The displacement of a particle varies with time according to the relation `y=a sin omega t +b cos omega t `.

Options

A. The motion is oscillatory but not S.H.M.
B. The motion is S.H.M. with amplitude a + b
C. The motion is S.H.M. with amplitude `a^(2) + b^(2)`
D. The motion is S.H.M. with amplitude `sqrt(a^(2) + b^(2))`

Answer

D

Explanation

Here `y = a sin omegat + b cos omegat`
This wave is due to superposition of two waves.
Let `a = A cos theta`
And `b = A sin theta`
`therefore a^(2) + b^(2) = A^(2)`
or `A = sqrt(a^(2) + b^(2))`
A is the amplitude of resultant wave
And `y = A cos theta sin omegat + A sin theta cos omegat`
or `y = A sin (omegat + phi)`
THis equation represents S.H.M.