Question

Figure 15-26shows three physical pendulums consisting of identical uniform spheres of the same mass that are rigidly connected by identical rods of negligible mass. Each pendulum is vertical and can pivot about suspension point O. Rank the pendulums according to their period of oscillation, greatest first.

Short answer

The ranking of pendulums according to period of oscillation is ${\mathrm{T}}_{\mathrm{b}}>{\mathrm{T}}_{\mathrm{c}}>{\mathrm{T}}_{\mathrm{a}}$.

Step-by-step solution

The given data 

The figure for the three pendulums is given.

Understanding the concept of SHM of a particle

We can predict the period of the pendulum B from torques acting on it. Then comparing arm lengths of A and C we can rank them according to their periods.

Formula:

The period of an oscillation of a pendulum, $T=2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}}}$ (i)

Calculation of the ranking of pendulums according to period of oscillation

Let’s denote pendulums in the given figure as A, B and C respectively from left to right.

Since equal torques is acting on both the sides of pendulum B, it will have infinite period of oscillation. So, from equation (i), we say that it will havethegreatest period amongst the three.

The time period of pendulum is directly proportional to its arm length. The pendulum C has greater arm length than that of the other pendulums.

Thus, the time period for pendulum C is greater than that for pendulum A.

Hence, we can rank the pendulums as ${\mathrm{T}}_{\mathrm{b}}>{\mathrm{T}}_{\mathrm{c}}>{\mathrm{T}}_{\mathrm{a}}$.