Determine if angular momentum is conserved given potential energy funtions?

[ENT]

Homework Statement

For each of the following potential energy functions in three dimensions, what quantities are conserved (energy, momentum, angular momentum)?

Homework Equations

U = k/2(x^2 + y^2)

U = α/r

U = β(z(hat) dotted with r)^2

U = α/r + β(z(hat) dotted with r)^2

Where z(hat) is the unit vector in the z direction.

The Attempt at a Solution

So I understand that in order for the angular momentum to be conserved the cross product of r and F, being the force obtained from the negative gradient of U, must be equal to zero. However I am not sure what to use for the position vector r.

 

[haruspex]

Suppose you measure moment about a point v for a force F acting at r. You know Fxr=0. What will the moment about v be?

 

[ENT]

We don't know that F x r = 0. We are trying to determine whether it is or not, and I am not sure what to use for r.

 

[haruspex]

We don't know that F x r = 0. We are trying to determine whether it is or not, and I am not sure what to use for r.

Yes, sorry, I was thinking specifically of the U = a/r case, but forgot to say so.
As I read that question, U is specified in relation to a coordinate system where r is the position vector, and the field it defines will be symmetric about the origin. Hence Fxr will be 0.
For any other reference point v, what is the vector from v to the point r? If the force of the field at r is F(r), what will be its moment about v?