Why Is the Total Angular Momentum of a Particle System Constant?

[ElDavidas]

hey everyone, I've been working through a past paper and I'm stuck on a question:

"(a) Consider a system consisting of n distinct particles P1, P2, : : :, Pn, with masses m1, m2, ..., mn and position vectors r1, r2, : : :, rn, relative to the origin O of an inertial frame, respectively.

For each i = 1, 2, : : :, n, suppose that the only forces acting on Pi are internal forces Fij , which always act along the line joining Pi and Pj , for j = 1, 2, : : :, n and j doesn't equal i.

Assume that Fij = ¡Fji for i = 1, 2, : : :, n, j = 1, 2, : : :, n and i doesn't equal j.

Define what is meant by the total angular momentum of the system about O.

Show, as a consequence of Newton's second law, that the total angular momentum of the system about O is constant."

I can define the total angular momentum of the system, it's the 2nd part of the Q I have trouble with.

Thanks

 

[Tide]

Use the definition of angular momentum:

$$\vec L = \vec r \times \vec p$$

Take the time derivative and sum over all particles noting that

$$\frac {d \vec p}{dt} = \vec F$$

You will find $d\vec L /dt = 0$.