A circular tube of unifonn cross-scction is filled with two...

A circular tube of unifonn cross-scction is filled with two liquids of densitics $\rho_{1}$ and $\rho_{2}$, such that half of each liquid occupies a quarter of volume of the tube. If the line joining the free surface of the liquids makes an angle $\theta$ with horizontal, find the value of $\theta$.

Key Concepts

Hydrostatic Pressure Variation

In fluid statics, pressure increases with depth according to the relation ?p = ?gh, where ? is the density of the fluid, g is the acceleration due to gravity, and h is the vertical height difference. This principle is fundamental when analyzing problems that involve fluids at rest, as it provides the means to relate differences in height to differences in pressure.

Free Surface Conditions

A free surface in a static fluid, which is usually exposed to the atmosphere, must be at constant pressure. This condition dictates that if a fluid is at rest, its free surface must adjust (often to become horizontal) to ensure that every point along it has the same pressure. When multiple fluids are involved, the requirement leads to specific geometric configurations of the fluid interfaces.

Interface Pressure Continuity

In systems involving two immiscible fluids, the pressure at the interface where the fluids meet must be continuous. This means that the hydrostatic pressure contributions from each fluid, which depend on their respective densities and vertical heights, must balance at points along the interface. This concept is crucial for determining the orientation of the boundary line between the fluids.

Geometric Analysis of Fluid Volumes in Containers

Describing the shape and position of fluid regions within a container often involves linking their volumes to geometric properties of the container’s cross-section. For containers with circular cross-sections, this typically requires analyzing circular segments and areas. Understanding such geometric relationships is key to solving problems where specific volume distributions lead to particular orientations or slopes of free surfaces.

Please give Ace some feedback