A U-tube open at both ends is partially filled with water...

A U-tube open at both ends is partially filled with water (Fig. P14.69a). Oil having a density 750 $\mathrm{kg} / \mathrm{m}^{3}$ is then poured into the right arm and forms a column $L=$ 5.00 $\mathrm{cm}$ high (Fig. Pl4.69b). (a) Determine the difference $h$ in the heights of the two liquid surfaces. (b) The right arm is then shielded from any air motion while air is blown across the top of the left arm until the surfaces of the two liquids are at the same height (Fig. Pl4.69 c). Determine the speed of the air being blown across the left arm. Take the density of air as 1.29 $\mathrm{kg} / \mathrm{m}^{3}$ .

Key Concepts

Bernoulli’s Equation and Dynamic Pressure

Bernoulli’s equation relates the pressure, velocity, and height within flowing fluids, identifying that an increase in fluid speed leads to a decrease in static pressure. In scenarios where a moving fluid (such as air) interacts with a fluid at rest, dynamic pressure (calculated as one half the density of the moving fluid times the square of its speed) can be used to counterbalance static pressure differences, thereby achieving a new equilibrium condition.

Pressure Equilibrium in Connected Fluid Systems

This concept involves understanding how pressure is balanced in connected vessels containing fluids. In a U-tube or manometer, even when different liquids are present, the pressures at points connected by the same continuous fluid must equalize. This principle is important when additional forces or fluids are introduced, requiring the system to adjust and reach a new equilibrium state.

Hydrostatic Pressure

This concept deals with the pressure exerted by a fluid at rest due to gravity. It explains that the pressure at a certain depth in a fluid is a function of the fluid’s density, the acceleration due to gravity, and the height of the fluid column above that point. This principle is used to understand and calculate how differences in fluid height translate into pressure differences across connected columns.

Fluid Density and Pressure Relationship

Fluid density plays a crucial role in determining the pressure exerted by a fluid column. In problems involving multiple liquids, the density of each fluid affects the hydrostatic pressure contribution. Recognizing that lower density fluids contribute less pressure per unit height than higher density fluids is essential in analyzing equilibrium conditions where different fluids interact.

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