A simple pendulum, in its simplest form, consists of heavy bob suspended at the end of a light in-extensible and flexible string. The other end of the string is fixed at O, as shown in fig:
Let, L = Length of the string
m = Mass of the bob in kg.
W = Weight of the bob in newtons = m.g
θ = Angle through which the string is displaced
When the bob is at A, the pendulum is in equilibrium position. If the bob is brought to B or C and released, it’ll start oscillating between the two position B and C, with A as the mean position. It has been observed that if the angle θ is very small. Now, the couple tending to restore the bob to the equilibrium position or restoring torque.
T = m.g sinθ x L
Since angle θ is very small, therefore sinθ = θ radians.
We know that the mass moment of inertia of the body about an axis through the point of suspension.
I = mass x (length)2 = mL2
∴ Angular Acceleration of the string,
α = T/I = m.g.L.θ/m.L2 = g.θ/L
or, θ/α = L/g
Angular displacement/Angular Acceleration = L/g
We know that the periodic time,
tp = 2π√(Displacement/Acceleration) = 2π√(L/g)
Frequency of Oscillation,
n = 1/tp = 1/2π√(g/L)
Laws of Simple Pendulum:
1. Law of Isochronism: It states “The time period (tp) of a simple pendulum does not depend upon its amplitude of vibration and remains the same. It provides the angular amplitude (θ) doesn’t exceed 4°”.
2. Law of Mass: It states “The time period (tp) of a simple pendulum doesn’t depend the mass of the body suspended at the free end of the string”.
3. Law of Length: It states “The time period (tp) of a simple pendulum is directly proportional to √L, where L is the length of the string”.
4. Law of Gravity: It states “The time period (tp) of a simple pendulum is inversely proportional to to √g, where g is the acceleration due to gravity”.