There is a circular tube in a vertical plane. Two liquids...

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 the centre. Radius joining their interface makes an angle $\alpha$ with vertical plane. Ratio $\frac{d_{1}}{d_{2}}$ is [2014](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) $\frac{1+\sin \alpha}{1-\cos \alpha}$

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 the centre. Radius joining their interface makes an angle $\alpha$ with vertical plane. Ratio $\frac{d_{1}}{d_{2}}$ is [2014](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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Key Concepts

Pressure Continuity at Fluid Interfaces

In a system of two immiscible liquids in contact, the pressure must be continuous along the interface for equilibrium to be maintained. This means that at any given point along the interface, the hydrostatic pressures derived from the different densities and corresponding vertical heights must be equal, leading to conditions that relate the densities to geometric parameters.

Circular Geometry and Angular Relations

When liquids are contained within a circular tube, the geometry of the tube—and in particular, how arc angles relate to vertical heights—plays a crucial role in determining the pressure distribution. Angles subtended by the liquids and the orientation of the interface (given by an angle relative to the vertical) allow one to convert between arc lengths and vertical displacements, which then link back to differences in hydrostatic pressure.

Hydrostatic Pressure

This fundamental principle states that in a fluid at rest, the pressure increases with depth due to the weight of the liquid above. Pressure differences are given by the product of density, gravitational acceleration, and height difference. This concept is essential when analyzing any system where fluids are subject to gravitational force.

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