# Angular momentum problem.

1. Aug 29, 2007

### cosmic_tears

Hi :). I'll just start describing the problem...
An ant with mass m is moving anti-clockwise on a Disk with a perpendicular rotation axis (with no friction). The disk's radius is R and it's moment of inertia - I.
The velocity of the ant *in relation to the Earth* is V while the disk is rotating *clock-wise* in an angular velocity W0 (that's omega zero).
The ant suddenly stops.
The questiong is - what's the angular velocity of the disk (with the ant on it) after it stops.

I know this problem involves angular momentum preservation equations, but I obviously do not conclude the right one, since there are four possible answers, and none fits my answer.

First of all, some questions:
1. The disk starts rotating because the ant start walking on it, right? As the ant continues walking - wouln't the disk's angular velocity increase? It apears it remains constant, since there's no time given.
2. if the ant is causing the rotation, why does the disk continue to rotate when it stops walking?
3. If I want to write the angular momentum preservation equation - should I relate to the ant's velocity compared to the Earth (an inertial system) or compared to the disk?

This is what I've tried:
While the ant's walking, the angular momentum of the disk + ant is:
I*W0 + R*m*V
After stopping, the angular momentum is:
I*W
Where W (omega) is the final angular velocity.
Therefore I*W0 + R*m*V = I*W.

Finding W from this equation did not satisfy any option :-\

I'd really appreciate help.
Tomer.

2. Aug 29, 2007

### Staff: Mentor

Could be--but the problem doesn't specify that disk and ant started from rest.
If the ant increases his speed, then the speed of the disk would increase. But if he keeps going at constant speed, so will the disk.
But you don't know that the disk started from rest. The ant could have dropped onto the disk while it was moving!
Measure velocities with respect to the inertial frame of the Earth.

Excellent. The only change I would make is with signs: the disk moves clockwise (+, say) and the ant moves counter-clockwise (-), so:
I*W0 - R*m*V
What happened to the ant??? What's the speed of the ant when he "stops"? (The ant stops walking with respect to the disk, but that doesn't mean his speed with respect to the Earth is zero.)
Redo this, taking into consideration the final speed of the ant. (Hint: Express the ant's speed in terms of W.)

3. Aug 29, 2007

### cosmic_tears

First of all, thank you.
I did what you suggested. Obviously, it worked :). I added r*m*wr to the right side of the equation and I got one of the answers I needed :)

But, theoratically, something still doesn't settles:
Say the Disk was rotating in a certain velocity. before the ant was walking on it. Then the ant starts walking in a const. velocity - it changes the angular velocity of the disk to preserve angular momentum.
But - say it suddenly stops. Why doesn't the angular velocity returns back to the original one? The one that was before the ant started walking? If the movement of the ant is the cause for the change of ang. velocity of the disk, why then, when the ant stops, it doesn't change the ang. velocity back to "normal" (I'm not assuming it rested before - I'm assuming it had a const. angular velocity)

Anyway, thanks again, that definitelly did the trick, and the "physics" is pretty clear to me.
:) Bless you.
Tomer.

4. Aug 29, 2007

### Staff: Mentor

OK. So the ant is "resting" on the disk: both are spinning with the same angular velocity.
Yes, now the ant starts walking.
Who says it doesn't? It does! If the ant and disk spun together at some angular speed before the ant started walking, it will return to that same speed when the ant stops walking.

5. Aug 29, 2007

### cosmic_tears

Doc Al,
I actually wrote another question now, and then deleted it, because I think I've realized what I'm missing.
W0 is the ang. velocity when the ant is walking on the disk, not the original ang. velocity (vefore the ant started walking). So what they're asking in the question is, in other words, what was the original ang. velocity of the disk, right? According to my intuition, and your agreement, it equals the final ang. velocity, when the ant stops...

So, the disk spins, an ant start walking on it, then it stops, and the disk is back to it's original speed. As if the ant didn't change anything, but is there only for calculations :)

I thank you so much :)

(oh, correct me if I'm wrong again please )

Tomer.

6. Aug 29, 2007

### Staff: Mentor

I think you have it exactly right.
Right.
If things started out with the ant just riding on the disk and not walking, spinning at some original angular speed, then yes, the final angular speed will equal the original angular speed. Makes sense to me!

But we don't really know how the ant got there or if he was not walking at some time. For example, he could have been dropped onto the disk after it was spinning! (Yes, I'm a nitpicker.)

All we know is that at some point the disk was spinning at some angular rate and the ant was walking at some speed. And we can calculate the final speed when the ant stops walking.

Yes. If the disk spins with the ant riding it, then the ant walks, then the ant stops again--the disk will be back at its first speed.

Note that we assume that the ant is walking along the edge of the disk. What would happen if the ant ended up at a different distance from the center? (Like R/2.)

7. Aug 29, 2007

### cosmic_tears

Good questions Al! What would happen? :)

No, seriously: I would guess that since the ant can only add (or do nothing) to the final angular velocity of the disk, that, in this case, the final angular velocity would be larger than the original. I'm too tired to analyze it physically, but... maybe tomorow :)

I feel I understand this problem completely now, from all aspects. Thank you so much!

8. Aug 29, 2007

### Staff: Mentor

A good way to view things, when ant and disk move together at the same angular speed, is in terms of total rotational inertia.

How does the total rotational inertia (of "ant + disk") change if the ant moves towards the center? Away from the center? That--and conservation of angular momentum--should tell you the answer.

My pleasure. Good job!