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Chapter 12: Problem 85

Water is flowing through a pipe of constant cross-section. At some point the pipe becomes narrow and the cross-section is halved. The speed of water is \mathrm{\\{} I U P ~ S E E ~ 2 0 0 5 ] ~ (a) reduced to zero (b) decreased by factor of 2 (c) increased by a factor of 2 (d) unchanged

Short Answer

Expert verified
(c) increased by a factor of 2

Step by step solution

01

Understanding the Continuity Equation

In fluid dynamics, the continuity equation states that the mass flow rate must remain constant from one cross-section of a pipe to another as long as there are no leaks or additional inputs. This is given by the equation: \( A_1v_1 = A_2v_2 \), where \( A \) is the cross-sectional area and \( v \) is the fluid velocity.
02

Identifying Known Values

Before the pipe narrows, let the cross-sectional area be \( A_1 \) and the velocity be \( v_1 \). After the pipe narrows, the cross-sectional area becomes \( A_2 = \frac{1}{2}A_1 \). We need to determine \( v_2 \), the velocity after the pipe narrows.
03

Setting Up the Equation

Substitute the known values into the continuity equation: \( A_1v_1 = \left(\frac{1}{2}A_1\right)v_2 \). This equation will allow us to solve for \( v_2 \), the new velocity after the narrowing of the pipe.
04

Solving for New Velocity

Simplify the equation: \( v_2 = 2v_1 \). This shows that the velocity of the water increases by a factor of 2 when the cross-sectional area is halved.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fluid Dynamics
Fluid dynamics is a branch of physics that focuses on the behavior of fluids (liquids and gases) in motion. It plays a crucial role in understanding how liquids like water flow through various channels, such as pipes, rivers, or even the bloodstream. At the heart of fluid dynamics is the study of how these flows behave under different conditions, like changes in pressure, volume, and temperature.
One fundamental principle in fluid dynamics is the continuity equation, which ensures that the total amount of fluid flowing into a system is equal to the amount flowing out, assuming there's no loss due to leakage. This principle is crucial because it helps predict and analyze fluid movement in a closed system, ensuring the conservation of mass.
Fluid dynamics is applied in various fields, including engineering, meteorology, marine biology, and even in designing everyday items like water pipes, car engines, and air conditioning systems. Understanding this concept helps make informed decisions regarding fluid flow in real-world applications.
Cross-sectional Area
The cross-sectional area of a pipe or channel is simply the surface area of the pipe's opening as seen from the direction of fluid flow. It is a crucial factor in determining how fluid moves through a channel.
A larger cross-sectional area allows more fluid to pass through, much like a wide doorway allowing more people to walk through at once. Conversely, in a narrow pipe, less fluid can flow through at the same time, affecting the flow speed.
  • For a circular pipe, the cross-sectional area ( \(A\)) is calculated using the formula: \(A = \pi r^2\), where \(r\) is the radius of the pipe.
The continuity equation showcases how the cross-sectional area interacts with velocity. If the cross-sectional area of a pipe reduces, as in the example exercise where the area is halved, the fluid's velocity increases proportionally to maintain a constant flow rate. This principle is pivotal for understanding how alterations in pipe dimensions can impact the behavior of fluid flow.
Velocity of Fluid Flow
The velocity of fluid flow refers to the speed at which fluid moves through a conduit, like a pipe. It is a key parameter in fluid dynamics, influencing factors such as pressure, flow rate, and energy exchange within systems.
Velocity can change based on several factors:
  • Cross-sectional Area: A change in the area where fluid flows causes a change in velocity. According to the continuity equation ( \(A_1v_1 = A_2v_2\)), if the area decreases, velocity must increase to maintain the same flow rate.
  • Pressure Differences: Fluid velocity is also affected by pressure differences along its path, with fluids moving from high-pressure areas to low-pressure areas, accelerating in the process.
In the exercise example, the pipe narrows, halving the cross-sectional area. Consequently, the velocity of water increases by a factor of two to compensate for the reduced area, ensuring the mass flow rate remains constant. Understanding how velocity interacts with other factors in fluid dynamics is vital for applications like pipeline design, water supply systems, and even in natural phenomena like river currents.

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Most popular questions from this chapter

Glycerine flows steadily through a horizontal tube of length \(1.5 \mathrm{~m}\) and radius \(1.0 \mathrm{~cm}\). If the amount of glycerine flowing per second at one end is \(40 \times 10^{-1} \mathrm{~kg} / \mathrm{s}\). What is the pressure difference between the two ends of the tube? (Density of glycerine \(=13 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) and viscosity of glycerine \(=083 \mathrm{~Pa}-\mathrm{s})\). (a) \(9.75 \times 10^{2} \mathrm{~Pa}\) (b) \(6.75 \times 10^{2} \mathrm{~Pa}\) (c) \(5.75 \times 10^{2} \mathrm{~Pa}\) (d) \(6.95 \times 10^{3} \mathrm{kPa}\)

A hemispherical bowl just floats without sinking in a liquid of density \(1.2 \times 10^{3} \mathrm{kgm}^{-3}\). If outer diameter and the density of the bowl are \(1 \mathrm{~m}\) and \(2 \times 10^{4} \mathrm{kgm}^{-3}\) respectively, then the inner diameter of the bowl will be (a) \(0.94 \mathrm{~m}\) (b) \(0.96 \mathrm{~m}\) (c) \(0.98 \mathrm{~m}\) (d) \(0.99 \mathrm{~m}\)

An incompressible fluid flows steadily through a cylindrical pipe which has radius \(2 R\) at a point \(A\) and radius \(R\) at a point \(B\). Further along the flow of direction if the velocity at point \(A\) is \(v\), its velocity at point \(B\) will be (a) \(v / 4\) (b) \(2 v\) (c) \(4 \underline{v}\) (d) \(-\frac{v}{2}\)

A capillary tube of radius \(R\) and length \(L\) is connected in series with another tube of radius \(R / 2\) and length \(L / 4\). If the pressure difference across the two tubes taken together is \(p\), then the ratio of pressure difference across the first tube to that across the second tube is (a) \(1: 4\) (b) \(1: 1\) (c) \(4: 1\) (d) \(2: 1\)

A plane is in level flight at a constant apeed and each wing has an area of \(25 \mathrm{~m}^{2}\). During flight the speed of the air is \(216 \mathrm{kmh}^{-1}\) over the lower wing surface and \(252 \mathrm{kmh}^{-1}\) over the upper wing surface of each wing of aeroplane. Take density of air \(=1 \mathrm{kgm}^{-3}\) and \(g=10 \mathrm{~ms}^{-2}\) The mass of the plane is (a) \(25 \mathrm{~kg}\) (b) \(250 \mathrm{~kg}\) (c) \(1750 \mathrm{~kg}\) (d) \(3250 \mathrm{~kg}\)
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