Question

In a horizontal water pipe that narrows to a smaller radius, the velocity of the water in the section with the smaller radius will be larger. What happens to the pressure? a) The pressure will be the same in both the wider and narrower sections of the pipe. b) The pressure will be higher in the narrower section of the pipe. c) The pressure will be higher in the wider section of the pipe d) It is impossible to tell.

Short answer

Answer: The pressure will be higher in the wider section of the pipe.

Step-by-step solution

Understanding the principle of conservation of mass

The principle of conservation of mass states that the total mass entering a section of the pipe must be equal to the total mass exiting that section. Mathematically, this can be expressed as: A1V1 = A2V2 Where A1 and A2 are the areas of the wider and narrower sections, and V1 and V2 are the velocities in the wider and narrower sections, respectively.

Applying Bernoulli's equation

Bernoulli's equation states that for an incompressible, steady flow and non-viscous fluid, the total energy (kinetic, potential, and pressure energy) along a streamline remains constant. Mathematically, this can be expressed as: P1 + 0.5 * ρ * V1^2 = P2 + 0.5 * ρ * V2^2 Where P1 and P2 are the pressures in the wider and narrower sections, ρ is the fluid density, and V1 and V2 are the velocities in the wider and narrower sections. In this exercise, the pipe is horizontal, so the potential energy due to height remains constant. Therefore, we can focus on the kinetic and pressure energy.

Solving for the pressure difference

From the principle of conservation of mass (A1V1 = A2V2), we know that when the pipe narrows, the velocity of the water (V2) will be larger than the velocity in the wider section (V1). We can use this information and plug it into Bernoulli's equation: P1 + 0.5 * ρ * V1^2 = P2 + 0.5 * ρ * V2^2 To solve for the pressure difference between the two sections, we can rearrange the equation to obtain: P1 - P2 = 0.5 * ρ * (V2^2 - V1^2) As V2 > V1, the expression (V2^2 - V1^2) will be positive. This means that the pressure difference (P1 - P2) will also be positive. Therefore, the pressure will be higher in the wider section of the pipe (P1) compared to the narrower section (P2). So, the correct answer is: c) The pressure will be higher in the wider section of the pipe.

Key concepts

Conservation of Mass

One of the fundamental principles in physics is the conservation of mass. This concept tells us that the mass of a fluid flowing through a system remains constant over time. For a fluid moving through a pipe, like water through a horizontal pipe, this means that the fluid coming into one end of the pipe must equal the fluid exiting the other end.
In mathematical terms, for a pipe with varying cross-sectional areas, this principle is expressed as:\[A_1V_1 = A_2V_2\]where \(A_1\) and \(A_2\) are the cross-sectional areas of the pipe in its wider and narrower parts, respectively, and \(V_1\) and \(V_2\) are the velocities of the fluid in those sections.
This equation indicates that if the pipe narrows (\(A_2\) is smaller than \(A_1\)), the velocity \(V_2\) must increase for the equation to hold true. This ensures that the same amount of mass passes through any two sections of the pipe in a given amount of time.

Fluid Dynamics

Fluid dynamics is a branch of physics that focuses on the behavior of fluids (liquids and gases) in motion. It encompasses various principles and laws, including Bernoulli's Principle, which helps us understand how fluids move and the forces acting on them.
In the context of our water pipe, fluid dynamics tells us that as the water flows and the pipe's cross-section reduces, the velocity of the water increases. This is related to the conservation of mass, as the fluid has to move faster through the narrower part to allow the same mass flow. Fluid dynamics principles reveal how the velocity and pressure of a fluid can change in different environments and under different conditions.

• Fluid movement can be streamlined or exhibit turbulence.
• The factors affecting fluid flow include pressure, velocity, density, and viscosity.
• Bernoulli's equation helps describe the trade-off between velocity and pressure.

In our scenario, the simplified fluid dynamics theory and Bernoulli's Principle together explain the changes in velocity and pressure as water travels through differently-sized sections of piping.

Pressure-Velocity Relationship

The pressure-velocity relationship within a fluid flow is crucial for understanding how energy is distributed in a moving fluid. Bernoulli's Principle, a cornerstone of fluid dynamics, explains this relationship succinctly.
According to Bernoulli's Principle, in a steady, incompressible fluid flow without friction, the total mechanical energy of the fluid remains constant along a streamline. This energy includes kinetic energy (related to velocity) and pressure energy. Therefore:\[P_1 + \frac{1}{2} \rho V_1^2 = P_2 + \frac{1}{2} \rho V_2^2\]where \(P_1\) and \(P_2\) are the pressures at two different points along the streamline, \(V_1\) and \(V_2\) are the velocities, and \(\rho\) is the fluid density.
When the fluid's velocity increases in a section of the pipe (as in the narrower part), its kinetic energy rises. Consequently, to conserve the total energy, the pressure energy must decrease. Thus, the pressure drops in the narrower section, demonstrating that pressure is inversely related to velocity in fluid flows without external energy input. This relationship is crucial for understanding fluid behavior in applications ranging from household plumbing to aerodynamics.