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

The Correct Option is 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 E (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.