Question

In an experiment to determine the period of a simple pendulum of length $1m$, it is attached to different spherical bobs of radii ${r_1}{\text{ and }}{{\text{r}}_2}$. The two spherical bobs have uniform mass distribution. If the relative difference in the periods, is found to be $5 \times {10^{ - 4}}s$ the difference in radii, $\left| {{r_1} - {r_2}} \right|$ is best given by:
(A) $0.01cm$
(B) $0.5cm$
(C) $1cm$
(D) $0.1cm$

Answer 1

Hint In this question a simple pendulum whose relative difference in the time periods of oscillation of two bobs of same mass is given. We have to find the difference in radii of the bobs. We know the relation of the time period of a simple pendulum with length of the pendulum. Use that relation to find the difference in radii of the bobs.

Complete step by step answer
A simple pendulum has a string with a very small mass (sphere ball). It is small but strong enough not to stretch. The mass suspended from the light string that can oscillate when displaced from its rest position.
Pendulums are in common usage such as in clocks, child’s swing and some are for fun. For small displacements, a pendulum is a simple harmonic oscillator.
The pendulum’s time period is proportional to the square root of the length of string and inversely proportional to the square root of acceleration due to gravity.
The time period of a simple pendulum is given by
$T \propto \sqrt {\dfrac{l}{g}} $
$T \propto \sqrt {\dfrac{1}{g}} {\text{ ; }}T \propto \sqrt l $
Where,
T is the time period
l is the length of the pendulum
g is the acceleration due to gravity
Let us take the relation,
$ \Rightarrow T \propto \sqrt l $
$ \Rightarrow T \propto {l^{\dfrac{1}{2}}}$
Differentiating on both sides we get
$ \Rightarrow \Delta T = \dfrac{1}{2}\Delta l{\text{ }} \to {\text{1}}$
Given that,
The relative difference in the periods, is $5 \times {10^{ - 4}}s$
$ \Rightarrow \Delta T = 5 \times {10^{ - 4}}s$
The pendulum is attached to different spherical bobs of radii ${r_1}{\text{ and }}{{\text{r}}_2}$
The change in length is the difference in radii
 $ \Rightarrow \Delta l = \left| {{r_1} - {r_2}} \right|$
Substitute the above in equation 1
$ \Rightarrow \Delta T = \dfrac{1}{2}\Delta l{\text{ }} \to {\text{1}}$
$ \Rightarrow 5 \times {10^{ - 4}} = \dfrac{1}{2}\left| {{r_1} - {r_2}} \right|$
$ \Rightarrow \left| {{r_1} - {r_2}} \right| = 2 \times 5 \times {10^{ - 4}}$
$ \Rightarrow \left| {{r_1} - {r_2}} \right| = 10 \times {10^{ - 4}}m$
$ \Rightarrow \left| {{r_1} - {r_2}} \right| = 0.1 \times {10^{ - 2}}m$
$ \Rightarrow \left| {{r_1} - {r_2}} \right| = 0.1cm$
The difference in radii, $\left| {{r_1} - {r_2}} \right|$ is $0.1cm$

Hence the correct answer is option (D) $0.1cm$

Note The time period of the simple pendulum is directly proportional to the length of the pendulum and inversely proportional to the acceleration due to gravity since acceleration due to gravity does not change the time period of the pendulum depends only on the length of the string of the pendulum.