Question

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities d₁​ and d₂​ are filled in the tube. Each liquid subtends 90° angle at centre. Radius joining their interface makes an angle α with vertical. Ratio ​d₁/d₂​​ is ______.

Options

• ${\displaystyle \frac{1+\sin \alpha }{1-\sin \alpha }}$

• ${\displaystyle \frac{1+\cos \alpha }{1-\cos \alpha }}$

• ${\displaystyle \frac{1+\tan \alpha }{1-\tan \alpha }}$

• ${\displaystyle \frac{1+\sin \alpha }{1-\cos \alpha }}$

Answer 1

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities d₁​ and d₂​ are filled in the tube. Each liquid subtends 90° angle at centre. Radius joining their interface makes an angle α with vertical. Ratio ​d₁/d₂​​ is ${\displaystyle \underline{\frac{1+\tan \alpha }{1-\tan \alpha }}}$.

Explanation:

(R cosα + R sinα)d₂g = (R cosα - R sinα)d₁g

⇒ ${\displaystyle \frac{{\text{d}}_1}{{\text{d}}_2}=\frac{\cos \alpha +\sin \alpha }{\cos \alpha -\sin \alpha }=\frac{1+\tan \alpha }{1-\tan \alpha }}$