Question

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_1$ and $d_2$ are filled in the tube. Each liquid subtends ${90}^{\circ }$ angle at centre. Radius joining their interface makes an angle $\alpha$ with vertical. Ratio of $\frac{d_1}{d_2}$ is

• A. $\frac{(1+tan\alpha )}{(1-tan\alpha )}$
• B. $\frac{(1+sin\alpha )}{(1-cos\alpha )}$
• C. $\frac{(1+sin\alpha )}{(1-sin\alpha )}$
• D. $\frac{(1+cos\alpha )}{(1-cos\alpha )}$

Answer 1

The correct option is A

$\frac{(1+tan\alpha )}{(1-tan\alpha )}$

Equating pressure at the bottommost point from both sides, $d_{1}gR(1-sin\alpha )=d_{1}gR(1-cos\alpha )+d_{2}gR(sin\alpha +cos\alpha )\Rightarrow \frac{d_1}{d_2}=\frac{cos\alpha +sin\alpha }{cos\alpha -sin\alpha }=\frac{1+tan\alpha }{1-tan\alpha }$