Angular Momentum of Multiparticle Systems

By Jonathan Peraza (jonperaza)

Main Idea

The Angular Momentum Principle may be extended to a multiparticle system to provide various insights, such as an explanation for the counter-intuitive behavior of Gyroscopes. This page details how one could derive the angular momentum principle of a multiparticle system and provides a few examples to illustrate the concept.

Mathematical Model

The derivation of the angular momentum principle of a multiparticle system is most easily understood when following a simple example, in this case we will work with a three particle system:

The angular momentum principle is written for each individual particle relative to location A, with position vectors r and external force vectors F.

The individual equation is then summed. Note that, due to the reciprocation of forces, the internal forces f cancel out.

The equation is now rewritten with the right side, representing net torque, as τ_net,A

Computational Model

Glowscript Trinket Example

Glowscript Example

Examples

Simple

Difficult

Each of the six particles in the figure have identical mass m and moves in the x-y plane with the same speed v as indicated in the diagram. Take the location of particle 5 as the reference point for all angular momentum calculations when answering the following questions.Be sure to express your answer as a vector.

What is the momentum of the individual particles?

Consider the particles above as a single multiparticle system. With respect to particle 5, what is the total angular momentum of the system? What is the translational angular momentum? What is the rotational angular momentum?

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