The function (sin , t -cos , t) represents - Vidyadip

MCQ

The function $(sin\,\omega t -cos\,\omega t)$ represents

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

✓

Answer

Correct option: B.

a simple harmonic motion with a period $\frac{2\pi }{\omega }$

b
$x=\sin \omega t-\cos \omega t$

$\Longrightarrow x=2 \cos \omega t=2 \sin \left(\omega t+\frac{\pi}{2}\right) \quad\left[\sin C-\sin D=2 \cos \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)\right]$

On comparing this with equation of $S H M: x=A \sin (\omega t+\phi)$

$\Longrightarrow A=2$ $\omega(\text { Angular frequency })$

$\Longrightarrow T=\frac{2 \pi}{\omega}$

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