Motion of Simple Pendulum

Simple Pendulum:

A simple pendulum is nothing but a small, heavy body suspended from a rigid support with the help of a long string. The heavy body remains in its lowest position A when the string is vertical. This position OA is called the position of equilibrium of the pendulum. The heavy body is called the pendulum bob. The support (O) from which the bob is suspended is called the point of suspension. The centre of gravity of the suspended bob is called the point of oscillation.

If a small, heavy body, suspended from rigid support by a long, weightless, and inextensible string, can be set into oscillation, then the arrangement is called a simple pendulum.

Motion of Simple Pendulum:

A simple pendulum of effective length L is oscillating with an angular amplitude not exceeding 4°. The bob of the pendulum is oscillating from B to C on either side of its position of equilibrium.

Let at any instant of motion, the bob of mass m be at P and its displacement from the position of equilibrium OP = x. If the angular displacement is θ, then θ = x/L rad, provided θ is small and sinθ ≈ θ ≈ tanθ.

At P, the weight mg of the bob acts vertically downwards. The component mgsinθ tries to bring the bob to the position of equilibrium. As this force acts in a direction opposite to that of displacement, it is the restoring force, expressed as,

F = -mgsinθ
= -mgθ [since θ is less than 4°]
= -mg.(x/L)

Now, the acceleration of the bob,

a = F/m

= (-g/L)x

= -ω2x [where, ω = √(g/L)]

As the motion of the bob obeys the equation, a = -ω2x, it can be said that the motion of a simple pendulum with an angular amplitude less than 4° is simple harmonic.