Two pendulums `X` and `Y` of time periods `4 s` and `4.2s` are made to vibrate simultaneously. They are initially in same phase. After how many vibration of `X`, they will be in the same phase again ?

Options

A. `30`
B. `25`
C. `21`
D. `26`

Explanation

To solve the problem of determining after how many vibrations of pendulum X will both pendulums X and Y be in the same phase again, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Time Periods**:
- Time period of pendulum X, \( T_X = 4 \, \text{s} \)
- Time period of pendulum Y, \( T_Y = 4.2 \, \text{s} \)

2. **Calculate the Angular Frequencies**:
- The angular frequency \( \omega \) is given by the formula \( \omega = \frac{2\pi}{T} \).
- For pendulum X:
\[ \omega_X = \frac{2\pi}{T_X} = \frac{2\pi}{4} = \frac{\pi}{2} \, \text{rad/s} \]
- For pendulum Y:
\[ \omega_Y = \frac{2\pi}{T_Y} = \frac{2\pi}{4.2} \approx \frac{2\pi}{4.2} \approx 1.49 \, \text{rad/s} \]

3. **Determine the Time for the Next Phase Coincidence**:
- The time \( T \) for both pendulums to be in the same phase again can be calculated using the formula:
\[ T = \frac{T_X \cdot T_Y}{T_Y - T_X} \]
- Substituting the values:
\[ T = \frac{4 \cdot 4.2}{4.2 - 4} = \frac{16.8}{0.2} = 84 \, \text{s} \]

4. **Calculate the Number of Vibrations of Pendulum X**:
- The number of vibrations \( N_X \) of pendulum X in time \( T \) is given by:
\[ N_X = \frac{T}{T_X} = \frac{84}{4} = 21 \]

5. **Conclusion**:
- Therefore, after 21 vibrations of pendulum X, both pendulums X and Y will be in the same phase again.

### Final Answer:
The number of vibrations of pendulum X after which both pendulums will be in the same phase again is **21**.