Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 79

A stream of atmospheric air is used to keep a ping-pong ball aloft by blowing the air upward over the ball. The ping-pong ball has a mass of \(2.5 \mathrm{g}\) and a diameter \(D_{1}=3.8 \mathrm{cm},\) and the air stream has an upward velocity of \(V_{1}=0.942 \mathrm{m} / \mathrm{s}\). This system is to be modeled by pumping water upward with a velocity \(V_{2}\) over a solid ball of diameter \(D_{2}\) and density \(\rho_{b_{2}}=2710 \mathrm{kg} / \mathrm{m}^{3} .\) In both cases, the net weight of the ball \(W_{b}\) is equal to the air drag, $$\mathrm{W}_{b}=\frac{\mathrm{C}_{\mathrm{D}} \rho A V^{2}}{2}$$where \(\mathrm{C}_{\mathrm{D}}=0.60, \rho\) is the fluid density, \(A\) the ball's projected area, and \(V\) the velocity of the fluid upstream from the ball. Determine all possible combinations of \(V_{2}\) and \(D_{2}\). [Hint: A force balance involving the drag on the ball, the buoyant force on the ball, and the weight of the ball is needed.]

Short Answer

Expert verified
To get all possible combinations of \(V_2\) and \(D_2\), adopt a numerical approach to solve the system of equations obtained in step 2.

Step by step solution

01

Setup the force balance equation

Given that the weight of the ball \(W_b\) is equal to the air drag, we set up the equation:\[W_b = \frac{C_D \cdot \rho \cdot A \cdot V^2}{2}\]where \(C_D = 0.60 (Drag Coefficient)\), \(\rho\) is the fluid density, \(A\) is ball's projected area, and \(V\) is the velocity of the fluid upstream from the ball. We know that the force on the ball is the drag force, which is balanced by the weight of the ball and the buoyant force. So, add the buoyant force \(B = \rho \cdot g \cdot V\) to the equation.
02

Substitute the values and simplify the equation

Substitute the given values into the equation and solve for \(V_2\) and \(D_2\). First, substitute for \(W_b = m_b \cdot g\), where \(m_b\) is the mass of the ball and \(g\) is the gravitational constant. Rearrange to express \(A\) in terms of diameter \(D\) and substitute (\(A=\pi \cdot (D/2)^2\)). Substitute the given values and solve for \(V_2\) and \(D_2\).
03

Find all possible combinations of \(V_2\) and \(D_2\)

The system of equations for \(V_2\) and \(D_2\) is generally difficult to solve analytically. But, it can be solved numerically using a numerical solver or a software utility, which can provide all possible combinations of \(V_2\) and \(D_2\) that satisfy the set-up equations.

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

The pressure rise, \(\Delta p=p_{2}-p_{1},\) across the abrupt expansion of Fig. \(\mathrm{P} 7.38\) through which a liquid is flowing can be expressed as $$\Delta p=f\left(A_{1}, A_{2}, \rho, V_{1}\right)$$ where \(A_{1}\) and \(A_{2}\) are the upstream and downstream cross-sectional areas, respectively, \(\rho\) is the fluid density, and \(V_{1}\) is the upstream velocity. Some experimental data obtained with \(A_{2}=1.25 \mathrm{ft}^{2}\) \(V_{1}=5.00 \mathrm{ft} / \mathrm{s},\) and using water with \(\rho=1.94\) slugs/ft \(^{3}\) are given in the following table: $$\begin{array}{l|l|l|l|l|r} A_{1}\left(\mathrm{ft}^{2}\right) & 0.10 & 0.25 & 0.37 & 0.52 & 0.61 \\ \hline \Delta p\left(\mathrm{lb} / \mathrm{ft}^{2}\right) & 3.25 & 7.85 & 10.3 & 11.6 & 12.3 \end{array}$$ Plot the results of these tests using suitable dimensionless parameters. With the aid of a standard curve fitting program determine a general equation for \(\Delta p\) and use this equation to predict \(\Delta p\) for water flowing through an abrupt expansion with an area ratio \(A_{1} / A_{2}=0.35\) at a velocity \(V_{1}=3.75 \mathrm{ft} / \mathrm{s}\).

A model hydrofoil is to be tested. Is it practical to satisfy both the Reynolds number and the Froude number for the hydrofoil when it is operating near the water surface? Support: your decision.

A 250 -m-long ship has a wetted area of \(8000 \mathrm{m}^{2} .\) A \(\frac{1}{100}\) -scale model is tested in a towing tank with the prototype fluid, and the results are: $$\begin{array}{|l|l|l|l|} \text { Model velocity }(\mathrm{m} / \mathrm{s}) & 0.57 & 1.02 & 1.40 \\ \hline \text { Model drag }(\mathrm{N}) & 0.50 & 1.02 & 1.65 \end{array}$$ Calculate the prototype drag at \(7.5 \mathrm{m} / \mathrm{s}\) and \(12.0 \mathrm{m} / \mathrm{s}\).

The speed of deep ocean waves depends on the wave length and gravitational acceleration. What are the appropriate dimensionless parameters?

The dimensional parameters used to describe the operation of a ship or airplane propeller (sometimes called a screw propeller) are rotational speed, \(\omega,\) diameter, \(D,\) fluid density, \(\rho\) speed of the propeller relative to the fluid, \(V\), and thrust developed, \(T .\) The common dimensionless groups are called the thrust coefficient and the advance ratio. Propose appropriate definitions for these groups.
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Waves Physics

Read Explanation

Dynamics

Read Explanation

Quantum Physics

Read Explanation

Turning Points in Physics

Read Explanation

Kinematics 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