A point mass is subjected to two simultaneous sinusoidal displacements in `x - direction, x_1 (t) = A sin (omega)t and x_2 (t) = A sin ((omega t + (2 pi)/(3))`. Adding a third sinusoidal displacement `x _3 (t) = B sin (omega t + phi)` brings the mas to a complete rest. The values of (B) and (phi) are.

Options

A. `sqrt2A, (3pi)/(4)`
B. `A, (4pi)/(3)`
C. `sqrt3A, (5pi)/(6)`
D. `A, pi/3`

Answer

B

Explanation

`x_1(t)+x_2(t)+x_3(t)`

Resultant is zero
`x: A+Bcosphi-Acospi//3=0`
`Bcosphi=-A//2` (i)
`y: Bsinphi+Asinpi//3=0`
`Bsinphi=-(sqrt3A)/(2)`

`B=sqrt((A/2)^2+((sqrt3A)/(2))^2)=A`
`tantheta=(sqrt3A//2)/(A//2)=sqrt3impliestheta=60^@=pi/3`
`B=A`, `phi=pi+theta=pi+pi/3=(4pi)/(3)`