Differential equation for a simple harmonic Motion

DIFFERENTIAL EQUATION FOR A SIMPLE HARMONIC MOTION

The differential equation for a simple harmonic motion shall be obtained from the expressions of displacement and velocity, as follows.

(i) Displacement: The displacement of vibrating particle at any instant is defined as the distance moved by the particle from its mean position of rest.

Let us consider a particle 'P' moving in a circular path of radius ‘A' with uniform velocity 'v' and angular velocity ω, with respect to the centre of the circle of reference 'O' as shown in Fig. 4.1.

When the particle 'P' moves around the circle, then the foot of the perpendicular “Q' vibrates along the diameter YY'. Further, if the motion of the particle 'P' is uniform, then the motion of 'Q' is also periodic. i.e., the particle will take the same time to vibrate between the points Y and Y'.

Therefore, if the particle 'P' completes one revolution, then the foot of the perpendicular 'Q' will complete one vertical oscillation.

Thus, the distance OQ is termed as displacement of the particle and is denoted by the letter 'y'.

If the particle moves from 'X' to 'P' in 't' seconds, then the angle between POX is given by

∟POX = ∟QPO = θ = ωt

From ΔQPO, we can write

sin θ (or) sin ωt = OQ / OP         ……….(1)

Since OQ = y and OP = A, we can write equation (1) as

sin ωt = y / A

(or)

y = A sin ωt                ………….(2)

Equation (2) represents the equation for the displacement of the vibrating particle at any instant 't'.

(ii) Velocity: Velocity of the vibrating particle is defined as the rate of change of displacement.

Velocity (v) = dy/dt                 ………….(3)

Substituting eqn (2) in eqn (3) we get

 v = d(A sin ωt) / dt

 v = A ω cosωt

Equation (4) represent the velocity of the vibrating particle at any instant 't'.

(iii) Acceleration : Acceleration of the vibrating particle is defined as the rate of change of velocity.

Acceleration = dv/dt         ………….(5)

Substituting eqn (4) is eqn (5) we get

Since Asin ωt = y, we can write the above equation as

 d²y / dt² = ‒ω²y          ………..(6)

Equation (6) represents the acceleration of the vibrating particle at any instant 't' (or) the so called differential equation for a simple harmonic motion.

This type of motion, where the acceleration is directed towards a fixed point (the mean position of rest) and is proportional to the displacement of the vibrating particle is called Simple Harmonic Motion (SHM).

Graphical Representation of SHM

Fig. 4.2 shows the change in displacement of the vibrating particle at various positions (say,P,Q,R,S) and angles in one complete vibration, which is also called as the displacement curve. From this curve we can see that the motion of the particle is a simple harmonic motion (SHM) and the curve is termed as the graphical representation of SHM.

Characteristics of SHM

The characteristics of simple harmonic motion are as follows.

(i) The motion of the particle is periodic.

(ii) The motion of the particle is along a straight line about its mean position.

(iii) The acceleration of the particle is proportional to the displacement and is directed towards its mean position.