A small uniform tube is bent into a circle of radius r whose...

A small uniform tube is bent into a circle of radius $r$ whose plane is vertical. The equal volumes of two fluids whose densities are $\rho$ and $\sigma(\rho>\sigma)$, fill half the circle. Find the angle that the radius passing through the interface makes with the vertical: (a) $\cot \theta=\frac{\rho-\sigma}{\rho+\sigma}$ (b) $\tan \theta=\frac{\rho-\sigma}{\rho+\sigma}$ (c) $\sin \theta=\frac{\rho+\sigma}{\rho-\sigma}$ (d) $\sin \theta=\frac{\rho}{\sigma}$

A small uniform tube is bent into a circle of radius $r$ whose plane is vertical. The equal volumes of two fluids whose densities are $\rho$ and $\sigma(\rho>\sigma)$, fill half the circle. Find the angle that the radius passing through the interface makes with the vertical:
(a) $\cot \theta=\frac{\rho-\sigma}{\rho+\sigma}$
(b) $\tan \theta=\frac{\rho-\sigma}{\rho+\sigma}$
(c) $\sin \theta=\frac{\rho+\sigma}{\rho-\sigma}$
(d) $\sin \theta=\frac{\rho}{\sigma}$

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Key Concepts

Circular Geometry Analysis

When dealing with a circular or curved container, the geometry of the container plays a crucial role. The circular structure introduces relationships involving arc lengths, central angles, and chords, which are essential for calculating the distribution of fluids along the curve under the influence of gravity.

Volume Conservation

The principle of volume conservation asserts that the total volume of fluid in a container remains constant under equilibrium conditions. When equal volumes of two fluids are used, this constraint interacts with the container's geometry and the fluids' densities, thereby influencing the position or orientation of the interface between the fluids.

Hydrostatic Pressure

This concept describes how the pressure in a fluid at rest increases with depth due to the weight of the fluid above. It is fundamental when analyzing fluids in gravitational fields, as it determines the pressure variation within each fluid based on its density and depth.

Interface Equilibrium

At the boundary between two immiscible fluids of differing densities, the pressure must be continuous. This condition ensures that at the interface, the pressures from the two fluids balance, and it serves as a key criterion for determining the location or angle of the interface between the fluids.

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