Two pendulums having time period `T` and `(5T)/(4)`. They start`S.H.M.` at the same time from mean position. what will be the phase difference between them after the bigger pendulum has completes one oscillation ?

Options

A. `45^(@)`
B. `90^(@)`
C. `60^(@)`
D. `30^(@)`

Explanation

To solve the problem of finding the phase difference between two pendulums with time periods \( T \) and \( \frac{5T}{4} \), we can follow these steps:

### Step 1: Understand the Time Periods
The time periods of the two pendulums are given as:
- Pendulum 1: \( T_1 = T \)
- Pendulum 2: \( T_2 = \frac{5T}{4} \)

### Step 2: Determine the Frequency of Each Pendulum
The frequency \( f \) of a pendulum is the reciprocal of the time period \( T \):
- For Pendulum 1:
\[ f_1 = \frac{1}{T} \]
- For Pendulum 2:
\[ f_2 = \frac{1}{\frac{5T}{4}} = \frac{4}{5T} \]

### Step 3: Calculate the Angular Frequencies
The angular frequency \( \omega \) is given by \( \omega = 2\pi f \):
- For Pendulum 1:
\[ \omega_1 = 2\pi f_1 = \frac{2\pi}{T} \]
- For Pendulum 2:
\[ \omega_2 = 2\pi f_2 = \frac{2\pi \cdot 4}{5T} = \frac{8\pi}{5T} \]

### Step 4: Determine the Phase Change After One Oscillation of the Bigger Pendulum
When the bigger pendulum (Pendulum 2) completes one full oscillation, it takes time \( T_2 = \frac{5T}{4} \). During this time, the smaller pendulum (Pendulum 1) will complete:
\[ \text{Number of oscillations of Pendulum 1} = \frac{T_2}{T_1} = \frac{\frac{5T}{4}}{T} = \frac{5}{4} \]
This means that Pendulum 1 completes \( 5/4 \) of an oscillation while Pendulum 2 completes 1 oscillation.

### Step 5: Calculate the Phase Difference
The phase difference \( \Delta \phi \) can be calculated based on the number of oscillations completed:
- Each complete oscillation corresponds to a phase change of \( 2\pi \) radians.
- Therefore, the phase change for \( \frac{5}{4} \) oscillations is:
\[ \Delta \phi = \frac{5}{4} \times 2\pi = \frac{5\pi}{2} \]
However, we need the phase difference relative to one complete oscillation of Pendulum 2:
\[ \Delta \phi \text{ (mod } 2\pi) = \frac{5\pi}{2} - 2\pi = \frac{\pi}{2} \]

### Conclusion
The phase difference between the two pendulums after the bigger pendulum has completed one oscillation is:
\[ \Delta \phi = \frac{\pi}{2} \text{ radians or } 90^\circ \]

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