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 SHM
B. The motion is SHM with an amplitude a+b
C. The motion is SHM with an amplitude `a^(2)+b^(2)`
D. The motion is SHM with an amplitude `sqrt(a^(2)+b^(2))`

Answer

D

Explanation

`y=a "sin" omega t + b "cos" omega t`
Let `a = A "sin" phi and b = A "cos" phi`
`therefore a^(2)+b^(2)=A^(2)("sin"^(2)phi+"cos"^(2)phi)=A^(2)`
`therefore A=sqrt(a^(2)+b^(2))`
and `y = A "sin" phi "sin" omega t + A "cos" phi "cos" omega t= A "sin" (omega t + phi )`
`therefore (dy)/(dt)=A omega "cos" (omegat + phi)`
and `(d^(2)y)/(dt^(2))= - omega^(2)A "sin" (omega t+ phi)= omega^(2)y`
`therefore (d^(2)y)/(dt^(2)) prop (-y)`
`therefore` It is a SHM of amplitude `sqrt(a^(2)+b^(2))`