bullet Angular Momentum of a Turning Rod

In this section, it is illustrated the angular momentum and its conservation when a rod turns with a given angular velocity with respect to a given axis. The rod has a total mass , length , and moment of inertia with respect to the center of mass .

 

Rotational Axis Crossing Center of Mass

The drawing on the left is a top view of the rod turning counterclockwise with respect to an axis crossing the center of mass. The forces acting on the rod are the weight, the normal force, and the forces exerted on the rod by the axis of rotation and . Here, and represent the centripetal forces exerted by the axis of rotation on the left and right sides of the rod. Since these two sides are equal, the magnitudes of  and are the same. The addition of these four forces is not only zero but also there is not net torque acting on the rod when calculated with respect to point .

As shown before, the angular momentum of a solid is independent of the point from where the angular momentum is calculated. Thus, it is constant (conserved) and equal to

from any reference point.

 

Rotational Axis at Point

Now, the drawing on the left represent the same rod turning with respect to point at the distance from the center of mass. If the angular momentum is calculated with respect to this point, , the value is

after using the parallel axis theorem. Still the torque with respect to point is zero and the previous angular momentum is conserved. The rod is turning on top of a table implying that the normal force can be considered to act on the center of mass of the rod. Thus, their contribution to the torque cancel. The forces and   are of different magnitude  because the smaller part of the rod needs an smaller centripetal force in order to maintain the circular motion as compared to the longer part. Nevertheless, these two forces do not produce a torque around an axis of rotation at point  . Notices that these forces are anti-parallel to the corresponding arms of the forces implying that their contribution to the torque vanishes. The last argument is not valid when the point of calculation for the torque is not at the axis of rotation.

In resume, for the rod rotating with constant angular velocity around an axis crossing through the center of mass, the angular momentum is conserved independent of the selected point for calculating the angular momentum and torque. On the other side, when the rotation of the rod is around an axis not going through the center of mass, the angular momentum is conserved only when calculated through a point at the axis of rotation. 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com