For a motion to be in simple harmonic, acceleration is directly proportional to displacement and it directs against the direction of displacement ,
so for the pendulum to be in simple harmonic it should satisfy these two conditions.
lets go for it.
Condition 1: Acceleration is directly proportional to displacement
1. Weight of the bob (W) acting vertically downward.
2. Tension in the string (T) acting along the string. The weight of the bob can be resolved into two rectangular components:
a. Wcosq along the string.
b. Wsinq perpendicular to string.
Since there is no motion along the string, therefore, the component Wcosq must balance tension (T)
i.e. Wcosq = T This shows that only Wsinq is the net force which is responsible for the acceleration in the bob of pendulum. According to Newton's second law of motion Wsinq will be equal to m x a i.e. Wsinq = m a
Since Wsinq is towards the mean position, therefore, it must have a negative sign.
i.e. m a = - Wsinq
But W =mg
m a = - mgsinq
a = - gsinq
In our assumption q is very small because displacement is small, in this condition we can take sinq = q
Hence a = - gq
Thus a is directly proportional to displacement.
Condition 2 : Acceleration directs against the direction of displacement
The pendulum travels to and fro motion thus, it acquires Inertia due to which the displacement is opposite to acceleration.
Thus it follows both the conditions and hence is in SHM