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

Answer

D

Explanation

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