Question

If a $SHM$ is given by $y=(\sin \omega t+\cos \omega t)m,$ Which of the following statements are true?

• A. The amplitude is $1m$
• B. The amplitude is $\left(\sqrt{2}\right)m$
• C. Time is considered from $y=1m$
• D. Time is considered from $y=0m$

Answer 1

The correct option is B The amplitude is $\left(\sqrt{2}\right)m$

Given that

$y=(\sin \omega t+\cos \omega t)m$

$y=\sin \omega t+\sin (90-\omega t)$

$y=2\sin \left(\frac{\omega t+90-\omega t}{2}\right)+\cos \left(\frac{\omega t-90+\omega t}{2}\right)$

$y=2\times \sin 45\cos (45-\frac{\omega t}{2})$

$y=\sqrt{2}\cos (45-\frac{\omega t}{2})$

So, the amplitude of the SHM is $\sqrt{2}m$