Two simple harmonic motions are given by `x_(1) = a sin omega t + a cos omega t and x_(2) = a sin omega t + (a)/(sqrt3) cos omega t`
The ratio of the amplitudes of first and second motion and the phase difference between them are respectively

Options

A. `sqrt((3)/(2)) and (pi)/(12)`
B. `(sqrt3)/(2) and (pi)/(12)`
C. `(2)/(sqrt3) and (pi)/(12)`
D. `sqrt(3/2) and (pi)/(6)`

Answer

A

Explanation

`x_(1)=asinomegat+acosomegat`
`A_(1)=sqrt(a^(2)+a^(2))=asqrt(2), phi_(1)=tan^(-1)((a)/(a))=(pi)/(4)`
`x_(2)=asinomegat+(a)/(sqrt3)cosomegat`
`A_(2)=sqrt(a^(2)+((a)/(sqrt3))^(2))=(2a)/(sqrt3),phi_(2)=tan^(-1)((1)/(sqrt3))=(pi)/(6)`
`(A_(1))/(A_(2))=(asqrt2)/(2a//sqrt3)=(sqrt3)/(sqrt2)" , "Deltaphi=(pi)/(4)-(pi)/(6)=(pi)/(12)`.