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Q.
The motion of a particle varies with time according to the relation
$ y=a(\sin \omega t+\cos \omega t) $
BHUBHU 2006Oscillations
Solution:
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.