# Determine if angular momentum is conserved given potential energy funtions?

• ENT
In summary, the conversation discusses the conservation of quantities such as energy, momentum, and angular momentum for various potential energy functions in three dimensions. The participants also consider the impact of using different reference points for measuring moments and how the position vector r plays a role in determining the conservation of angular momentum.
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.

Last edited:
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?

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.

ENT said:
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?

To determine if angular momentum is conserved, we need to look at the potential energy functions and see if they have any dependence on angular position or angular velocity. If they do, then angular momentum would not be conserved.

Let's look at each potential energy function individually:

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

This potential energy function does not have any dependence on angular position or angular velocity. Therefore, angular momentum would be conserved.

2. U = α/r

This potential energy function does have a dependence on angular position, as it is inversely proportional to the distance r from the origin. Therefore, angular momentum would not be conserved.

3. U = β(z(hat) dotted with r)^2

This potential energy function also has a dependence on angular position, as it involves the dot product of the position vector r and the unit vector in the z direction. Therefore, angular momentum would not be conserved.

4. U = α/r + β(z(hat) dotted with r)^2

This potential energy function has both a dependence on angular position and angular velocity. Therefore, angular momentum would not be conserved.

In summary, only the first potential energy function would conserve angular momentum. The other three would not conserve angular momentum due to their dependence on angular position or velocity.

## 1. What is angular momentum?

Angular momentum is a physical quantity that measures the rotational motion of a system. It is the product of the moment of inertia and the angular velocity of an object.

## 2. How is angular momentum conserved?

Angular momentum is conserved in a closed system, meaning that it does not change in the absence of external torque. This is due to the law of conservation of angular momentum, which states that the total angular momentum of a system remains constant.

## 3. How do potential energy functions affect angular momentum conservation?

Potential energy functions can affect angular momentum conservation by changing the rotational motion of a system. For example, if the potential energy function is asymmetric, it can cause a change in the direction of rotation of a system, affecting the angular momentum.

## 4. Can angular momentum be conserved in the presence of external forces?

Yes, angular momentum can still be conserved in the presence of external forces as long as the net external torque acting on the system is zero. In this case, the angular momentum will remain constant.

## 5. How can we determine if angular momentum is conserved given potential energy functions?

To determine if angular momentum is conserved, we need to calculate the total angular momentum of a system before and after a potential energy function is applied. If the values are the same, then angular momentum is conserved. However, if there is a change in the rotational motion, it indicates that angular momentum is not conserved.

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