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Ch. 13 - Fluids
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 13 - FluidsProblem 102
Chapter 13, Problem 102

A drinking fountain shoots water about 12 cm up in the air from a nozzle of diameter 0.60 cm (Fig. 13–67). The pump at the base of the unit (1.1 m below the nozzle) pushes water into a 1.2-cm-diameter supply pipe that goes up to the nozzle. What gauge pressure does the pump have to provide? Ignore the viscosity; your answer will therefore be an underestimate.
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Step 1: Identify the key principles involved. This problem involves fluid dynamics and can be solved using Bernoulli's equation, which relates pressure, velocity, and height in a fluid system. Additionally, the continuity equation will help relate the velocities in the supply pipe and the nozzle.
Step 2: Write Bernoulli's equation for the system. The equation is: P1 + 12ρv1^2 + ρgh1 = P2 + 12ρv2^2 + ρgh2 Here, ρ is the density of water, g is the acceleration due to gravity, and h is the height. Subscripts 1 and 2 refer to the supply pipe and the nozzle, respectively.
Step 3: Use the continuity equation to relate the velocities in the supply pipe and the nozzle. The continuity equation is: A1v1 = A2v2 Here, A is the cross-sectional area of the pipe or nozzle. Calculate the areas using the formula for the area of a circle: A = π4d2 where d is the diameter.
Step 4: Solve for the velocity at the nozzle. The water is shot 12 cm into the air, so use the kinematic equation to find the velocity at the nozzle: v2 = 2gh where h is the height of 12 cm (converted to meters).
Step 5: Substitute the known values into Bernoulli's equation. Set the pressure at the nozzle to atmospheric pressure (gauge pressure is zero at the nozzle). Solve for the gauge pressure at the pump, P1, considering the height difference of 1.1 m and the velocities calculated using the continuity equation and kinematics.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bernoulli's Principle

Bernoulli's Principle states that in a flowing fluid, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle is essential for understanding how the water is propelled upward from the nozzle, as it relates the pressure at the pump to the kinetic energy of the water exiting the nozzle.
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Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. In this scenario, the pressure at the pump must overcome the hydrostatic pressure created by the height difference (1.1 m) between the pump and the nozzle, which is crucial for determining the required gauge pressure.
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Continuity Equation

The Continuity Equation states that for an incompressible fluid, the mass flow rate must remain constant from one cross-section of a pipe to another. This concept helps in understanding how the change in diameter from the supply pipe to the nozzle affects the velocity of the water, which is necessary for calculating the pressure needed at the pump.
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Related Practice
Textbook Question

You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are holding the hose horizontally, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening did you block?

Textbook Question

Estimate the total mass of the Earth’s atmosphere, using the known value of atmospheric pressure at sea level.

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Textbook Question

A ship, carrying fresh water to a desert island in the Caribbean, has a horizontal cross-sectional area of 2240 m² at the waterline. When unloaded, the ship rises 8.55 m higher in the sea. How much water (m³) was delivered?