Show that simple harmonic motion may be regarded as the projection of uniform circular motion along a diameter of the circle.

Explanation

Tie a ball to the end of a string and make it move in a horizontal plane about a fixed point with a constant angular speed.
The ball then perform cirular motion in the horizontal phane.
Observe the motion of ball in a plane by lighting torch in a dark room. The ball will appear to execute to and fro motion along a horizontal line with the point of rotation as the mid-point. Also observe the shadow of the ball on a wall which is perpendicular to the plane of the circle.
Here the motion of the ball on a diameter of the circle normal to the direction of viewing. This is shown in below figure.

Suppose position of ball moving in circular motion are ABCDEFGA. Observing this plane perpendicularly, the shadow of ball appear on wall will be OPQPORSRO respectively.
The motion of this shadow seems around point O (up and down) which is simple harmonic motion.