Differential Equation of the Simple Harmonic Motion

Differential Equation of the Simple Harmonic Motion

Differential Equation of the simple harmonic motion

Simple harmonic motion is produced due to the oscillation of a spring. Find out the differential equation for this simple harmonic motion. Suppose mass of a particle executing simple harmonic motion is ‘m’ and if at any moment its displacement and acceleration are respectively x and a, then according to definition,

a = – (K/m) x, K is the force constant.

But a = d2x/dt2

So, d2x/dt2 = – (K/m) x … … … (1)

In order to solve any differential equation, a general procedure is to assume a solution and it is observed whether the given differential equation can be derived from it or not. Suppose the solution of the equation (1) is –

x(t) = a sin ωt, here a and ω are constants.

dx/dt = a ω cos ωt and d2x/dt2 = – a ω2 sin ωt

From equation we get,

– a ω2 sin ωt = – (K/m) a sin ωt

So, if the value of the constant is, ω = √(K/m) … … … (2)

then, the relation x (t) = a sin ωt satisfies the differential equation.

So, a solution of the equation is x(t) = a sin ωt.

Similarly, we can prove that the relation, x(t) = b cos ωt is also a solution. Hence, x(t) = a sin ωt or, x(t) = b cos ωt is the solution of the simple harmonic motion.

It can be proved that,

x(t) = a sin ωt + b cos ωt … … … (3)

This equation is the general solution of the differential equation (1), as,

dx/dt = a ω cos ωt – b ω sin ωt

or, d2x/dt2 = – a ω2 sin ωt – b ω2 cos ωt

= – ω2 (a sin ωt + b cos ωt) = – ω2x

Then, d2x/dt2 = – ω2x                (here = a sin ωt + b cos ωt)

So, d2x/dt2 + ω2x = 0 … … … (4)

So, equation (4) is the differential equation of the simple harmonic motion.

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