Question

A circular tube of uniform cross section is filled with two liquids densities `rho_(1)` and `rho_(2)` such that half of each liquid occupies a quarter to volume of the tube. If the line joining the free surfaces of the liquid makes an angle `theta` with horizontal find the value of `theta.`

A. `tan^(-1)((1)/(5))`
B. `tan^(-1)((3)/(2))`
C. `tan^(-1)((2)/(3))`
D. zero

Answer 1

Correct Answer - A
In equilibrium, pressure of same liquid at same level will be same.
Therefore, `P_(1) =P_(2)`
or `P+(1.5 rho gh_(1)) =P+ (rho gh_(2))`
`(P =` pressure of gas in empty part of the tube)
`:. 1.5 h_(1) =h_(2)`
`1.5 [R cos theta -R sin theta] =(R cos theta +R sin theta)`
or `3cos theta - 3 sin theta =2 cos theta +2 sin theta`
or `5 tan theta =1 theta= tan^(-1) ((1)/(5))`