The displacement of a particle along the $x-$ axis is given by $x=asin^2$$\omega t$ . The motion of the particle corresponds to
AIPMT 2010, Medium
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$x=a \sin ^{2} \omega t=a\left(\frac{1-\cos 2 \omega t}{2}\right)$
$\left(\because \cos 2 \theta=1-2 \sin ^{2} \theta\right) $
$= \frac{a}{2}-\frac{a \cos 2 \omega t}{2}$
$\therefore \quad$ Velocity, $v=\frac{d x}{d t}=\frac{2 \omega a \sin 2 \omega t}{2}=\omega a \sin 2 \omega t$
Acceleration, $a=\frac{d v}{d t}=2 \omega^{2} a \cos 2 \omega t$
For the given displacement $x=a \sin ^{2} \omega t,$
$a \propto-x$ is not satisfied.
Hence, the motion of the particle is non simple harmonic motion.
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