# Conservation of angular momentum - tetherball vs charged particle

1. Dec 26, 2011

### gabdolce

Hey all,

I was wondering if one of you could help me out with a debacle I'm having.

I'm having trouble reconciling what exactly happens between an orbiting tetherball and the separate scenario of a orbiting particulate charge.

With the particulate charge: you can, given the velocity and magnetic field magnitude, determine the radius of the orbit of the charged particle.

F=ma=mv^2/r=qvB
r=mv/qB

In the case of varying B-field magnitudes, the only thing that changes is the radius of the orbit . The magnetic force is ALWAYS perpendicular to the velocity vector and thus can NEVER do work and NEVER (de)accelerate the spinning particle. Velocity of this particle must remain constant when the only perturbation can be altering the strength of the magnetic field.

Here is my confusion: As far as I can see, there is NO net torque on the system, therefore, there must be conservation of angular momentum.

Let's consider an INCREASING B field magnitude: According to L=mvr, since r is consequently getting smaller, shouldn't the velocity of this particle be increasing as well?!?!

This would make sense, especially since when looking at a combination of F=mv^2/r and mvr, you get F=L/r^3 which further corroborates that the centripetal force here is getting larger (as we expected from the increasing of B)...

Is anyone else confused by this as well?

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Let's now look at a mechanical scenario: a tetherball circling the pole and getting shorter and shorter.

Can I use the conservation of angular here too and say that since the radius of orbit is getting shorter and shorter, then the tetherball must also be accelerating as well?

2. Dec 26, 2011

### rcgldr

In the case of the moving charge, whatever is generating the field will eventually experience an equal but opposing force to the force the field applies to the moving charge. I'm not sure how to take into account that the moving or accelerating charge is generating it's own field.

In the case of the tetherball, angular momentum is conserved only if you consider the torque being applied to the pole that the rope winds or unwinds from. Assuming the pole is attached to the earth, you'd have to include the earth as part of the closed system for angular momentum to be conserved.

If you change the tetherball case to a two dimensional situation, such as a puck sliding on a frictionless plane (and no aerodynamic drag) attached to a string that winds or unwinds from a pole, then the path is involute of circle, with the string always perpendicular to the path of the puck, so it's speed remains constant. Link to wiki article that includes an animated unwinding example:

http://en.wikipedia.org/wiki/Involute

3. Dec 26, 2011

### gabdolce

I'm not quite sure how your comment on the charge is relevant? The opposing forces you're talking about which I assume to be instances of Faraday's/Lenz's Laws apply to fluxes varying in time. Let's consider a situation where you instantaneously increased the [uniform] B field so that the radius of the orbiting charge grows smaller. My original question still remains.

With the tetherball, if you do include the torque from the pole transmitted through the rope, then angular momentum would NOT be conserved...

4. Dec 27, 2011

### rcgldr

I'm not sure about a situation where the B field changes instantly. If you assume that angular momentum is preserved in a closed system, then if the charged particle is losing angular momentum, then the device generating the field and whatever the device is attached to (the earth), must be gaining angular momentum, so somehow there would be a torque feed back via the field.

Angular momentum is conserved if you include whatever the pole is attached to as part of a closed system. Instead of the earth, imagine the pole is attached to a large disc resting on a frictionless plane in a vacuum (no aerodynamic drag). The disc will accelerate in the same direction as the rope is wrapped around the pole as long as there is tension in the rope. The angular momentum of this closed system, disc, pole, rope, ball, would be conserved.

Last edited: Dec 27, 2011
5. Dec 27, 2011

### granpa

a changing magnetic field will produce an electric field that will accelerate the electron

6. Dec 28, 2011

### gabdolce

I've already mentioned that the electron cannot be accelerated because there is no work done in the direction of motion. F produced by the magnetic field B will always be perpendicular to the particle (let's just say it's a positron for clarity of discussion).

I'm going to be assuming that the magnetic field produced by the circling positron will be big enough to exactly offset the increase in applied B (judging from the equation of the B=Mi/2r form)... [Does anyone know if this has any merit? Doesn't this necessarily have to be true to avoid the induced circular emf granpa mentioned, that would ultimately indeed accelerate the positron?]

So I guess what I'm trying to resolve is how there is no change in velocity of the circling positron even when the radius has changed (gotten smaller when subjected to a stronger B field).

So are you saying for the first example: let's say the thing creating the B field in the first place is a giant solenoid so that the variation between B from the center of the solenoid to a distance along the radius is negligible. You're saying that the solenoid must now be moving in a way that compensates for the change in angular momentum on the positron?

7. Dec 28, 2011

### gabdolce

So then can you say whether or not this tetherball will be accelerating or not?

8. Dec 28, 2011

### rcgldr

The closed system would include the solenoid and whatver it was attached to (the earth). If the solenoid and positron were in outer space free from any external forces, then the closed system would consist of the solenoid and the positron, and angular momentum would be conserved.

I'm not sure about a tetherball. I don't know if there is a component of tension in the rope in the direction of the spiral path of the ball in this 3d case. In the 2d case of a puck sliding on a frictionless plane attached to a string that winds or unwinds from a post, then the speed of the puck remains constant unless it collides with the post.

Last edited: Dec 29, 2011
9. Dec 28, 2011

### granpa

and I just told you that that is only true for an unchanging magnetic field

10. Dec 29, 2011

### gabdolce

Are you referring to the induced emf that would accompany a *changing* magnetic field?