Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 12: Fluid Mechanics

문제 40

12장, 문제 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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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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