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# If the mass of the bob of a simple pendulum increases its time period will :(A) Increases(B) Decreases(C) Remain unaffected (D) Neither of these

Last updated date: 20th Jun 2024
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Hint: The time period of the pendulum can be calculated and we can get form the expression of the time period as that the time period is independent of the mass of the pendulum and hence the value of the time period remains unaffected. Analogous to how mass of a body doesn’t affect its acceleration due to gravity.

Complete Step by step solution
A simple pendulum is a mechanical arrangement that demonstrates periodic motion.
Generally, a pendulum consists of a small bob of mass m suspended by a thin string that is tied at its upper end of length L.

The time period of a simple pendulum is defined as the time required by the pendulum to complete one oscillation and it is denoted by T.

By using the equation of motion we get $T - mg\cos \theta = m{v^2}L$
The torque acting to bring the mass to its equilibrium position is given as
Torque $t = mgL \times \sin \theta = mg\sin \theta \times L = I \times \alpha \;$
Therefore $I\alpha = \; - mgL\theta$ , since for small angle of oscillation $\sin \theta = \theta$

Then $\alpha = \; - \dfrac{{mgL\theta }}{I} = \; - {\omega ^2}\theta$
Hence from the above equation, we get ${\omega ^2} = \dfrac{{mgL}}{I}$
Also since $I = M{L^2}\;$ we get $\omega = \;\sqrt {\dfrac{g}{L}\;}$
Therefore the time period of the simple pendulum is given by $T = \dfrac{{2\pi }}{\omega } = 2\pi \sqrt {\dfrac{L}{g}}$
From the equation of the time period of a simple pendulum, we can say that the time period of a simple pendulum does not depend on the mass of the bob.
Therefore the time period of the simple pendulum does not get affected by the mass of the bob.
Hence if the mass of the bob increases, the time period of the pendulum remains unaffected.

Hence the correct option is C.
Note: We may think that as the mass of the pendulum increases the value of the time period may increase as the weight is increased but it is not true and we have derived the time period of the pendulum and finally cleared a common misconception
Not just here but the mass of the body doesn’t play a role in a lot of other scenarios. In general, for the bodies in free fall the time period involved in the motion is independent of the mass of the bodies. Eg for the body going in an orbit around another body because of gravity.