bullet Angular Momentum of a Particle moving in a Circle

Considering a particle of mass moving in a circle of radius at a constant rate. That is maintaining constant the magnitude of the velocity but changing the direction. Such motion requires of a force, , directed to the center of the circle, the centripetal force.

In this motion, if the angular momentum is calculated with respect to the center of the circle, , the angular momentum is unchanged, . The angular momentum is given by

where the radius of the circle, the magnitude of the linear momentum of the particle, and is the unit vector pointing out of the page. Every one of the quantities involved are constant and, consequently, the angular momentum is unchanged or conserved (constant). Because and are anti-parallel the torque with respect to point is zero at all times, confirming that the angular momentum should be unchanged when calculated respect to point .

However, the same calculation with respect to another point, such as point , results in a changing angular momentum. In this case,

where is given above and is calculated below.

From the diagram, , with and .

Thus, and which implies that the angular momentum with respect to point is

but and the angular momentum becomes .

This angular momentum correspond to a torque with respect to point obtained from where as discussed above. Therefore, the torque becomes with , , implying that .  The result for the torque with respect to point is

 

 

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