Angular momentum conserved for central forces not at origin?

In summary: So, in summary, for a central force at the origin, the angular momentum is constant and conserved due to the derivative rxF being zero. However, if the force is at a different point, the angular momentum must be calculated with respect to that point and may not be conserved if there is a torque exerted by the force. This can be determined through the use of Noether's theorem and the definition of angular momentum with respect to a different point.
  • #1
OmegaKV
22
1
My textbook says that for a central force at the origin, the angular momentum is constant, because the derivative rxF is zero since F points radially outwards so it is in the same direction as r. Ok, but what about the angular momentum about a point other than the origin, or the angular momentum with respect to the origin of a central force that is not located at the origin? Would angular momentum still conserved in these cases?
 
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  • #2
What would it mean if angular momentum were not conserved in those cases?

Can you do the maths to see what would happen?
 
  • #3
Of course then you have to relate the angular momentum to the new point of rotational symmetry. So you have to define
$$\vec{J}=(\vec{r}-\vec{r}_0) \times \vec{p},$$
if ##\vec{r}_0## is the position vector of the point around which the situation is rotatinal symmetric (isotropic). It's very obvious in view of Noether's theorem.
 
  • #4
OmegaKV said:
Angular momentum with respect to the origin of a central force that is not located at the origin?
Momentum is conserved in a closed or isolated system. If there is a central force not located at the origin, then from the frame of the origin, that force must be exerting a torque on something with respect to the origin, and would need to be taken into account.
 

1. What is angular momentum?

Angular momentum is a physical quantity that describes the rotational motion of an object around a fixed point or axis. It is often represented by the symbol L and is equal to the product of an object's moment of inertia and its angular velocity.

2. What does it mean for angular momentum to be conserved?

Conservation of angular momentum means that the total angular momentum of a system remains constant, even as individual components may change. In other words, the angular momentum before an interaction or event is equal to the angular momentum after the interaction or event.

3. How is angular momentum conserved for central forces?

Angular momentum is conserved for central forces because these types of forces act on an object in such a way that the direction of the force is always perpendicular to the direction of motion. This results in no torque being applied to the object, which means its angular momentum remains constant.

4. Why is angular momentum not conserved for non-central forces?

Non-central forces can apply a torque on an object, which can change its angular momentum. This is because the direction of the force may not always be perpendicular to the direction of motion, causing a change in the object's rotational motion.

5. Is angular momentum always conserved for central forces not at the origin?

No, angular momentum is not always conserved for central forces not at the origin. If the central force is not symmetrical or changes over time, then the direction of the force may not always be perpendicular to the direction of motion, resulting in a change in angular momentum. However, if the central force is symmetrical and remains constant, then angular momentum will still be conserved.

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