Question

The function $\left(sin\right)^{2}\left(\omega t\right)$ represents

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

Answer 1

Answer: (B)

$y=\left(sin\right)^{2}\left(\omega t\right)=\frac{1 - cos \left(2 \omega t\right)}{2}=\frac{1}{2}-\frac{cos \left(2 \omega t\right)}{2}$
$\Rightarrow y=\frac{1}{2}-\frac{1}{2}cos\left(2 \omega t\right)$
It is an SHM with amplitude $\frac{1}{2}$ .
$\therefore $ Angular speed = $2 \omega $
$\therefore $ Time period, $\textit{T} = \frac{2 \pi }{\text{angular speed}} = \frac{2 \pi }{2 \omega } = \frac{\pi }{\omega }$