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. Pl4.71a). Oil having a density of 750 $\mathrm{kg} / \mathrm{m}^{3}$ is then poured into the right arm and forms a column $L=5.00 \mathrm{cm}$ high (Fig. Pl4.71b). (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. Pl4.71c). 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

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It increases with depth and is given by the product of the fluid’s density, gravitational acceleration, and the height of the fluid column. This concept is crucial for understanding how pressures balance in connected columns of liquid.

U-Tube Fluid Statics

The analysis of a U-tube involves applying the principle of hydrostatic equilibrium across connected fluid columns. In a U-tube, different liquids or additional columns of a second liquid can shift the balance, leading to differences in height that reflect the differences in density and pressure. This concept is used to determine how height differences develop in a system with multiple fluids.

Bernoulli’s Principle

Bernoulli’s principle relates the pressure, velocity, and elevation in a flowing fluid, stating that an increase in the speed of the fluid occurs simultaneously with a decrease in its pressure or potential energy. This principle is applied in scenarios where fluid motion (such as air blown over a liquid surface) affects the pressure at the interface and, consequently, the equilibrium of the system.

Dynamic Pressure

Dynamic pressure represents the kinetic energy per unit volume of a moving fluid and is calculated as one half the product of the fluid density and the square of its velocity. This is important in situations where the motion of a fluid (like wind) impacts the static pressure at a surface, altering the balance of forces in a fluid system.

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