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 $\omega / \pi$
• B. simple harmonic motion of frequency $3 \omega / 2 \pi$
• C. non simple harmonic motion
• D. simple harmonic motion of frequency $\omega / 2 \pi$

Answer 1

Answer: (C)

non simple harmonic motion

Answer 2

\begin{array}{l}
\text { Given, } X = a \sin ^{2} \omega t\\
\text { or } X=a\left(\frac{1-\cos 2 \omega t}{2}\right)\left[\because \cos 2 \theta=1-2 \sin ^{2} \theta\right]\\
\text { or } X =\frac{a}{2}-\frac{ a \cos 2 \omega t }{2}\\
\text { Now, } V =\frac{ d x }{ d t }= a \omega \sin 2 \omega t\\
\text { and } a = d v / d t =2 a \omega^{2} \cos 2 \omega t\\
\text { Here, a is not directly proportional to }(- x ) \text { which is a condition for SHM. }
\end{array}