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 $\mathrm{period}\pi \omega$
• 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, $y={\sin }^2\omega t$

$\frac{dy}{dt}=2\omega \sin \omega t\cos \omega t=\omega \sin 2\omega t$
$\frac{d^{2}y}{d{t}^2}=2{\omega }^2\cos 2\omega t$
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
$t=\frac{\pi }{\omega }$