A small uniform tube is bent into a circle of radius r whose plane is vertical. Equal volumes of two fluids whose densities are`rho` and `sigma(rhogtsigma)` fill half the circle. Find the angle that the radius passing through the interface makes with the vertical.

Options

A. ` tan ^(-1) ((rho _1 + rho _ 2 )/(rho_2 - rho _1 )) `
B. `tan ^(-1) ((rho_2 - rho_1 )/(rho_1 + rho_2 )) `
C. `sin^(-1)((rho_1 + rho_2 )/(rho_2 - rho _1 ))`
D. `cos ^(-1) ((rho_1 + rho_2 )/(rho_2- rho_1 )) `

Answer

B

Explanation

Let the radius of the tube be r
Equating pressure at the bottom most point of the tube due to the liquid on the left and the liquid on the right `P_("left") = P _("right")`
`rho 2 gr (1 - sin theta ) = rho2 gr (1 - cos theta ) + rho 1 gr cos theta + rho 1 g r sin theta rArr theta = tan ^(-1) ((rho 2 - rho 1 ) /(rho 1 + rho 2 )) `