Question
The displacement of a particle along the $$x$$-axis is given by $$x = a\,{\sin ^2}\omega t.$$ The motion of the particle corresponds to
A.
simple harmonic motion of frequency $$\frac{\omega }{\pi }$$
B.
simple harmonic motion of frequency $$\frac{{3\omega }}{{2\pi }}$$
C.
non-simple harmonic motion
D.
simple harmonic motion of frequency $$\frac{{\omega }}{{2\pi }}$$
Answer :
non-simple harmonic motion
Solution :
For a particle executing $$SHM$$
$$\eqalign{
& {\text{Acceleration}}\left( a \right) \propto - {\omega ^2}{\text{displacement}}\left( x \right)\,......\left( {\text{i}} \right) \cr
& {\text{Given}}\,x = a{\sin ^2}\omega t\,......\left( {{\text{ii}}} \right) \cr} $$
Differentiating the above equation w.r.t, we get
$$\frac{{dx}}{{dt}} = 2a\omega \left( {\sin \omega t} \right)\left( {\cos \omega t} \right)$$
Again differentiating, we get
$$\eqalign{
& \frac{{{d^2}x}}{{d{t^2}}} = a = 2a{\omega ^2}\left[ {{{\cos }^2}\omega t - {{\sin }^2}\omega t} \right] \cr
& = 2a{\omega ^2}\cos 2\omega t \cr} $$
The given equation does not satisfy the condition for $$SHM$$ (Eq. (i)]. Therefore, motion is not simple harmonic.