Question

Two pendulums of time periods \(3\,s\) and \(7\,s\), respectively, start oscillating simultaneously from opposite extreme positions. After how much time will they be in same phase?

• A. \( \frac{21}{8}\,s \)
• B. \( \frac{21}{4}\,s \)
• C. \( \frac{21}{2}\,s \)
• D. \( \frac{21}{10}\,s \)

Hint:

Opposite extreme start $\Rightarrow$ initial phase difference \(=\pi\).

Answer 1

The Correct Option is A

Concept: Phase difference: \[ \Delta \phi = 2\pi \left(\frac{t}{T_1} - \frac{t}{T_2}\right) \] Since they start from opposite extremes, initial phase difference = \( \pi \) For same phase: \[ \Delta \phi = 2\pi n \]

Step 1: \[ 2\pi \left(\frac{t}{3} - \frac{t}{7}\right) = 2\pi n - \pi \] \[ 2\pi t \left(\frac{4}{21}\right) = \pi(2n-1) \]

Step 2: \[ t = \frac{21}{8}(2n-1) \] Minimum time at \(n=1\): \[ t = \frac{21}{8}\,s \]