Question

If the amplitude of the oscillation of a simple pendulum is increased by $30\%$, then the percentage change in its time period will be:
(A). $90\%$
(B). $60\%$
(C). $30\%$
(D). $\text{zero}$

Answer 1

Hint: Time period of oscillation of a simple pendulum is the time taken by a pendulum to finish one full oscillation. The only thing that affects the time period is length of string and acceleration due to gravity. It can be calculated using the formula $T=2\Pi \sqrt{\dfrac{l}{g}}$.

Formula used:
$T=2\Pi \sqrt{\dfrac{l}{g}}$

Complete step by step solution:
Time period of a simple pendulum is defined as the time taken by a pendulum to complete one full oscillation and the Amplitude of the pendulum is the distance travelled by the pendulum from equilibrium position to one extreme side.
Time period of oscillation of a simple pendulum is given by the formulae $T=2\Pi \sqrt{\dfrac{l}{g}}$
Where, $l$=length of string of pendulum
$g$=acceleration due to gravity
Time period of oscillation of a simple pendulum does not depend upon the mass of the oscillating object and amplitude of oscillation. It only depends upon the length of the pendulum and acceleration due to gravity.
Percentage change in time period of the oscillation will be zero.
Hence, the correct option is D.

Additional information:
From a simple pendulum equation, we can calculate maximum velocity and maximum acceleration at the equilibrium position.
$\begin{aligned} & {{V}_{\text{max}}}=A\omega \\\ & {{a}_{\text{max}}}=-A{{\omega }^{2}} \\\ \end{aligned}$
Here $A$ is the amplitude of oscillation
If $A$ is increased by $30\%$ then percentage change in the maximum velocity ${{V}_{\text{max}}}$ will be $30\%$ and percentage change in maximum acceleration will also be $30\%$ because from the above equations we can see that maximum velocity and maximum acceleration of oscillation of a simple pendulum is linearly dependent on the amplitude of the oscillation.

Note: Students should not get confused between the amplitude of oscillation and displacement covered by the mass while oscillation. Increasing the amplitude means there is a larger distance to cover but at the same time restoring force experienced by the mass also increases, due to which acceleration increases proportionally. This implies that mass can travel a greater distance at a greater speed and hence changing the amplitude of oscillation leaves no effect on the time period of oscillation of a simple pendulum.