20.6 Conservation of angular momentum (application)

Questions and their answers are presented here in the module text format as if it were an extension of the theoretical treatment of the topic. The idea is to provide a verbose explanation of the solution, detailing the application of theory. Solution presented here, therefore, is treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

Hints on solving problems

• The first thing in attempting questions, based on conservation of angular momentum, is to know the torques operating on the system. If there is no torque on the system, then we can apply law of conservation of angular momentum in any convenient direction as we choose in accordance with the inputs given in the problem.
• In most of the situations, we find that there is external torque in certain direction. In general, however, we can apply law of conservation of momentum in a particular direction, if net torque on the system has no component in that direction. In the case of rotation, we can apply law of conservation, if the torque is perpendicular to the axis of rotation.
• Application of law of conservation of angular momentum in component form can be used in scalar form as there are only two directions, which can be represented by appropriate sign convention.
• We need to specify direction (assign sign) of angular quantities like angular velocity, momentum and torque with the help of right hand rule vector product rule.
• When the system involves both rotation and translation, then we should assign angular momentum with respect to an axis for rotation and a point for non-rotational motion of particles.
• The force at the axis of rotation is external force on the system. However, this force does not have moment arm and, therefore, does not constitute a torque on the system.

Representative problems and their solutions

We discuss problems, which highlight certain aspects of the conservation of angular momentum. For this reason, questions are categorized in terms of the characterizing features pertaining to the questions :

• Change in the distribution of mass about the axis of rotation
• Conservation of angular momentum about two parallel axes
• System consisting of both rotational and translational motion

Change in the distribution of mass about the axis of rotation

Example 1

Problem : A person sitting on a turn-table is free to rotate about a vertical axis. Initially, he rotates with angular velocity, “ω”, holding two weights in his outstretched hands. The person, then, folds his hand horizontally to move weights towards axis of rotation. In the process, MI of the system of "turn-table, person and weights" is reduced from "I" to "I/2". Find (i) the final angular velocity and (ii) work done by the hands in moving the weights (neglect friction and air resistance).

Rotation of the system