SIMPLE PENDULUM

“ A system of a small massive body suspended by a light, inextensible string from a rigid ( fixed ) support and capable of oscillating in one vertical plane only is known as a simple pendulum.”

Mass of the pendulum, m, is supposed to be concentrated at the center of the suspended body called bob of the pendulum as shown in the figure. 

The distance of the center of the bob from the point of suspension A is called the length ( l ) of the simple pendulum. At some instant, the bob of the pendulum is at B and the string makes an angle θ with the vertical.

The pendulum oscillates on the circular arc of radius l in a vertical plane as shown in the figure.

Two forces act on the bob of the pendulum.

( 1 ) Weight of the bob = mg, in the downward direction

( 2 ) tension in the string T’, in the direction BA.

The torque about A due to T’ is zero as its line of action passes through A. The torque due to weight, mg, about A is

 t = l × m g = - l mg sin θ

But, t = I α = m l ²α and α = dω₀ / dt = d²θ / dt²

Therefore m l ² [d²θ / dt²] = - l mg sin θ … … … … … … ( 1 )

For small θ ( in radian ), linear displacement of the bob on the curved path is x and sin θ ≈ θ = x / l

Putting this value of sin θ in equation ( 1 ), we get

m l ² {d ² ( x / l )} / dt²  = - l mg ( x / l )

Therefore d ²x / dt ² = - gx / l This is the differential equation of SHM

Hence g / l  = ω₀ ² = 4 p² / T ²

Therefore T = 2p Ö l / g

This is the expression of the period of the simple pendulum. Its value does not depend on the mass of the bob of the pendulum.

•           The period of the spring-block type of SHO does not change when taken to a different planet, as the values of m and k appearing in the expression of its period do not change.

•           The period of simple pendulum increases on a planet where the value of ‘g’ is less and the pendulum clock taken there loses time, whereas its period decreases on the planet where the value of ‘g’ is more and the pendulum clock gains time when taken to that planet.