A cylinder with moment of inertia I1 rotates about a...

A cylinder with moment of inertia $I_{1}$ rotates about a vertical, frictionless axle with angular speed $\omega_{i}$ . A second cylinder, this one having moment of inertia $I_{2}$ and initially not rotating, drops onto the first cylinder (Fig. Pl1.29). Because of friction between the surfaces, the two eventually reach the same angular speed $\omega_{f} .$ (a) Calculate $\omega_{f}$ (b) Show that the kinetic energy of the system decreases in this interaction, and calculate the ratio of the final to the initial rotational energy.

Key Concepts

Moment of Inertia

The moment of inertia is the rotational analogue of mass in linear motion and quantifies how resistant a body is to changes in its rotational state about a given axis. In problems involving multiple rotating bodies, the individual moments of inertia combine to influence the system's final angular velocity after an interaction or collision.

Inelastic Rotational Collisions

In such collisions, the colliding bodies stick together and end up rotating as a single system. Although angular momentum is conserved in these processes, the collision is inelastic because some of the initial kinetic energy is lost, typically converted into heat due to friction between the bodies.

Conservation of Angular Momentum

This principle states that in an isolated system where no external torques act, the total angular momentum remains constant. In the scenario described, friction acts internally between the two cylinders when they come into contact, so the angular momentum before and after they stick together must be equal, even though the kinetic energy changes.

Rotational Kinetic Energy

Rotational kinetic energy is the energy a rotating object possesses due to its rotation, calculated as one half times the moment of inertia times the square of the angular velocity. In inelastic rotational interactions, some of this kinetic energy is dissipated (typically as heat) despite the conservation of angular momentum, resulting in a decrease in the system’s total rotational kinetic energy.

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