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, $\text{T}=\frac{2\pi }{\text{angular speed}}=\frac{2\pi }{2\omega }=\frac{\pi }{\omega }$