As you can see, when the weight of the pendulum bob is resolved, the tension of the string, T, and the mg cos x cancel each other out, leaving mg sin x as the net force, as shown above. This force is responsible for bringing the bob down in a curved path.
Using F = ma for the bob,
mg sin x = ma, where a is the acceleration of the bob.
If the pendulum swings through a small angle and is measured in radians, sin x is almost equal to x.
mg. x = m a
gx = a
g d/l = a ( x = d / l radians)
a = (g/l) d
a = k d
a α d
The acceleration of the bob is directly proportional to the distance from the centre point. Therefore, the motion of a simple pendulum is simple harmonic.
k = ω2 where ω is the angular speed.
a = ω2 d
ω2 = g/l
ω = √g/l
If the time period is T,
T = 2π/ω
T = 2π √l/g
So, the time period of a simple pendulum depends only on its length; it does not depend on the mass of the bob. The formula only works for the oscillations through small angles, as it was something we assumed in the process of proof.
You can see the importance of the angle between the string and the vertical line being very small in the following animation:
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