If simple harmonic motion is a linear motion , then how can motion of pendulum be simple harmonic motion , as its not linear ....?

The motion of a pendulum is indeed not linear, but it can still be considered as simple harmonic motion under certain conditions. Here's why:

1. Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It is true that SHM is often represented as a linear motion back and forth along a line, but this is not a requirement.

2. A pendulum swings back and forth in an arc, which is a segment of a circle. This is not linear motion. However, if we consider small angles of displacement, the arc of the swing starts to look more and more like a straight line.

3. For small angles of displacement (usually less than about 15 degrees), the restoring force acting on the pendulum is approximately proportional to the displacement. This is because for small angles, sin(theta) is approximately equal to theta (when theta is measured in radians).

4. Therefore, under the condition of small angles, the motion of a pendulum approximates Simple Harmonic Motion.

5. It's important to note that this is an approximation. The motion of a pendulum is not exactly simple harmonic because the restoring force is not exactly proportional to the displacement. But for small angles, the approximation is very close.

So, while the motion of a pendulum is not linear, it can still be considered as simple harmonic motion under the condition of small angles of displacement.