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The length of a simple pendulum executing simple harmonic motion is increased by 21%. The percentage increase in the time period of the pendulum of increased length is.
A. 11%
B. 21%
C. 10.5%
D. 10%

Last updated date: 26th Feb 2024
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IVSAT 2024
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Hint: A basic pendulum is made up of a mass m suspended from a string with a length L and a pivot point P. When the pendulum is displaced to an original angle and released, it will swing back and forth with intermittent motion at the normal frequency.

Complete step by step answer:
In physics, simple harmonic motion is described as repeated movement back and forth through an equilibrium, or centre, location with the maximum displacement on one side equal to the maximum displacement on the other side. Each full vibration has the same time interval.

Simple harmonic motion can be used to model a multitude of motions, but it is best exemplified by the oscillation of a mass on a spring as it is subjected to Hooke's law's linear elastic restoring force. The motion has a single resonant frequency and is sinusoidal in time.

Other phenomenon, such as the motion of a single pendulum, can be modelled using simple harmonic motion, but in order for it to be a reliable model, the net force on the point at the end of the pendulum must be equal to the displacement (and even so, it is only a good approximation when the angle of the swing is small; see small-angle approximation).

Given, \[\dfrac{{{l_2}}}{{{l_1}}} = 1.21\]
The time period of the simple pendulum executing simple harmonic motion is given by the formula
\[{\bf{T}} = 2\pi \sqrt {\dfrac{1}{{\;{\rm{g}}}}} \]
We deduce
\[T \propto \sqrt l \]
From given
\[\dfrac{{{{\bf{T}}_2}}}{{{{\bf{T}}_1}}} = \sqrt {\dfrac{{{{\bf{l}}_2}}}{{{{\bf{l}}_1}}}} = \sqrt {{\bf{1}}.{\bf{21}}} \]
\[\Rightarrow {{\bf{T}}_2} = {\bf{1}}.{\bf{1}}{{\bf{T}}_{\bf{2}}}\]
The % increase in time period is given as = \[\dfrac{{{{\bf{T}}_2} - {{\bf{T}}_1}}}{{{{\bf{T}}_1}}} \times {\bf{100}}\]
\[\therefore {\rm{ increase = }}\dfrac{{{\bf{1}}.{\bf{1}}{{\bf{T}}_1} - {{\bf{T}}_1}}}{{{{\bf{T}}_1}}} \times {\bf{100}} = {\bf{10}}\% \]

Hence, option D is correct.

Note: Simple harmonic motion is a form of periodic motion in dynamics and physics in which the restoring force on a moving object is directly proportional to the degree of the object's displacement and operates towards the object's equilibrium state. It causes an oscillation that, if not interrupted by friction or other energy dissipation, will last forever.
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