Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 25

Water flows through a constant diameter pipe with a uniform velocity given by \(\mathbf{V}=(8 / t+5) \hat{\mathbf{j}} \mathrm{m} / \mathrm{s},\) where \(t\) is in seconds. Determine the acceleration at time \(t=1,2,\) and \(10 \mathrm{s}\)

Short Answer

Expert verified
The acceleration at \(t=1, 2, 10 \mathrm{s}\) are \(-8 \mathrm{m/s}^{2}\), \(-2 \mathrm{m/s}^{2}\), and \(-0.08 \mathrm{m/s}^{2}\) respectively.

Step by step solution

01

Determine the velocity function

Given the velocity function is \(\mathbf{V}=(8 / t+5) \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}\), where \(t\) is in seconds.
02

Find the acceleration function

The acceleration of the water is derived from the velocity function by taking the derivative with respect to time. The derivative of \((8 / t+5)\) with respect to \(t\) results to \(-8 / t^{2} \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}^{2}\). That’s the acceleration function.
03

Find acceleration at given times

Substitute the given time \(t = 1, 2, 10 \mathrm{s}\) into the acceleration function, \(-8 / t^{2} \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}^{2}\). The acceleration at \(t = 1 \mathrm{s}\) is \(-8/1^{2} = -8 \mathrm{m/s}^{2}\). The acceleration at \(t = 2 \mathrm{s}\) is \(-8 / 2^{2} = -2 \mathrm{m/s}^{2}\). The acceleration at \(t = 10 \mathrm{s}\) is \(-8 / 10^{2} = -0.08 \mathrm{m/s}^{2}\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

A fluid flows along the \(x\) axis with a velscity given by \(\mathbf{V}=(x / t) \hat{\mathbf{i}},\) where \(x\) is in feet and \(t\) in seconds. (a) Plot the speed for \(0 \leq x \leq 10\) ft and \(t=3\) s. (b) Plot the speed for \(x=7\) ft and \(2 \leq t \leq 4 \mathrm{s},\) (c) Determine the local and convective acceleration. (d) Show that the acceleration of any fluid particle in the flow is zero. (e) Explain physically how the velocity of a particle in this unsteady flow remains constant throughout its motion.

As is indicated in Fig. \(P 4.42,\) the speed of exhaust in a car's cxhaust pipe varies in time and distance because of the periodic nature of the engine's operation and the damping effect with distance from the engine. Assume that the speed is given \\[ \text { by } \quad V=V_{0}\left[1+a e^{-b x} \sin (\omega t)\right], \quad \text { where } \quad V_{0}=8 \text { fps, } a=0.05 \\] \(b=0.2 \mathrm{ft}^{-1},\) and \(\omega=50 \mathrm{rad} / \mathrm{s} .\) Calculate and plot the fluid acceleration at \(x=0,1,2,3,4,\) and 5 ft for \(0 \leq t \leq \pi / 25\) s.

The following pressures for the air flow in Problem 4.24 were measured: $$\begin{array}{cccc} & x=0 & x=10 \mathrm{m} & x=20 \mathrm{m} \\ \hline t=0 \mathrm{s} & p=101 \mathrm{kPa} & p=101 \mathrm{kPa} & p=101 \mathrm{kPa} \\ t=1.0 \mathrm{s} & p=121 \mathrm{kPa} & p=116 \mathrm{kPa} & p=111 \mathrm{kPa} \\ t=2.0 \mathrm{s} & p=141 \mathrm{kPa} & p=131 \mathrm{kPa} & p=121 \mathrm{kPa} \\ t=3.0 \mathrm{s} & p=171 \mathrm{kPa} & p=151 \mathrm{kPa} & p=131 \mathrm{kPa} . \end{array}$$ Find the local rate of change of pressure \(\partial p / \partial t\) and the convective rate of change of pressure \(V\) on \(/ \partial x\) at \(t=2.0 \mathrm{s}\) and \(x=10 \mathrm{m}.\)

Air flows steadily through a circular, constant-diameter duct. The air is perfectly inviscid, so the velocity profile is flat across cach flow area. However, the air density decreases as the air flows down the duct. Is this a one-, two-, or three-dimensional flow?

The components of a velocity ficld are given by \(u=x+y\) \(v=x y^{3}+16,\) and \(w=0 .\) Determine the location of any stagnation points \((\mathbf{V}=0)\) in the flow field.
See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Quantum Physics

Read Explanation

Scientific Method Physics

Read Explanation

Physical Quantities And Units

Read Explanation

Mechanics and Materials

Read Explanation

Fields in Physics

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.

Sign-up for free