The U-tube in Fig. P2.19 has a 1-cm ID and contains mercury...

The U-tube in Fig. P2.19 has a $1-\mathrm{cm}$ ID and contains mercury as shown. If $20 \mathrm{cm}^{3}$ of water is poured into the right hand leg, what will the free-surface height in each leg be after the sloshing has died down?

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the gravitational force acting on its mass. In a fluid at rest, pressure increases with depth in direct proportion to the density of the fluid and the gravitational acceleration (P = ?gh). This principle is foundational for analyzing fluid columns in connected systems, ensuring that the pressures at the same horizontal level are equal when the system reaches equilibrium.

Manometer Principle

The manometer principle is used to measure pressure differences by comparing the heights of fluid columns in a U-tube or similar setup. In such systems, fluids of different densities adjust until the pressure at the interface is equalized. The resulting height differences depend on the densities involved and allow for the calculation of unknown pressures based on known fluid properties.

Conservation of Volume

Conservation of volume in fluid statics asserts that the total volume of fluid remains constant within a closed system. When additional fluid is introduced into one part of a system, it causes displacement in another, but the overall volume remains unchanged. This concept is crucial for relating the change in fluid heights to the volume added, given the cross-sectional area of the container.

Density Effects in Fluid Columns

Density plays a critical role in determining how the height of a fluid column relates to the pressure exerted at its base. Fluids with higher density require a shorter column to exert the same pressure as a taller column of a lower-density fluid. This relationship is key when analyzing systems with fluids of differing densities, as the height differences adjust so that the pressures at the interface match.

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