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Ch 12: Fluid Mechanics
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 12: Fluid MechanicsProblema 46
Capitolo 12, Problema 46

At one point in a pipeline the water's speed is 3.00 m/s and the gauge pressure is 5.00×104 Pa. Find the gauge pressure at a second point in the line, 11.0 m lower than the first, if the pipe diameter at the second point is twice that at the first.

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Identify the known variables: initial speed \( v_1 = 3.00 \text{ m/s} \), initial gauge pressure \( P_1 = 5.00 \times 10^4 \text{ Pa} \), height difference \( h = 11.0 \text{ m} \), and the relationship between the diameters of the pipe at the two points.
Apply the principle of conservation of mass, which states that the mass flow rate must be constant. This can be expressed as \( A_1 v_1 = A_2 v_2 \), where \( A \) is the cross-sectional area of the pipe. Since the diameter at the second point is twice that at the first, \( A_2 = 4A_1 \). Solve for \( v_2 \) to find the speed at the second point.
Use Bernoulli's equation to relate the pressures and velocities at the two points: \( P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \). Here, \( \rho \) is the density of water, and \( g \) is the acceleration due to gravity. Substitute \( h_1 = 0 \) and \( h_2 = -11.0 \text{ m} \) to account for the height difference.
Rearrange Bernoulli's equation to solve for the gauge pressure at the second point \( P_2 \): \( P_2 = P_1 + \frac{1}{2} \rho (v_1^2 - v_2^2) + \rho g h \).
Substitute the known values into the equation to calculate \( P_2 \). Remember to use the density of water \( \rho = 1000 \text{ kg/m}^3 \) and the acceleration due to gravity \( g = 9.81 \text{ m/s}^2 \).

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Bernoulli's Equation

Bernoulli's Equation relates the speed, pressure, and height of a fluid in steady flow. It states that the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline. This principle helps determine how changes in speed and height affect pressure in a fluid system.
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Kinematics Equations

Continuity Equation

The Continuity Equation is a principle of fluid dynamics that states the mass flow rate must remain constant from one cross-section of a pipe to another. It is expressed as A1V1 = A2V2, where A is the cross-sectional area and V is the fluid velocity. This concept is crucial for understanding how changes in pipe diameter affect fluid speed.
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Flow Continuity

Hydrostatic Pressure

Hydrostatic Pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth in a fluid and is given by the equation P = ρgh, where ρ is the fluid density, g is the acceleration due to gravity, and h is the height difference. This concept is essential for calculating pressure changes due to elevation differences in a fluid system.
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Pressure and Atmospheric Pressure
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