__To show the motion of simple pendulum is simple harmonic motion__

The x-component mgcosθ is opposite to the tension T in the string. Since there is no motion along any directions of T, therefore, mgcosθ must be equal to T and will be cancelling one another.

The y-component has the same direction as that of the motion of the pendulum. Therefore, this must be causing the motion of the pendulum. Therefore, the restoring force which drives the pendulum is,

Now, weight W is parallel to OP and T and mgcosθ are collinear, therefore,

Put this value in equation (1),

Now m, g and ℓ are constants, therefore,

This is condition for Simple Harmonic Motion. Therefore, if the displacement is small, the motion of a simple pendulum is simple harmonic one.

Similarly, by Newton’s Second Law

Compare equation (3) with equation (2),

Simplify

Now g and ℓ are also constants, therefore,

This acceleration is caused by the component mgsinθ of the weight W of the bob and shows the motion of the pendulum is simple harmonic motion.