Question

In an experiment with a bar pendulum having four holes, the same time period is recorded when it is suspended at distances $ 12cm $ , $ 24cm $ , $ 40cm $ and $ 52cm $ respectively from one end. The length of the bar pendulum is
(A) $ 84cm $
(B) $ 72cm $
(C) $ 64cm $
(D) $ 60cm $

Answer 1

Hint : If the same time period is recorded for two different lengths measured from one end of the bar pendulum, then the lengths of the pendulum from those points will be equal when measured from the opposite sides of the pendulum. Use this idea to approach the problem and find the length of the pendulum.

Complete step by step answer
The time period of oscillation of a pendulum is given by
 $\Rightarrow T=2\pi \sqrt{\dfrac{l}{g}} $
Here, $ l $ is the distance from the pivoted end to the base of the pendulum and $ g $ is the acceleration due to gravity.
From this equation, we can see that the time period is dependent on the length from the pivoted end alone. So, when it is given that the time period for different lengths of the bar pendulum measured from the same end of the bar pendulum is equal, it means that the distance from the pivoted end to the base of the pendulum is equal for those measurements. So, we can say that the distance $ 12cm $ measured from one end is equivalent to the distance $ 52cm $ measured from the other end, and the distance $ 24cm $ measured from one end is equivalent to the distance $ 40cm $ measured from the other end. If this is the case, then
The total length of the pendulum,
 $ \Rightarrow L=52cm+12cm $
Or it can also be written as
 $ \Rightarrow L=40cm+24cm $
And both result in the total length value $ 64cm $ .
So, as the total length of the bar pendulum is computed to be $ 64cm $ , option (C) is the correct answer.

Note
Here, we have assumed the bar pendulum to act as a simple pendulum. The bar pendulum can be used to study compound pendulum behavior also. A compound pendulum is a pendulum where the dimensions of the whole suspended body cannot be neglected and the center of mass of the suspended body is not concentrated at the end of the suspended part of the pendulum.