# Tension in a rotating ring

• ShakyAsh
In summary, the conversation discusses two methods of calculating tension in a ring rotating with constant angular velocity. The first method uses the relationship 2Tsin(dθ/2) = dmωr^2 and the second method uses the work-energy theorem, resulting in the same tension value of T = (mrω^2)/2π. There are doubts about the validity of the second method, but it is justified by dividing the ring into N discrete points and taking the limit as N approaches infinity. The conversation also mentions using this approach for calculating tension in situations involving electro-magnetic fields and rotation.

#### ShakyAsh

Hello people,

So i found out the tension in a ring rotating with constant angular velocity (in gravity free space)

Considering a small element of mass dm - tension will provide the centripetal force,
2Tsin(dθ/2) = dmrω^2
sindθ ≈ dθ
dm = m/2πr ds
ds = rdθ

T = (mrω^2)/2πNow, the other method
K.E. = K = 1/2 Iω^2 = 1/2 mr^2 ω^2

If we increase the radius from r to r+dr, then work done by tension
dW = T d(2πr) = dK
T = 1/2π dK/dr
T = (mrω^2)/2πEven though i get the same result, i have a doubt whether the second method is correct
I know that F=-dU/dr , but whether T=dK/ds , i don't know

Also, i want to know the general approach of calculating tension in situations like electro-magnetic fields, rotation & all.

Regards

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ShakyAsh said:
2Tsin(dθ/2) = dmωr^2
Typo: you mean dmrω^2
Now, the other method
K.E.= K = 1/2 Iω^2 = 1/2 mr^2 ω^2

If we increase the radius from r to r+dr, then work done by tension
dW = T d(2πr) = dK
T = 1/2π dK/dr
T = (mrω^2)/2π
I can't think of a justification for that method. Can you describe your reasoning here?

Yeah dmrω^2 , sorry

As i said, if radius of the ring is increased by dr, then work done by the tangential force tension will be T*(change in circumference) which will be equal to the change in kinetic energy which in this case is the rotational energy.

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Yes, the approach is valid. If you have doubts, you can replace the ring by N discrete points with tension T between them and derive dW=T ds for N->Inf.

Dividing it into N discrete points, why didn't i think of that?

Anyways, thank you very much.
I think i understand it now.

## 1. What is tension in a rotating ring?

Tension in a rotating ring refers to the force that holds the ring together as it rotates. This tension is caused by the centripetal force, which is directed towards the center of the ring.

## 2. How is tension related to the angular velocity of the ring?

Tension in a rotating ring is directly proportional to the angular velocity of the ring. This means that as the angular velocity increases, the tension in the ring also increases.

## 3. What happens to tension if the radius of the ring is increased?

If the radius of the ring is increased, the tension in the ring will decrease. This is because a larger radius means a larger circumference, which requires a lower tension to maintain the same angular velocity.

## 4. How does tension affect the stability of a rotating ring?

Tension plays a crucial role in the stability of a rotating ring. If the tension is too low, the ring may collapse due to the centrifugal force. On the other hand, if the tension is too high, the ring may break due to the high force acting on it.

## 5. Can tension be manipulated in a rotating ring?

Yes, tension in a rotating ring can be manipulated by changing the angular velocity or the radius of the ring. By adjusting these parameters, the tension can be increased or decreased as desired.