Properties of Matter 2 Question 3
3. 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 $d _1 / d _2$ is
(2014 Main)
(a) $\frac{1+\sin \alpha}{1-\sin \alpha}$
(b) $\frac{1+\cos \alpha}{1-\cos \alpha}$
(c) $\frac{1+\tan \alpha}{1-\tan \alpha}$
(d) $\frac{1+\sin \alpha}{1-\cos \alpha}$
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Answer:
Correct Answer: 3. (c)
Solution:
- Equating pressure at $A$, we get
$R \sin \alpha d _2+R \cos \alpha d _2+R(1-\cos \alpha) d _1$ $=R(1-\sin \alpha) d _1$
$(\sin \alpha+\cos \alpha) d _2=d _1(\cos \alpha-\sin \alpha)$
$\Rightarrow \quad \frac{d _1}{d _2}=\frac{1+\tan \alpha}{1-\tan \alpha}$
