Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 12: Fluid Mechanics

Problem 40

Chapter 12, Problem 40

Home Repair. You need to extend a 2.50-inch-diameter pipe, but you have only a 1.00-inch-diameter pipe on hand. You make a fitting to connect these pipes end to end. If the water is flowing at 6.00 cm/s in the wide pipe, how fast will it be flowing through the narrow one?

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Identify the principle to use: This problem involves fluid flow through pipes of different diameters, so we will use the principle of conservation of mass, specifically the continuity equation for incompressible fluids.

Write the continuity equation: The continuity equation states that the product of the cross-sectional area and the velocity of the fluid must be constant along the pipe. Mathematically, this is expressed as A1 * v1 = A2 * v2, where A1 and v1 are the area and velocity in the wide pipe, and A2 and v2 are the area and velocity in the narrow pipe.

Calculate the cross-sectional areas: The area of a circle is given by A = π * (d/2)^2, where d is the diameter. Calculate A1 for the 2.50-inch pipe and A2 for the 1.00-inch pipe using this formula.

Substitute known values into the continuity equation: You know the velocity in the wide pipe (v1 = 6.00 cm/s) and the areas you just calculated. Substitute these into the equation A1 * v1 = A2 * v2 to solve for v2, the velocity in the narrow pipe.

Solve for the unknown velocity: Rearrange the equation to solve for v2, which will give you the velocity of the water in the narrow pipe. This involves dividing both sides of the equation by A2.

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Key Concepts

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

Continuity Equation

The continuity equation in fluid dynamics states that the mass flow rate must remain constant from one cross-section of a pipe to another. This is expressed as A1V1 = A2V2, where A is the cross-sectional area and V is the velocity of the fluid. For incompressible fluids, this principle ensures that a decrease in pipe diameter results in an increase in fluid velocity.

Cross-Sectional Area of a Pipe

The cross-sectional area of a pipe is crucial for determining flow characteristics. It is calculated using the formula A = π(d/2)^2, where d is the diameter of the pipe. In this problem, the areas of the two pipes must be calculated to apply the continuity equation and find the velocity in the narrower pipe.

Incompressible Fluid Flow

Incompressible fluid flow assumes that the fluid density remains constant throughout the flow. This assumption simplifies the analysis of fluid dynamics, allowing the use of the continuity equation. Water is typically treated as incompressible, meaning its flow rate and velocity can be analyzed without accounting for changes in density.

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Textbook Question

A small circular hole 6.00 mm in diameter is cut in the side of a large water tank, 14.0 m below the water level in the tank. The top of the tank is open to the air. Find (a) the speed of efflux of the water and (b) the volume discharged per second.

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Textbook Question

A shower head has 20 circular openings, each with radius 1.0 mm. The shower head is connected to a pipe with radius 0.80 cm. If the speed of water in the pipe is 3.0 m/s, what is its speed as it exits the shower-head openings?

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Textbook Question

Water is flowing in a pipe with a varying cross-sectional area, and at all points the water completely fills the pipe. At point 1 the cross-sectional area of the pipe is 0.070 m², and the magnitude of the fluid velocity is 3.50 m/s. (c) Calculate the volume of water discharged from the open end of the pipe in 1.00 hour.

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Textbook Question

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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Textbook Question

A cubical block of wood, 10.0 cm on a side, floats at the interface between oil and water with its lower surface 1.50 cm below the interface (Fig. E12.33). The density of the oil is 790 kg/m³. (a) What is the gauge pressure at the upper face of the block? (b) What is the gauge pressure at the lower face of the block?

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Textbook Question

BIO. Artery Blockage. A medical technician is trying to determine what percentage of a patient's artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is 1.20×10⁴ Pa, while in the region of blockage it is 1.15×10⁴ Pa. Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at 30.0 cm/s, and the specific gravity of this patient's blood is 1.06. What percentage of the cross-sectional area of the patient's artery is blocked by the plaque?