The displacement of a particle along the $x$ axis is given by $\mathrm{x}=\mathrm{a} \sin ^2 \omega…
The displacement of a particle along the $x$ axis is given by $\mathrm{x}=\mathrm{a} \sin ^2 \omega \mathrm{t}$. The motion of the particle corresponds to
simple harmonic motion of frequency $\omega / \pi$
simple harmonic motion of frequency $3 \omega / 2 \pi$
non simple harmonic motion
simple harmonic motion of frequency $\omega / 2 \pi$
Solution
For a particle executing SHM acceleration (a) $\propto-\omega^2$ displacement $(x)$
Given
$\mathrm{x}=\mathrm{a} \sin ^2 \omega \mathrm{t}$
Differentiating the above equation w.r.t, we get $\frac{\mathrm{dx}}{\mathrm{dt}}=2 \mathrm{a} \omega(\sin \omega \mathrm{t})(\cos \omega \mathrm{t})$
Again differentiating, we get
$\begin{aligned}
& \frac{\mathrm{d}^2 \mathrm{x}}{\mathrm{dt}^2}=\mathrm{a}=2 \mathrm{a} \omega^2\left[\cos ^2 \omega \mathrm{t}-\sin ^2 \omega \mathrm{t}\right] \\
& =2 \mathrm{a} \omega^2 \cos 2 \omega \mathrm{t}
\end{aligned}$
The given equation does not satisfy the condition for SHM [Eq. (i)] . Therefore, motion is not simple harmonic.
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