/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Assume that the drag on a small ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 23

Assume that the drag on a small sphere placed in a rapidly moving stream of fluid depends on the fluid density but not the fluid viscosity. Use dimensional analysis to determine how the drag is affected if the velocity of the fluid is doubled.

Short Answer

Expert verified
Doubling the fluid velocity quadruples the drag force.

Step by step solution

01

Identify Physical Quantities

We need to identify the physical quantities involved in the problem: drag force \( F \), fluid density \( \rho \), velocity \( v \), and the radius of the sphere \( r \). The problem states that drag does not depend on fluid viscosity.
02

Write Dimensional Equations

The dimensions for each physical quantity are as follows: Drag force \( F \) has dimensions \([M][L][T]^{-2}\), fluid density \( \rho \) is \([M][L]^{-3}\), velocity \( v \) is \([L][T]^{-1}\), and radius \( r \) is \([L]\).
03

Formulate a Dimensional Equation

Assume that the drag force \( F \) is proportional to \( \rho^a \), \( v^b \), and \( r^c \). This gives the equation: \( F = C \rho^a v^b r^c \), where \( C \) is a dimensionless constant.
04

Substitute Dimensions

Substitute the dimensions into the equation: \([M][L][T]^{-2} = [M]^a [L]^{-3a} [L]^b [T]^{-b} [L]^c\). This simplifies to: \([M]: a, [L]: -3a + b + c, [T]: -b\).
05

Solve for Exponents Using Dimensional Homogeneity

By comparing dimensions, we set up the following system of equations: 1. \( a = 1 \) for mass dimensions. 2. \( -3a + b + c = 1 \) for length dimensions. 3. \( -b = -2 \) for time dimensions.
06

Solve the System of Equations

From Equation 3, \( b = 2 \). Using \( a = 1 \) and substituting into Equation 2 gives \( -3(1) + 2 + c = 1 \), resulting in \( c = 2 \).
07

Write Final Equation Relating Drag Force

The drag force equation is \( F \propto \rho v^2 r^2 \). It shows that the drag force is proportional to \( v^2 \).
08

Calculate Effect of Doubling Velocity

If the velocity \( v \) is doubled, the new drag force \( F' \) becomes \( F' = \rho (2v)^2 r^2 \). Simplifying gives \( F' = 4 \rho v^2 r^2 = 4F \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Drag Force
Drag force is a significant concept in the field of Fluid Dynamics. It refers to the force exerted by a fluid, such as air or water, against an object moving through it. This force is crucial in various applications, from designing cars to understanding natural phenomena.
  • Drag force acts in the opposite direction of the object's movement, attempting to slow it down.
  • It depends on several factors, including the object's shape, size, and speed, as well as the fluid's density and velocity.
  • In the exercise provided, we see that drag force is not influenced by the fluid's viscosity, which can simplify analysis.
Understanding how parameters like velocity affect drag force, as demonstrated, is vital for optimizing designs and improving performance in engineering tasks.
Fluid Dynamics
Fluid dynamics is the study of how fluids (liquids and gases) move and the forces acting on them. It is a branch of fluid mechanics that is fundamental in predicting how liquids and gases behave in different situations.
One of the core aspects of fluid dynamics is understanding the behavior of flow, which can be:
  • Laminar, where the fluid flows in parallel layers with minimal mixing.
  • Turbulent, where there is chaotic fluid motion.
In the context of the given problem, fluid dynamics helps determine how the velocity impacts the drag force on a sphere. The faster the fluid moves, the greater the impact on the drag force due to increased momentum transfer. Solutions often involve using equations like the Navier-Stokes equations, which are key in fluid dynamics to predict future behavior of fluids under various forces.
Dimensional Homogeneity
Dimensional homogeneity is a principle used in dimensional analysis to ensure that equations are physically meaningful. It states that every term in a physical equation must have the same dimension, which allows for the comparison and solving of complex problems.
In the exercise, we applied dimensional homogeneity to equate the dimensions of involved quantities such as mass \([M]\), length \([L]\), and time \([T]\). This approach led us to:
  • Identify relationships between variables like drag force, fluid density, and velocity.
  • Ensure the derived equation is consistent in its dimensions.
  • Create a proportionality between these variables indicating that the drag force is proportional to velocity squared, among other parameters.
Using dimensional homogeneity helps in verifying that a physical equation is correct and can lead to solutions in engineering and physics, particularly when dealing with scaling scenarios and complex systems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

7.62 A thin rectangular plate is towed through seawater at an average velocity of 5 mph. The plate is held in a vertical position and projects above the undisturbed level of the water to a height \(z . A 1: 4\) scale model is to be used to predict the drag on the plate, and the model fluid is also seawater. (a) Assuming that Froude number similarity must be maintained, determine the required model velocity. (b) What is the required value of \(z_{m} / z ?\) (c) A measured drag of 1 lb on the model will correspond to what drag on the prototype?

A liquid flows with a velocity \(V\) through a hole in the side of a large tank. Assume that \\[V=f(h, g, \rho, \sigma)\\] where \(h\) is the depth of fluid above the hole, \(g\) is the acceleration of gravity, \(\rho\) the fluid density, and \(\sigma\) the surface tension. The following data were obtained by changing \(h\) and measuring \(V,\) with a fluid having a density \(=10^{3} \mathrm{kg} / \mathrm{m}^{3}\) and surface tension \(=0.074 \mathrm{N} / \mathrm{m}\) \begin{tabular}{l|l|l|l|l|l} \(V(\mathrm{m} / \mathrm{s})\) & 3.13 & 4.43 & 5.42 & 6.25 & 7.00 \\ \hline\(h(\mathrm{m})\) & 0.50 & 1.00 & 1.50 & 2.00 & 2.50 \end{tabular} Plot these data by using appropriate dimensionless variables. Could any of the original variables have been omitted?

For a certain model study involving a 1: 5 scale model it is known that Froude number similarity must be maintained. The possibility of cavitation is also to be investigated, and it is assumed that the cavitation number must be the same for model and prototype. The prototype fluid is water at \(30^{\circ} \mathrm{C},\) and the model fluid is water at \(70^{\circ} \mathrm{C}\). If the prototype operates at an ambient pressure of \(101 \mathrm{kPa}\) (abs), what is the required ambient pressure for the model system?

Assume that the drag, \(\mathscr{D},\) on an aircraft flying at supersonic speeds is a function of its velocity, \(V\), fluid density, \(\rho\) speed of sound, \(c,\) and a series of lengths, \(\ell_{1}, \ldots, \ell_{i},\) which describe the geometry of the aircraft. Develop a set of pi terms that could be used to investigate experimentally how the drag is affected by the various factors listed. Form the pi terms by inspection.

A cone and plate viscometer consists of a cone with a very small angle \(\alpha\) which rotates above a flat surface as shown in Fig. \(\mathrm{P} 7.17 .\) The torque, \(\mathscr{T},\) required to rotate the cone at an angular velocity, \(\omega,\) is a function of the radius, \(R,\) the cone angle, \(\alpha,\) and the fluid viscosity, \(\mu,\) in addition to \(\omega .\) With the aid of dimensional analysis, determine how the torque will change if both the viscosity and angular velocity are doubled.
See all solutions

Recommended explanations on Physics Textbooks

Particle Model of Matter

Read Explanation

Turning Points in Physics

Read Explanation

Wave Optics

Read Explanation

Electromagnetism

Read Explanation

Thermodynamics

Read Explanation

Force

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.