A ring and a disc of different masses are rotating with the same kinetic energy. If we apply a retarding torque `tau` on the ring, it stops after making n revolution. After how many revolutions will the disc stop, if the retarding torque on it is also `tau` ?

Options

A. `(n)/(2)`
B. `n`
C. `2n`
D. Data insufficient

Explanation

To solve the problem, we need to analyze the situation involving the ring and the disc that are rotating with the same kinetic energy and are subjected to the same retarding torque.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
- The kinetic energy (KE) of a rotating object is given by the formula:
\[ KE = \frac{1}{2} I \omega^2 \]
- Where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

2. **Setting Up the Problem**:
- Let the kinetic energy of the ring be \(KE_R\) and the kinetic energy of the disc be \(KE_D\).
- According to the problem, \(KE_R = KE_D\).

3. **Applying Retarding Torque**:
- When a retarding torque \(\tau\) is applied, the work done by the torque is equal to the change in kinetic energy.
- For the ring:
\[ \text{Work done} = \tau \cdot \theta_R = \Delta KE_R = KE_R - 0 = KE_R \]
- For the disc:
\[ \text{Work done} = \tau \cdot \theta_D = \Delta KE_D = KE_D - 0 = KE_D \]

4. **Relating Angular Displacement to Revolutions**:
- The angular displacement in terms of revolutions is given by:
\[ \theta = n \cdot 2\pi \]
- Therefore, for the ring:
\[ \tau \cdot (n \cdot 2\pi) = KE_R \]
- And for the disc, let the number of revolutions be \(n'\):
\[ \tau \cdot (n' \cdot 2\pi) = KE_D \]

5. **Equating Work Done**:
- Since \(KE_R = KE_D\), we can equate the work done by the torque on both the ring and the disc:
\[ \tau \cdot (n \cdot 2\pi) = \tau \cdot (n' \cdot 2\pi) \]

6. **Canceling Common Terms**:
- We can cancel \(\tau\) and \(2\pi\) from both sides:
\[ n = n' \]

7. **Conclusion**:
- Therefore, the number of revolutions \(n'\) that the disc makes before stopping is equal to \(n\):
\[ n' = n \]

### Final Answer:
The disc will also stop after making \(n\) revolutions.

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