Question

Two pendulums with identical bobs and lengths are suspended from a common support such that in rest position the two bobs are in contact (Figure). One of the bobs is released after being displaced by 10° so that it collides elastically head-on with the other bob.

1. Describe the motion of two bobs.
2. Draw a graph showing variation in energy of either pendulum with time, for 0 ≤ t ≤ 2T, where T is the period of each pendulum.

Answer 1

a. Consider the adjacent diagram in which the bob B is displaced through an angle θ and released.

At t = 0, suppose bob B is displaced by θ  = 10° to the right. It is given potential energy E₁ = E. Energy of A, E₂ = 0

When B is released, it strikes A at t = T/4. In the head-on elastic collision between B and A comes to rest and A gets the velocity of B. Therefore. E₁ = 0 and E₂ = E. At t = 27T/4, B reaches its extreme right position when KE of A is converted into PE = E₂ = E. Energy of B, E₁ = 0.

At t = 3T/4, A reaches its mean position. when its PE is converted into KE = E₂ = E. It collides elastically with B and transfers the whole of its energy to B. Thus, E₂ = 0 and E₁ = E. The entire process is repeated.

b. The values of energies of B and A at different time intervals are tabulated here. The plot of energy with time 0 ≤ t ≤ 2T is shown separately for 8 and A in the figure below.

Time (t)	Energy of A (E1)	Energy of B (E2)
0	E	0
T/4	0	E
2T/4	0	E
3T/4	E	0
4T/4	E	0
5T/4	0	E
6T/4	0	E
7T/4	E	0
8T/4	E	0