Two identical pendulums of length $L$ are connected by a light spring. They oscillate **in opposite phases** (antiphase). The time period is:
A$2\pi\sqrt{L/g}$
B$2\pi\sqrt{L/(g + 2kL/m)}$ — coupled mode
C$2\pi\sqrt{L/(g - 2kL/m)}$
DEqual to in-phase mode
Answer & Solution
Correct answer: B. $2\pi\sqrt{L/(g + 2kL/m)}$ — coupled mode
Coupled pendulums have 2 normal modes: (1) in-phase (T = $2\pi\sqrt{L/g}$ — spring uninvolved); (2) antiphase (T smaller because spring restoring force adds). For antiphase: effective restoring per unit mass = $g/L + 2k/m$; angular freq $\omega = \sqrt{g/L + 2k/m}$, $T = 2\pi/\omega = 2\pi\sqrt{L/(g + 2kL/m)}$. Classic JEE problem.
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