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 gyroscpoes. 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

Question:

What is the total angular momentum L_tot of a system of particles with individual angular momenta L₁ = <0,0,10>, L₂ = <0,0,-5>, & L₃ = <0,0,15>?

Answer:

L_tot = <0,0,20>

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?

Connectedness

This concept is highly interesting because of its part in describing highly complex systems and interactions on various scales, as multi-component systems exist on all scales from quantum to cosmic. Being in MSE, it may be of interest to investigate materials as multi-component systems, and to apply the angular momentum principle in this manner.

History

See: History of the Angular Momentum Principle

References

Matter & Interactions, Vol. I: Modern Mechanics, 4nd Edition by R. Chabay & B. Sherwood (John Wiley & Sons 2015)