Name: Condition for Resonance
Uploaded: 2019-07-19
Duration: 5 min 17 s
Description: Do you know you can shatter the glass goblet without touching it? Yes, it is possible when the sound of a specific frequency is produced. Let's find out the...

Question

Define S.H.M. Hence derive an expression for the displacement of a particle executing S.H.M.

Accepted Answer

Simple Harmonic Motion or S.H.M is defined as a motion in which the restoring force is directly proportional to the displacement of the body from its mean position. The direction of this restoring force is always towards the mean position.

For displacement and velocity of a body executing S.H.M we have to use this formula.

$x = Asin (ωt + ϕ),$

Where, $x$ Is the displacement from equilibrium position, $A$ Is amplitude, $ω$ Is angular frequency, $t$ Is time and $ϕ$ Is initial face angle. Here, $(ωt + ϕ)$ Is called phase.

$v = \frac{dx}{dt}$

Where, $v$ Is the velocity of particle, $x$ Is the displacement from equilibrium position and $t$ Is time.

$a = \frac{dv}{dt}$

Where, $a$ Is the acceleration of the particle, $v$ Is the velocity of particle and $t$ Is time.

We know that the displacement from equilibrium position is given by: $x = Asin (ωt + ϕ)$ .

$v = \frac{dx}{dt}$

$∴ v = \frac{d}{dt} (Asin (ωt + ϕ)) = ωAcos (ωt + ϕ)$

Now, at the equilibrium, initial phase angle is zero which means $ϕ = 0$

$v = ωAcosωt = ωA \sqrt{1 - \sin^{2} ωt}$

Also at equilibrium, $x = Asinωt \Rightarrow \sin^{2} ωt = \frac{x^{2}}{A^{2}}$

Putting this value in the equation of velocity, we get

$v = Aω \sqrt{1 - \frac{x^{2}}{A^{2}}}$ $= ω \sqrt{A^{2} - x^{2}}$

Thus, the expression of the velocity of the particle executes simple harmonic motion is

$v = ω \sqrt{A^{2} - x^{2}}$

Now, let us find the expression of acceleration of the particle executing simple harmonic motion by differentiating its velocity with respect to time.

$a = \frac{dv}{dt}$

$∴ a = \frac{d}{dt} (ωAcos (ωt + ϕ)) = - ω 2 Asin (ωt + ϕ)$

But, we know that $x = Asin (ωt + ϕ)$

$∴$ $a = - ω^{2} x$

Thus, the expressions for velocity and acceleration of a particle executing simple harmonic motion are $v = ω \sqrt{A^{2} - x^{2}}$ and $a = - ω^{2} x$ Respectively.

On squaring both sides

$v^{2} = ω^{2} (A^{2} - x^{2})$

$\frac{v^{2}}{ω^{2}} = A^{2} - x^{2}$

$\frac{v^{2}}{ω^{2} A^{2}} = 1 - \frac{x^{2}}{A^{2}}$

$\frac{x^{2}}{A^{2}} + \frac{v^{2}}{ω^{2} A^{2}} = 1$

This is the equation of an ellipse. Therefore, we can say that the curve between displacement and velocity of a particle executing the simple harmonic motion is an ellipse.

Define S.H.M. Hence derive an expression for the displacement of a particle executing S.H.M.

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