Question

The function $si{n}^2(\omega t)$ 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 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, $y-si{n}^2\omega t$
$\frac{dy}{dt}=2\omega sincos\omega t=\omega 2\omega t$
$\frac{d^{2}y}{d{t}^2}=2{\omega }^{2}cos2\omega t$
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
$T=\frac{\pi }{\omega }$