Four simple harmonic vibrations
`y_(1)=8 sin omega t`, `y_(2)= 6 sin (omega t+pi//2)`, `y_(3)=4 sin (omega t+pi)`, `y_(4)=2sin(omegat+3pi//2)`
are susperimposed on each other. The resulting amplitude and phase are respectively.

Options

A. `sqrt(45)` and `tan^(-1)((1)/(2))`
B. `sqrt(45)` and `tan^(-1)((1)/(3))`
C. `sqrt(75)` and `tan^(-1)(2)`
D. `sqrt(75)` and `tan^(-1)((1)/(3))`

Answer

A

Explanation

Resultant displacement along x-axis is
`x=y_(1)-y_(3)=8-2=6`
Resultant displacement along y-axis is
`y=y_(2)-y_(4)=4-1=3`
`A=sqrt(x^(2)+y^(2))` and `tantheta=(y)/(x)`