Two pendulums have time periods `T` and `5T//4`. They start `SHM` at the same time from the mean position After how many oscillations of the smaller pendulum they will be again in the same phase:

Options

A. `5`
B. `4`
C. `11`
D. `9`

Answer

To solve the problem, we need to determine after how many oscillations of the smaller pendulum both pendulums will be in the same phase again.

### Step-by-Step Solution:

1. **Identify the Time Periods**:
- Let the time period of the smaller pendulum be \( T \).
- The time period of the larger pendulum is given as \( \frac{5T}{4} \).

2. **Calculate the Phase Change in One Oscillation**:
- In one complete oscillation, the smaller pendulum completes a phase of \( 2\pi \) radians in time \( T \).
- The larger pendulum completes a phase of \( 2\pi \) radians in time \( \frac{5T}{4} \).
- To find out how much of a complete oscillation the larger pendulum completes in time \( T \):
\[ \text{Phase covered by larger pendulum in time } T = \frac{2\pi}{\frac{5T}{4}} \cdot T = \frac{8\pi}{5} \]

3. **Find the Phase Difference After One Oscillation**:
- After one oscillation of the smaller pendulum, the phase difference between the two pendulums is:
\[ \text{Phase difference} = 2\pi - \frac{8\pi}{5} = 2\pi - 1.6\pi = 0.4\pi \]

4. **Determine the Condition for Same Phase**:
- For both pendulums to be in the same phase again, the total phase difference must be a multiple of \( 2\pi \). This means we need to find \( n \) such that:
\[ n \times 0.4\pi = m \times 2\pi \quad \text{for some integer } m \]
- Rearranging gives:
\[ n \times 0.4 = m \times 2 \quad \Rightarrow \quad n = \frac{m \times 2}{0.4} = 5m \]
- This means \( n \) must be a multiple of 5.

5. **Find the Smallest \( n \)**:
- The smallest positive integer \( n \) that satisfies this condition is \( n = 5 \).

6. **Conclusion**:
- Therefore, after 5 oscillations of the smaller pendulum, both pendulums will be in the same phase again.

### Final Answer:
The smaller pendulum will oscillate **5 times** before both pendulums are in the same phase again.

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