The U-tube in Fig. P2.19 has a 1-cm ID and contains mercury as shown. If 20 cm^3 of water is pou

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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 righthand leg, what will the free-surface height in each leg be after the sloshing has died down?

Key Concepts

Volume Displacement and Conservation

When a fluid is added to one arm of a U-tube, it causes displacement of the existing fluid such that the total volume of each fluid is conserved. The added fluid increases the height of the liquid column in one limb while causing a corresponding displacement, or drop, in the other limb. The relationship between the cross-sectional area of the tube and the volume change is central to quantifying the height variations in the connected arms.

Density Differences in Immiscible Fluids

In problems involving immiscible fluids, such as water and mercury, the significant difference in fluid densities plays a crucial role in the final arrangement of the fluid levels. Because mercury is much denser than water, a small change in the mercury column height corresponds to a larger pressure change compared to water. Recognizing the impact of these density differences is essential to correctly applying the hydrostatic equilibrium condition at the interface and solving for the height levels in each leg.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. The pressure within a fluid increases with depth according to the relationship p = p? + ?gh, where ? is the fluid density, g is the gravitational acceleration, and h is the height of the fluid column. This concept is key to understanding how different heights of fluid in connected vessels provide the necessary pressure balance in a static fluid system.

Equilibrium in U-tubes

The equilibrium condition in a U-tube requires that the pressures at the same horizontal level in the connected arms must be equal. When an additional volume is introduced in one limb, the fluid levels adjust until the hydrostatic pressures are balanced across both limbs. This principle allows us to determine how much each fluid column will rise or fall in response to added volumes and differences in fluid density.

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