Simple Harmonic Motion and Uniform Circular Motion

Remember: A particle is said to be in Simple Harmonic Motion if it moves to and fro about its mean position, such that, its acceleration is directly proportional to displacement in magnitude but opposite in direction and is always directed towards the mean position.

Consider a particle P moving around a reference circle with the centre, O, radius, A = OZ = OY, and uniform angular velocity, ω, in an anti-clockwise direction.

BD and YZ are the diameters of the circle.

The angular velocity, ω, is related to the speed v of particle, P, by the equation:

v = ωA

Let Q be the projection of P on the diameter YZ. (This is a perpendicular line from P to YZ)

As the particle moves on the circumference of the circle, Q moves to and fro about a fixed point along the diameter YZ. It moves with a maximum speed as it passes through O the centre of diameter YZ, and is momentarily at rest at Y and Z.

While the particle P moves around the circle with a constant velocity v, the motion of Q along the diameter is therefore said to be simple harmonic motion.

Equation of Displacement of the Particle: