Two pendulum of time periods 3 s and 7 s respectively start oscillating simultaneously from two opposite extreme positions. After how much time they will be in same phase?

Options

A. `(21)/(8) s`
B. `(21)/(4) s`
C. `(21)/(2)` s
D. `(21)/(10) s`

Explanation

To solve the problem of finding the time after which two pendulums with time periods of 3 seconds and 7 seconds will be in the same phase, we can follow these steps:

### Step 1: Understand the Problem
We have two pendulums that start oscillating from opposite extreme positions. Their time periods are given as:
- \( T_1 = 3 \) seconds
- \( T_2 = 7 \) seconds

### Step 2: Determine Angular Frequencies
The angular frequency \( \omega \) of a pendulum is related to its time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
For the first pendulum:
\[ \omega_1 = \frac{2\pi}{3} \]
For the second pendulum:
\[ \omega_2 = \frac{2\pi}{7} \]

### Step 3: Set Up the Phase Condition
Since the pendulums start from opposite extreme positions, their phases can be represented as:
- For pendulum 1: \( \phi_1 = \omega_1 t + \frac{\pi}{2} \)
- For pendulum 2: \( \phi_2 = \omega_2 t - \frac{\pi}{2} \)

To find when they are in the same phase, we set:
\[ \phi_1 = \phi_2 \]
This gives us:
\[ \omega_1 t + \frac{\pi}{2} = \omega_2 t - \frac{\pi}{2} \]

### Step 4: Rearranging the Equation
Rearranging the equation, we get:
\[ \omega_1 t - \omega_2 t = -\pi \]
This can be simplified to:
\[ (\omega_2 - \omega_1) t = \pi \]
Thus, we can solve for \( t \):
\[ t = \frac{\pi}{\omega_2 - \omega_1} \]

### Step 5: Substitute the Angular Frequencies
Substituting the values of \( \omega_1 \) and \( \omega_2 \):
\[ t = \frac{\pi}{\left(\frac{2\pi}{7} - \frac{2\pi}{3}\right)} \]

### Step 6: Simplify the Expression
Calculating the difference:
\[ \frac{2\pi}{7} - \frac{2\pi}{3} = 2\pi \left(\frac{1}{7} - \frac{1}{3}\right) \]
Finding a common denominator (which is 21):
\[ \frac{1}{7} = \frac{3}{21}, \quad \frac{1}{3} = \frac{7}{21} \]
Thus:
\[ \frac{1}{7} - \frac{1}{3} = \frac{3}{21} - \frac{7}{21} = -\frac{4}{21} \]
Substituting back:
\[ t = \frac{\pi}{2\pi \left(-\frac{4}{21}\right)} = \frac{21}{8} \]

### Step 7: Final Answer
Since time cannot be negative, we take the absolute value:
\[ t = \frac{21}{8} \text{ seconds} \]

### Conclusion
The time after which both pendulums will be in the same phase is \( \frac{21}{8} \) seconds.

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