Shown below is a small particle of mass 20 g that is moving...

Shown below is a small particle of mass 20 g that is moving at a speed of $10.0 \mathrm{m} / \mathrm{s}$ when it collides and sticks to the edge of a uniform solid cylinder. The cylinder is free to rotate about its axis through its center and is perpendicular to the page. The cylinder has a mass of $0.5 \mathrm{kg}$ and a radius of $10 \mathrm{cm},$ and is initially at rest. (a) What is the angular velocity of the system after the collision? (b) How much kinetic energy is lost in the collision?

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy describes the energy of a rotating system and is given by the expression ½ times the moment of inertia multiplied by the square of the angular velocity. It is used to assess the energy associated with rotation and to identify any loss of mechanical energy during events like inelastic collisions.

Inelastic Collision

An inelastic collision is one in which the colliding bodies stick together after the impact, meaning that while momentum (or angular momentum in rotational cases) is conserved, kinetic energy is not. This concept is fundamental when analyzing collisions where some energy is transformed into other forms rather than remaining entirely as mechanical energy.

Conservation of Angular Momentum

This principle states that if no external torques act on a system, the total angular momentum remains constant before and after an event like a collision. It is crucial in analyzing rotational collision problems, as it allows the computation of final angular speeds based on the initial conditions of the colliding bodies.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion and depends on the mass distribution relative to the axis of rotation. In systems involving composite objects, such as a rigid body combined with a point mass, the total moment of inertia is determined by summing the individual moments, which plays a key role in relating angular momentum to angular velocity.

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