Question

Two identical simple pendulums each of length L are connected by a weightless spring as shown in figure. The force constant of the spring is k. In equilibrium, the pendulums are vertical and the spring is horizontal and un-deformed. Find the time of small oscillations of the linked pendulums, when they are deflected from their equilibrium positions through equal displacements in the same vertical plane

(a) in the same direction,

(b) in opposite direction and released.

Answer 1

(a) When both the pendulums are displaced in the same direction by same amount, the spring will neither compress nor stretch, so the restoring torque on each pendulum about the point of suspension will be due to its own weight only.

(b) When both the pendulums are displaced in opposite directions by equal amount (say y), the restoring torque on each pendulum will be due to its own weight and also by elastic force of the spring which is stretched by 2y (=2L sinθ). So the restoring torque on a pendulum about the point of suspension will be