Question

The function $\sin ^2(\omega t)$ represents

• A. a periodic, but not simple harmonic motion with a period $2 \pi / \omega$
• B. a periodic, but not simple harmonic motion with a $\operatorname{period} \pi \omega$
• C. a simple harmonic motion with a period $2 \pi / \omega$
• D. a simple harmonic motion with a period $\pi / \omega$

Answer 1

Answer: (B)

Here, $y=\sin ^2 \omega t$

$ \frac{d y}{d t} =2 \omega \sin \omega t \cos \omega t=\omega \sin 2 \omega t $
$\frac{d^2 y}{d t^2} =2 \omega^2 \cos 2 \omega t$
For SHM, $\frac{d^2 y}{d t^2} \propto(-y)$
Hence, function is not SHM, but periodic.
From the $y-t$ graph, time period is
$t=\frac{\pi}{\omega}$