conserved. Initially, the system had a higher moment of inertia and a lower angular velocity. As the mouse reaches the center, the moment of inertia decreases, and to conserve angular momentum, the angular velocity increases.

Real-World Application

Ice skaters use this principle to control their spin. When a skater pulls in their arms, they reduce their moment of inertia and spin faster to conserve angular momentum. This illustrates the inverse relationship between the moment of inertia and angular velocity in the absence of external torques.

Just as objects in linear motion have kinetic energy due to their mass and velocity (\( KE = \frac{1}{2}mv^2 \)), objects in rotational motion have rotational kinetic energy because of their moment of inertia and angular velocity. The formula for rotational kinetic energy is \( KE = \frac{1}{2}I\omega^2 \), where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

In the turntable scenario, the change in the mouse's position does not only alter the system's angular velocity due to conservation of angular momentum; it also changes the system's rotational kinetic energy. As the turntable's rotation speeds up, its rotational kinetic energy increases, even though no external forces or energies are added to the system. The work done by the mouse, which involves moving against rotational inertia, is what changes the energy of the system. Therefore, the mouse is essentially converting its own energy to increase the turntable's rotational kinetic energy.

Understanding Energy Transfer

For deeper comprehension, consider a diver tucking into a spin during a dive. Their tucked position minimizes the moment of inertia, increasing angular velocity, and thus increasing rotational kinetic energy as they spin towards the water. This conservation and transformation of energy are at the heart of rotational dynamics and critical to understanding motion in a rotating frame.