# Characteristics Of Simple Harmonic Motion

Contents

Some of the important parameters which define the characteristics of a simple harmonic motion are given below.

## (I) Displacement

The displacement of a particle executing SHM at an instant is given by the distance of the particle from the mean position at that instant.

The value of displacement as a continuous function of time can be represented as a graph given below.

The function containing sine or cosine term is known as sinusoidal function.

## (II) Amplitude

The magnitude of maximum displacement of a particle executing SHM is called amplitude of the oscillation of that particle.

Amplitude is measured on either side of mean position.

The displacement varies between the extremes +A and -A because the sinusoidal function of time varies from +1 to -1.

Two SHM may have same angular frequency(ω) and same phase constant(Ø) but different amplitude A and B.

## (III) Phase

The phase of a vibrating particle at any instant gives the state of the particle as regards its position and the direction of motion at that instant.

It is equal to the argument of sine or cosine function occurring in the displacement equation of the SHM.

Let a simple harmonic equation is represented by

x = A Cos(ωt+Ø₀), then

Phase of the particle is Ø = ωt+Ø₀ clearly, the phase Ø is a function of time t. It is usually expressed either as the fraction of the time period T or fraction of angle 2π.

For t=0 phase Ø = ωt+Ø₀. Thus, Ø is called phase constant or phase angle.

Two SHM may have the same amplitude(A) and angular frequency(ω) but different phase angle(Ø).

This can be shown in the below figure.

In above graph, the curve 1 has phase constant of π/2 and amplitude A. The curve 2 has phase constant of π/4 and amplitude A.

## (IV) Angular Frequency

Angular frequency of a body executing periodic motion is equal to the product of frequency of the particle with factor 2π.

It is denoted by ω and its SI unit is radian/second.

ω = 2πf = 2π/T

Angular frequency, ω = 2π/T

Two SHM may have same amplitude (A) and phase angle(Ø) but different angular frequency(ω).

This can be represented on the graph as below.

The above wave diagram, curve(b) has half the period and twice the frequency of the curve(a).

## (V) Velocity

The velocity of a particle executing SHM at any instant is defined as the time rate of change of its displacement at that instant.

## (VI) Acceleration

The acceleration of the particle executing SHM at any instant is defined as the time rate of change of its velocity at that instant.

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