Simple Pendulum

What is called a Pendulum?

A pendulum is defined as a weight tied at one end of a string which is suspended from a pivot so that it can swing freely under the force of gravity.

Weight attached to the string is called bob of pendulum. The point from which the pendulum is suspended is called point of suspension. Centre of mass of the bob is called point of oscillation. The distance between the point of suspension to the point of oscillation is called length of pendulum.

The Italian scientist Galileo first noticed the periodic nature of swinging lamp in Pisa cathedral which becomes the basis for development of a simple pendulum.

When a pendulum is in rest, its bob remains in equilibrium position with its string being vertical. When the bob is displaced sideways from equilibrium position, it is subject to a restoring force due to gravity, that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the bob causes it to oscillate about the equilibrium position.

Types of Pendulum

There are various types of pendulum in use. These are as follows –

1. Simple pendulum – A simple pendulum consists of a bob suspended from a support and the bob oscillates in one dimension. This type of pendulum is the most common and can be seen in metronomes and seismo-meters.
2. Seconds pendulum – A pendulum used in wall clocks is called a seconds pendulum because its time period is of 2 seconds.
3. Foucault pendulum – Foucault pendulum is a type of simple pendulum in which bob swings in two dimensions. This is used for demonstration of the rotation of earth.
4. Double pendulum – A double pendulum consists of two simple pendulums, one suspended from the other. It is also called a chaotic pendulum. Double pendulums are used primarily in mathematical simulations.
5. Conical Pendulum – If a pendulum is fixed at one end and the bob is rotating in a horizontal circle, then it is called a conical pendulum. This pendulum is used as a model to analyze the motion of planets.
6. Compound pendulum – A compound pendulum has an extended mass like a swinging bar and is free to oscillate about a horizontal axis.
7. Kater’s pendulum – It is a special type of compound pendulum used to measure the value of earth’s acceleration due to gravity.

Simple Pendulum

A point mass when suspended from a rigid support by a long, flexible, inelastic and weightless string and is made to oscillate within a small angle of oscillations in circular arc is called a simple pendulum.

The oscillations of bob of a simple pendulum are simple harmonic motion.

Characteristics of Pendulum

A simple pendulum must possess certain essential characteristics so that, it can oscillate with simple harmonic motion. These characteristics are –

1. The suspended mass or bob is spherical in shape and should be made of homogeneous material. The effective weight of the bob is considered to be acting at a point located at the geometrical centre of mass of the bob.
2. The string of the pendulum should be fine with negligible mass so that the centre of mass of the pendulum coincides with centre of mass of the bob.
3. String of the bob should be flexible so that, no other force acts on the bob and oscillations should be considered only due to the force of gravity.
4. Angle of oscillations of the bob are kept very small so that the path of motion of the bob can assumed as a straight line.
5. Oscillations of the bob must be in one vertical plane.

Oscillations of Simple Pendulum

Consider about a simple pendulum as shown in figure. In the equilibrium position, the bob lies vertically below the point of suspension.

If the bob is slightly displaced on either side and released, it begins to oscillate about its equilibrium position.

Suppose, at any instant the bob lies at extreme position A . Then forces acting on the bob are –

1. Weight ( mg ) of the bob which is acting vertically downwards.
2. Tension ( T ) of the string acting along the string.

Weight of the bob, ( mg )  has two components –

1. Component ( mg \cos \theta ) is providing tension in the string.
2. Component ( mg \sin \theta ) provides net force on the bob and tends to bring it back to equilibrium position.

   

Therefore, restoring force acting on the bob will be –

F = - m g \sin \theta

= - m g \left ( \theta - \frac { \theta ^ 3 }{ 3 ! } + \frac { \theta ^ 5 }{ 5 ! } - .... \right )

= - m g \ \theta \left ( 1 - \frac { \theta ^ 2 }{ 6 } + \frac { \theta ^ 4 }{ 120 } - .... \right ) where \theta is in radians.

If, ( \theta ) is very small, then its higher powers are neglected.

Therefore, \quad F = - m g \ \theta ……… (1)

But for the bob, ( mg ) is a constant.

Hence, \quad F \propto \theta

Let, ( l ) is the length of the pendulum and ( x ) is the displacement of bob at an instant. Then swing angle \quad \theta = \frac { arc }{ radius } = \left ( \frac { x }{ l } \right )

From equation (1), we will get –

F = - m g \left ( \frac { x }{ l } \right )

= - \left ( \frac { m g }{ l } \right ) x ………. (2)

Therefore, \quad F \propto x

Because, \left ( \frac {m g}{l} \right ) is a constant term for a given pendulum.

From equations (1) and (2) we see that, the force on the bob is proportional to the displacement and acts in opposite direction. Hence, the pendulum executes simple harmonic motion.

Force constant

For a simple harmonic motion, restoring force is given as –

F = - k x

But, from equation (2), we will get –

F = - \left ( \frac { m g }{ l } \right ) x

Therefore, force constant is given by –

k = \left ( \frac {m g}{l} \right )

Time period

By definition, time period of oscillations will be –

T = 2 \pi \sqrt { \frac { m }{ k }}

Or, \quad T = 2 \pi \sqrt { \frac { m }{ \left ( \frac {m g}{l} \right ) }}

Or, \quad T = 2 \pi \sqrt { \frac { l }{ g }}

Acceleration

Force acting on bob is –

F = - m g \left ( \frac { x }{ l } \right )

But, from definition of force ( F = m a ) where ( a ) is the acceleration.

Therefore, \quad a = - \left ( \frac { g }{ l } \right ) x = \omega ^ 2 x

Therefore, acceleration ( a ) of the bob is proportional to the displacement ( x ) and directed opposite to it.

See numerical problems based on this article.

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