Simple Pendulum

A simple pendulum consists of a concentrated mass known as the ‘bob’, which is free to swing from side to side at the end of a light cord or rod. When bob is given a small angular displacement and released, the pendulum bob appears to describe simple harmonic motion (SHM). It doesn’t travel along a perfectly straight line, but for small angular displacements it is approximately straight, with a central equilibrium position.

If the mass is given a small displacement (r) from its equilibrium position and then released, it will oscillate about the equilibrium position with amplitude ‘r’ as shown in figure 1 (a). At some instant when the bob is at distance ‘x’ from its equilibrium position, and accelerating towards it, the forces acting on it will be as shown in figure 1 (b). It is assumed that the bob oscillates with free or natural vibrations, i.e. that the air resistance is negligible and that there are no energy losses at the point of suspension.

Equating forces in the direction of motion at the instant shown gives:

Inertia force = Component of weigh in direction of motion

ma  = mg sin θ
a = g sin θ

Simple Pendulum

Figure 1: Simple pendulum

But

1
This is of the form:

Acceleration  = Constant × Displacement
It shows that for small displacements, the simple pendulum describes SHM.
Acceleration of the system is also given by the general formula:

Acceleration equation in terms of angular velocity and distance

The natural circular frequency of the system ω rad s-1 is thus given by the formula:

angular velocity equation in terms of acceleration due to gravity and length

The periodic time of the natural vibrations is given by the formula:

time period equation in terms of acceleration due to gravity
The natural frequency of the vibrations is the reciprocal of periodic time is given by the formula:

natural frequency in terms of acceleration due to gravity and length

In the mass-spring systems there is always some air resistance.
There is also some energy losses at the point of suspension in the simple pendulum and the combined effect would cause the vibrations to die away.

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