A circular disc is rotating about its own axis at uniform angular velocity `omega`. The disc is subjected to uniform angular retardation by which its angular velocity is decreased to `omega//2` during 120 rotations., The number of rotations further made by it before coming to rest is

Options

A. 120
B. 60
C. 40
D. 20

Answer

To solve the problem step by step, we will follow the principles of angular motion.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Initial angular velocity, \( \omega \)
- Final angular velocity after 120 rotations, \( \omega_f = \frac{\omega}{2} \)
- Number of rotations during the retardation, \( \theta = 120 \) rotations

2. **Convert Rotations to Radians:**
- Since we are dealing with angular motion, we need to convert rotations to radians.
- \( \theta = 120 \text{ rotations} = 120 \times 2\pi \text{ radians} = 240\pi \text{ radians} \)

3. **Use the Angular Motion Equation:**
- We can use the equation of motion for angular displacement:
\[ \omega_f^2 = \omega_i^2 + 2\alpha\theta \]
- Plugging in the values:
\[ \left(\frac{\omega}{2}\right)^2 = \omega^2 + 2\alpha(240\pi) \]
- This simplifies to:
\[ \frac{\omega^2}{4} = \omega^2 + 480\pi\alpha \]

4. **Rearranging the Equation:**
- Rearranging gives:
\[ 480\pi\alpha = \frac{\omega^2}{4} - \omega^2 \]
- This simplifies to:
\[ 480\pi\alpha = -\frac{3\omega^2}{4} \]
- Therefore, we can express angular retardation \( \alpha \):
\[ \alpha = -\frac{3\omega^2}{1920\pi} \]

5. **Finding Further Rotations Until Rest:**
- Now, we need to find how many additional rotations the disc makes before coming to rest.
- Initial angular velocity for this phase is \( \omega_i = \frac{\omega}{2} \) and final angular velocity \( \omega_f = 0 \).
- Using the same angular motion equation:
\[ 0 = \left(\frac{\omega}{2}\right)^2 + 2\alpha\theta' \]
- Substituting \( \alpha \):
\[ 0 = \frac{\omega^2}{4} + 2\left(-\frac{3\omega^2}{1920\pi}\right)\theta' \]

6. **Rearranging to Find \( \theta' \):**
- Rearranging gives:
\[ 2\left(-\frac{3\omega^2}{1920\pi}\right)\theta' = -\frac{\omega^2}{4} \]
- This simplifies to:
\[ \theta' = \frac{\frac{\omega^2}{4}}{\frac{6\omega^2}{1920\pi}} = \frac{1920\pi}{24} = 80\pi \]

7. **Convert Radians Back to Rotations:**
- Converting \( \theta' \) back to rotations:
\[ \theta' = \frac{80\pi}{2\pi} = 40 \text{ rotations} \]

### Final Answer:
The number of additional rotations made by the disc before coming to rest is **40 rotations**.

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