What's the angular velocity of the disk

In summary, the question posed is what is the angular velocity of the disk with an ant on it after the ant suddenly stops walking on it. The answer can be found by using the angular momentum preservation equation, taking into consideration the speed and direction of the ant's movement in relation to the disk. This will result in one of the four possible answers given. Additionally, it is important to measure velocities with respect to the inertial frame of the Earth. Ultimately, the final angular velocity of the disk will be the same as its original velocity before the ant started walking on it.
  • #1
cosmic_tears
49
0
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.
Thanks for reading.
Tomer.
 
Physics news on Phys.org
  • #2
cosmic_tears said:
First of all, some questions:
1. The disk starts rotating because the ant start walking on it, right?
Could be--but the problem doesn't specify that disk and ant started from rest.
As the ant continues walking - wouln't the disk's angular velocity increase? It apears it remains constant, since there's no time given.
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.
2. if the ant is causing the rotation, why does the disk continue to rotate when it stops walking?
But you don't know that the disk started from rest. The ant could have dropped onto the disk while it was moving!
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?
Measure velocities with respect to the inertial frame of the Earth.

This is what I've tried:
While the ant's walking, the angular momentum of the disk + ant is:
I*W0 + R*m*V
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
After stopping, the angular momentum is:
I*W
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.)
Where W (omega) is the final angular velocity.
Therefore I*W0 + R*m*V = I*W.
Redo this, taking into consideration the final speed of the ant. (Hint: Express the ant's speed in terms of W.)
 
  • #3
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
cosmic_tears said:
But, theoratically, something still doesn't settles:
Say the Disk was rotating in a certain velocity. before the ant was walking on it.
OK. So the ant is "resting" on the disk: both are spinning with the same angular velocity.
Then the ant starts walking in a const. velocity - it changes the angular velocity of the disk to preserve angular momentum.
Yes, now the ant starts walking.
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?
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
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.


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
I think you have it exactly right.
cosmic_tears said:
W0 is the ang. velocity when the ant is walking on the disk, not the original ang. velocity (vefore the ant started walking).
Right.
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...
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! :wink: (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.

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
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
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
cosmic_tears said:
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 :)
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.

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

What is angular velocity?

Angular velocity is a measure of how fast an object is rotating around a fixed point. It is usually represented by the Greek letter omega (ω) and is measured in radians per second or degrees per second.

How is angular velocity calculated?

Angular velocity is calculated by dividing the change in angular displacement by the change in time. It can also be calculated by multiplying the rotational speed by the radius of the object.

Is angular velocity the same as linear velocity?

No, angular velocity and linear velocity are two different types of velocity. Linear velocity is a measure of how fast an object is moving in a straight line, while angular velocity is a measure of how fast an object is rotating around a fixed point.

What is the difference between angular velocity and angular acceleration?

Angular velocity is a measure of how fast an object is rotating, while angular acceleration is a measure of how fast an object's angular velocity is changing over time. It is represented by the Greek letter alpha (α) and is measured in radians per second squared or degrees per second squared.

How does angular velocity affect the motion of a disk?

The angular velocity of a disk determines how fast it is rotating and therefore affects its rotational motion. A higher angular velocity will result in a faster rotation, while a lower angular velocity will result in a slower rotation. Additionally, changes in angular velocity can also impact the stability and balance of the disk.

Similar threads

  • Introductory Physics Homework Help
2
Replies
45
Views
2K
  • Introductory Physics Homework Help
Replies
9
Views
891
  • Introductory Physics Homework Help
Replies
30
Views
2K
  • Introductory Physics Homework Help
Replies
3
Views
706
  • Introductory Physics Homework Help
Replies
1
Views
990
  • Introductory Physics Homework Help
Replies
1
Views
844
  • Introductory Physics Homework Help
Replies
28
Views
3K
  • Introductory Physics Homework Help
Replies
8
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
18
Views
1K
Back
Top