Question

The motion of a particle varies with time according to the relation
$y=a(\sin \omega t+\cos \omega t)$

• A. the motion is oscillatory but not SHM
• B. the motion is SHM with amplitude a
• C. the motion is SHM with amplitude $a\sqrt{2}$
• D. the motion is SHM with amplitude 2a

Answer 1

Answer: (C)

The equation of particle varying with time is
$y=a(\sin \omega t+\cos \omega t)$
Or $y=a\sqrt{2}\left(\frac{1}{\sqrt{2}}\sin \omega t+\frac{1}{\sqrt{2}}\cos \omega t\right)$
or $y=a\sqrt{2}\left(\cos \frac{\pi }{4}\sin \omega t+\sin \frac{\pi }{4}\cos \omega t\right)$
or $y=a\sqrt{2}\sin \left(\omega t+\frac{\pi }{4}\right)$ ..(i)
This is the equation of simple harmonic motion with amplitude
$a\sqrt{2}$ .
Note: We can represent the resultant Eq. (i) in angular SHM as
$\theta ={\theta }_0\sin \left(\omega t+\frac{\pi }{4}\right)$
where ${\theta }_0$
is amplitude of angular SHM of particle.