Mercury is poured into a U-tube, as shown in Figure P15.18a....

Mercury is poured into a U-tube, as shown in Figure P15.18a. The left arm of the tube has a cross-sectional area $A_{1}$ of $10.0 \mathrm{~cm}^{2}$, and the right arm has a cross-sectional area $A_{2}$ of $5.00 \mathrm{~cm}^{2}$. One-hundred grams of water are then poured into the right arm, as shown in Figure P15.18b. (a) Determine the length of the water column in the right arm of the U-tube. (b) Given that the density of mercury is $13.6 \mathrm{~g} / \mathrm{cm}^{3}$, what distance $h$ does the mercury rise in the left arm?

Key Concepts

Pressure Balance

Pressure balance involves setting the pressure due to one fluid column equal to the pressure from another fluid column or combination of fluids. This principle is essential when dealing with systems where multiple fluids or additional weights are present, ensuring static equilibrium throughout the system.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It depends on the fluid’s density, gravitational acceleration, and the height of the fluid column above the point of measurement. This concept is used to determine how pressure changes with depth in any fluid-filled container.

Fluid Statics in U-Tubes

Fluid statics in U-tubes refers to the study of fluids at rest in connected containers, particularly how pressure equilibrates in different branches of the tube. In a U-tube, the pressure at corresponding points in the fluid must be the same, leading to predictable differences in fluid level when different fluids or additional masses are introduced.

Cross-Sectional Area Consideration

Considering the cross-sectional area is crucial when analyzing fluid displacement in systems such as a U-tube. A difference in area affects the relationship between the volume of fluid added and the resulting change in fluid height. Smaller areas yield larger height changes for the same volume, while larger areas show smaller changes in height.

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