If we put a particle into a box and allow it to freely move, why isn't angular momentum conserved?

Question

Whereas if the particle were in a sphere, angular momentum *would* be conserved...?

Answer 1

Yes - good observation, but note that this applies to angular momentum about the center of the sphere only (it's not necessarily conserved about any other point).

Angular momentum conservation is a direct result of rotational symmetry. More precisely this means that the laws of motion for the particle look the same in every cartesian coordinate system centered at the same origin (the point around which the angular momentum is calculated). So clearly it's conserved in the case of the sphere but not in the case of the box. The connection between symmetry and conservation laws is best understood in the context of quantum mechanics.

A more pedestrian way of looking at it is to see whether or not there's a torque when the particle collides with the wall: The force on the particle is always normal to the wall (assuming frictionless collisions). As long as this direction is parallel to the radius vector from the origin, there's no torque (since torque is r x F) and consequently no change in angular momentum.

In the case of the sphere, the normal to the wall is always along the radius vector so there's no torque and no change in angular momentum. In the case of the box, it's clearly possible for the line connecting the collision point and the coordinate origin to be in different directions so the torque can be nonzero and almost always is.

If any of this isn't clear or you have any other questions, send me an e-mail.

Answer 2

Why would you suppose that angular momentum is not conserved? Actually, angular momentum is ALWAYS conserved.

Answer 3

Please do your own homework....