Four simple harmonic motions`, c_(1)=8sinomegat,x_(2)=6sin(omegat=pi//2) , x_(3)=4sin(omegat+pi) and x_(4)=2sin (omegat+3pi//2)` are superimposed on each other. The resuslting amplitude and its phase difference with `x_(1)` are respectively

Options

A. `20, tan^(-1)(1//2)`
B. `4sqrt(2), pi//2`
C. `20, tan^(-1)(2)`
D. `4sqrt(2), pi//4`

Answer

D

Explanation

We represent the various motion in a vector diagram, as shown in figure.
Effective amplitude along x-direction
`A_(x)=A_(1)-A_(3)=8-4=4`
Effective amplitude along y direction
`A_(y)=A_(2)-A_(4)=6-2=4`
`:. `Resulting amplitude,
`A=sqrt(A_(x)^(2)+A_(y)^(2))=sqrt(4^(2)+4^(2))=4sqrt(2)`
If `phi` is the phase difference with `x_(1)`, then
`tanphi=(A_(y))/(A_(x))=(4)/(4)=1=tan((pi)/(4)) or phi=(pi)/(4)`