## Simple harmonic motion

In simple harmonic motion (SHM), the displacement x(t) of a particle from its equilibrium position is given by,

A term that means the periodic motion is a sinusoidal function of time. Equation 3, in which the sinusoidal function is a cosine function, is plotted in Fig. 14.5.

in which *A *is the amplitude of the displacement, the quantity (ω*t + ϕ*) is the phase of the motion, and ϕ is the phase constant.

The angular frequency ω is related to the period and frequency of the motion by,

The **SI unit** of angular frequency is **radians per second**.

To illustrate the significance of period T, sinusoidal function with two different periods are plotted in below graph.

*Fig. Plots of Eq. for ? = 0 for two different periods.*

In this plot, the SHM represented by curve a, has a period T and that represented by curve b, has a period T’ = T/2.

**Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion takes place***.*

### Velocity and Acceleration in simple Harmonic Motion

The magnitude of velocity, v, with which the reference particle P is moving in a circle is related to its angular speed, ω as

where A is the radius of the circle describe by the particle P. its projection on the x-axis at any time t, as shown in above Fig. is

The negative sign appears because the velocity component of P is directed towards the left, in the negative direction of x. Eq. (4) expresses instantaneous velocity of the particle P’ (projection of P). Therefore, it expresses the instantaneous velocity of a particle executing SHM. Eq. (4) can also be obtained by differentiating Eq. (3) w.r.t as,

We have seen that a particle executing a uniform circular motion is subjected to a radial acceleration **a **directed towards the centre. The magnitude of the radial acceleration of P is ω^{2}*A. *Its projection on the *x*-axis at any time *t *is, which is the acceleration of the particle P′ (the projection of particle P).

Eq. (5) expresses the acceleration of a particle executing SHM. It is shown that in SHM, the acceleration is proportional to the displacement and is always directed towards the mean position.

The inter-relationship between the displacement of a particle executing simple harmonic motion, its velocity and acceleration can be seen in below Fig.

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