Angular Momentum of a Falling Particle

An application of the definition of angular momentum is to calculate the angular momentum of a falling particle under the influence of gravity.

 Let us select the vertical axis along the direction of motion of the falling particle as shown in the graph of the left, . The angular momentum of the particle of mass with respect to point is where the linear momentum of the particle. Point is located at the coordinate of the positive -axis. The direction of the corresponding angular momentum vector is pointing out of the page, . However, if  point is located on the negative side of the -axis, , the corresponding direction of the vector angular momentum is directed inside the page, . On the other hand, if the selected location for point is at the origin of coordinates, , the corresponding angular momentum is zero. The magnitude of the angular momentum with respect to point is given by with but . Therefore, the magnitude of the angular momentum is but from the diagram and the magnitude of the angular momentum becomes In free fall, the velocity of the particle changes due to the acceleration of gravity. Consequently, the angular momentum also changes. Notices that, if the angular momentum is calculated with respect to the origin, the angular momentum is zero at all times.

In all the other cases, the change in the magnitude of the angular momentum is due to the torque acting on the particle. The torque is with . When point is located in the positive side of the -axis, the direction of the torque is out of the page, into the page when point is in the negative side of the -axis, and zero when placed at the origin of coordinates. The magnitude of the torque is

but the force acting on the particle is the weight of it, . Thus, the torque is

This result corresponds to the change with the time associated with the angular momentum,

where the value corresponding to the location of point is independent of the time.

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