Question

Obtain an expression for the fraction of a complete period that a simple harmonic oscillator spends within a small interval delta_x at a position x. What is its physical significance?

Answer 1

We know that,

The general equation of the simple harmonic motion

$x= A\sin\omega t$

Now, taking the differentiation with respect to t,

$\dfrac{dx}{dt}=A\omega \cos \omega t$

taking the second derivative of the equation

$\dfrac{d^2x}{dt^2}=-A\omega^2\sin \omega t$

when sin will be maximum =1

$\dfrac{d^2x}{dt^2}=-A\omega^2$