when two displacements represented by `y_(1) = a sin(omega t)` and `y_(2) = b cos (omega t)` are superimposed the motion is

Options

A. Not a simple harmonic
B. Simple harmonic with amplitude `(a)/(b)`
C. Simple harmonic with amplitude `sqrta^(2)+b^(2)`
D. Simple harmonic with amplitude `((a)+(b))/(2)`

Answer

C

Explanation

`y = y_(1) + y_(2)`
` = a sin omegat + b cos omegat`
`=sqrt(a^(2) + b^(2))((a)/(sqrt((a^(2) + b^(2))))sinomegat + (b)/(sqrt(a^((2)) + b^(2)))cosomegat)`
` = sqrt(a^(2) + b^(2)) [cos theta sin omegat + sin theta cos omegat]`
` = sqrt(a^(2) + b^(2)) sin(omegat +theta)`
`y = A sin(omegat + theta)`
It executes S.H.M. with amplitude `sqrt(a^(2) + b^(2))`