The displacement of a particle varies with time according to the relation `y=asinomegat+bcoasomegat`.

Options

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

Explanation

According to the question, the displacement
`y=asinomegat+bcosomegat`
Let `a=Asinphi` `b=Acosphi`
Now, `a^(2)+b^(2)=A^(2)sin^(2)phi+A^(2)cos^(2)phi`
`=A^(2)rArrA=sqrt(a^(2)+b^(2))`
`y=Asinphi.sinomegat+Acosphi.cosomegat`
`=Asin(omegat+phi)`
`(dy)/(dt)=Aomegacos(omegat+phi)`
`(d^(2)y)/(dt^(2))=-Aomega^(2)sin(omegat+phi)=-Ayomega=(-Aomega^(2))y`
`rArr(d^(2)y)/(dt^(2))prop(-y)`
Hence, it is an equation of SHM with amplitude `A=sqrt(a^(2)+b^(2))`