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# Simple Pendulum

The concept of periodic motion is due to the elasticity and constructed by simple harmonic motion. We know that simple harmonic motion consists with their restoring force relates to displacement of mass. The periodic motion includes pendulum and explained by using the terms like frequency and given period of time. The pendulum can be a physical pendulum or simple pendulum.

A simple pendulum is tiny mass which is suspended with string or rod. The suspended rod is also massless. It is similar to resonant system possess resonant frequency. When it is pulled and released then it freely swings with the force of gravity due to inertia which shows motion condition of the object. We know that any object cannot change its position without any exerted force. It can also compound pendulum in which the weight is in an even form on the whole length of the pendulum.

Let’s discuss about simple pendulum, picture representation of a simple pendulum, it’s all properties like frequency, period, gravity and also its period formula with some problems.

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## Simple Pendulum Definition

Simple pendulum is a hypothetical apparatus which consists of a inextensible, light and flexible string having a heavy but small sized sphere called bob tied to its one end. The upper free end is tied to a rigid support (S), called point of suspension. The distance between point of suspension and center of gravity of the bob, is called the length (L) of simple pendulum. The simple pendulum is said to have been suspended. The bob rests in mean position O. The diagram of the simple pendulum is shown below.

## Simple Harmonic Motion Pendulum

When an object oscillates with constant time period even if the amplitude varies, we say it is moving with simple harmonic motion (SHM). The regular oscillation of a pendulum through a small angle are approximately simple harmonic.

The motion of the pendulum has explained here, when the bob is slightly moved to one side (say right), to extreme position A (above figure) and left free, it dose not stay there. It moves towards mean position O with increasing speed. Its speed become maximum at mean position. Due to inertia the bob dose not stay at O but over shot to the other side (left) and continues moving ahead with decreasing speed. The speed become zero at extreme position B, where the bob comes to rest momentarily. From B, the bob return back to O and continues moving towards right extreme A.

From A, the motion is repeated as before. In this way, the bob continues to and fro between A and B with O as mean position. The motion of the bob become an oscillatory motion. We say that the simple pendulum is oscillating.

Due to friction at rigid support and air resistance for the motion of the bob, the extreme points shift inwardly and the oscillation seem to die out and finally the bob comes to rest at mean position O.
• Displacement : At any moment, the distance of bob from mean position, is called displacement. It is a vector quantity.
• Amplitude : Maximum displacement on either side of the mean position, is called amplitude. In the above figure OA and OB measure amplitude.
• Vibration : Motion from the mean position to one extreme, then to other extreme and then back to mean position, is called one vibration.
• Oscillation : Motion from one extreme to other extreme, is called one oscillation. Thus, one oscillation is half vibration.

## Period of a Simple Pendulum

The time taken by the bob to complete one vibration, is called the time period of the pendulum. It is represented by the symbol $T$. The time period of a simple pendulum is expressed by the formula

$T$ = $2\pi \sqrt{\frac{l}{g}}$

where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity.

## Frequency of a Simple Pendulum

The number of vibrations made by the bob in one second, is called the frequency of the pendulum. It is represented by the symbol $\nu$. Its S.I unit is per sec. Which has special name hertz (Hz). Hence,

$\nu$ = $\frac{1}{T}$
or
$\nu T = 1$

## Simple Pendulum Equation

The equation for time period and frequency are

Time period, $T$ = $2\pi \sqrt{\frac{l}{g}}$

Frequency, $\nu$ = $\frac{1}{T}$ = $\frac{1}{2\pi}\sqrt{\frac{g}{l}}$

## Pendulum Gravity

The pendulum is the instrument choice for gravity measurement. A simple pendulum is a mass on the end of a cord or string. Put into motion by raising the mass to the side, the mass is attracted by the earth. The variation in the length and time are depends on the local attraction of gravity.

## Simple Pendulum Problems

The following are the problems of simple pendulum.

### Solved Examples

Question 1: If a pendulum takes 4 sec to swing in each direction, find the length of the pendulum.
Solution:

Solution:
Given $T = 4$ sec
$g = 9.8$ m/s

We know that,  $T$ = $2\pi \sqrt{\frac{l}{g}}$
we can obtain the equation for length ($l$),
$l$ = $\frac{gT^{2}}{4 \pi^{2}}$
=$\frac{9.8 \times 4^{2}}{4 \pi^{2}}$ = $3.97$m

The length of the pendulum is $3.97$.

Question 2: What will be the value of the frequency of simple pendulum in a gravity free space?
Solution:

We know that  $T$ = $2\pi \sqrt{\frac{l}{g}}$

Frequency $\nu$ = $\frac{1}{T}$ = $\frac{1}{2\pi}\sqrt{\frac{g}{l}}$

In gravity free space the value of $g$ is zero.

Hence $\nu = 0$

The frequency in the gravity free space is zero.

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