The oscillations of a harmonic oscillator can be represented by terms containing sine and cosine of an angle. If the displacement of an oscillatory particle from its mean position can be represented by an equation

y = a sinθ or y = a cosθ or y = A sinθ + B cosθ,

where a, A and B are constants, the particle executes simple harmonic motion.

A particle is said to execute **simple harmonic motion** if it moves to and fro about a fixed point periodically, under the action of a force F which is directly proportional to its displacement x from the fixed point and the direction of the force is opposite to that of the displacement.

Mathematically,

F = – kx

where k is constant of proportionality.

To derive the equation of simple harmonic motion, consider a point M moving with a constant speed v in a circle of radius a with centre O. At t = 0, let the point be at X. The position vector OM specifies the position of the moving point at time t.

The position vector OM, also called the phaser, rotates with a constant angular velocity ω = v/a. The acceleration of the point M is v^{2}/a = aω^{2} towards the centre O.

At time t, the component of this acceleration along OY = aω^{2} sin ωt. Draw MP perpendicular to YOY′. Then P can be regarded as a particle of mass m moving with an acceleration aω^{2} sin ωt. The force on the particle P towards O is given by

F = maω^{2} sin ωt

But sin ωt = y/a. Therefore

F = mω^{2}y

The displacement is measured from O towards P and force is directed towards O. Therefore,

F = – mω^{2}y

Since this force is directed towards O, and is proportional to displacement ‘y’ of P from O, the particle P is executing simple harmonic motion.

Let mω^{2} = k, a constant.

**F = – ky**

The constant k, which is force per unit displacement, is called **force constant**. The angular frequency of oscillations is given by

**ω ^{2} = k/m **

In one complete rotation, OM describes an angle 2π and it takes time T to complete one rotation. Hence

**ω = 2π/T**

**T = 2π/km **

This is the time taken by P to move from O to Y, then through O to Y′ and back to O.During this time, the particle moves once on the circle and the foot of perpendicular from its position is said to make an oscillation about O.