A particle has simple harmonic motion along an axis *OX* when its displacement *x* relative to the origin of the coordinate system is given as a function of time by the relation

*x* = *A* cos (w*t* + a) …(i)

The quantity (w*t* + a) is called the **phase angle **or simply the phase of the SHM, and a is the initial phase, i.e. the phase at *t* = 0. Although we have defined simple harmonic motion in terms of a cosine function, it may just as well as expressed in terms of a sine function. The only difference between the two forms is an initial phase difference of p/2. Since the cosine (or sine) function varies between a value of –1 and +1, the displacement of the particle varies between *x* = – *A* and *x* = + *A*. The maximum displacement from the origin, *A*, is the **amplitude** of the simple harmonic motion. The cosine function repeats itself every time the angle wt increases by 2p. Thus the displacement of the particle repeats itself after a time interval of 2p/w. Therefore, simple harmonic motion is periodic, and its **period** is

*P* = 2p/w

The frequency *v* of a simple harmonic motion is equal to the number of complete oscillations per unit time;

*v* = 1/P

frequency is measured in hertz. The quantity w, called the **angular frequency **of the oscillating particle, is related to the frequency by a relation similar to circular motion, namely

w = 2p/*P* = 2p*v *…(ii)

The velocity of the particle is

…(iii)

which varies periodically between the values wA and –wA. Similarly, the acceleration, is given by

…(iv)

and therefore varies periodically between the values + w2A and –w2A. This expression also indicates that : *in simple harmonic motion the acceleration is proportional and opposite to the displacement.*

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