Question

A pendulum is hung from the roof of a sufficiently high building and is moving freely to and fro like a simple harmonic oscillator. The acceleration of the bob of the pendulum is $20m{s^{ - 2}}$ at a distance of 5m from the moving position. The time period of oscillation is?
(A) 2s
(B) $2\pi s$
(C) 1s
(D) $\pi s$

Answer 1

Hint: Pendulum is a weight suspended from a fixed point in such a way that it can swing. A simple harmonic oscillator consists of a mass, which experiences a force $F$ which pulls the mass in its equilibrium position.
A simple pendulum is a simple harmonic oscillator for a small angle.

Formula used:
$a = {\omega ^2}x$ ;
This is the equation of motion for a simple harmonic oscillator. Where ‘$a$’ is acceleration, $\omega $ is angular frequency and $x$ is the displacement from the mean position.
$\omega = \dfrac{{2\pi }}{T}$ ;
This is the relation between angular frequency and time period of oscillation.

Complete step by step solution:
So according to the question the pendulum is displaced with a small angle so that it performs harmonic oscillation. The equation of motion for harmonic oscillation is given below.
$a = {\omega ^2}x$
Where ‘$a$’ is acceleration, $\omega $ is the angular frequency and $x$ is the displacement from the mean position.
Following information is given in the question, $a=20m{s^{ - 2}}$ and $x=5m$.
Now let us use the formula $a = {\omega ^2}x$ and substitute the values in it.
$20 = {\omega ^2}5$
Let us further simplify it.
${\omega ^2} = \dfrac{{20}}{5} = 4$
$ \Rightarrow \omega = 2{s^{ - 1}}$
As we got the value of angular frequency, let us use the formula $\omega = \dfrac{{2\pi }}{T}$ to find the time period.
The time period is the time taken by the pendulum to complete one complete rotation or cycle.
Substitute the values and on solving we get a time period.
$2{s^{ - 1}} = \dfrac{{2\pi }}{T}$
$ \Rightarrow T = \pi s$

$\therefore $ The time period of oscillation is $\pi s$. Option (D) is the correct answer.

Additional information:
When a body is suspended from a fixed point with the help of a string in such a way that it can move to and fro, we call it a pendulum.
In the case of a simple pendulum, we assume all its mass is in the bob. The string with which it is hung is mass-less. With this pendulum, it is easy to study harmonic motion.

Note:
A simple pendulum is a simple harmonic oscillator when the restoring force acting on it is directly proportional to the displacement. Also for a pendulum to act as a simple harmonic oscillator displacement angle should be small.