A particle of mass m is describing a circular path of radius...

A particle of mass $m$ is describing a circular path of radius $r$ with uniform speed. If $L$ is the angular momentum of the particle about the axis of the circle, the kinetic energy of the particle is given by [CPMT 1995] (a) $L^{2} / m r^{2}$ (b) $L^{2} / 2 m r^{2}$ (c) $2 L^{2} / m r^{2}$ (d) $m r^{2} L$

A particle of mass $m$ is describing a circular path of radius $r$ with uniform speed. If $L$ is the angular momentum of the particle about the axis of the circle, the kinetic energy of the particle is given by [CPMT 1995]
(a) $L^{2} / m r^{2}$
(b) $L^{2} / 2 m r^{2}$
(c) $2 L^{2} / m r^{2}$
(d) $m r^{2} L$

Key Concepts

Relating Angular Momentum to Kinetic Energy

This involves expressing the kinetic energy of a particle in terms of its angular momentum. By recognizing that for a particle in circular motion the angular momentum (L) and linear velocity (v) are related by L = mvr, one can substitute v = L/(mr) into the kinetic energy expression. This substitution provides a direct connection between rotational and translational forms of energy, enabling calculations when angular momentum is the known quantity.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, typically given by the expression (1/2) times mass times the square of the speed. In the context of circular motion, it quantifies the energy associated with the uniform speed of the particle and is essential in energy conservation analyses.

Angular Momentum

Angular momentum is a measure of the rotational motion of an object about a point or an axis. For a particle moving in a circle, it is defined as the product of the mass, tangential speed, and the radius of the path. This concept links the linear motion of the particle to its rotational effects and is a conserved quantity in isolated systems where no external torques act.

Uniform Circular Motion

This is the motion of a particle along a circular path at constant speed. In such motion, although the speed remains constant, the direction of velocity changes continuously, resulting in centripetal acceleration directed toward the center of the circle. Understanding this concept is crucial to analyzing forces, velocities, and angular quantities in curved trajectories.

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