/*! 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 60 A hot-air balloon has a volume o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 60

A hot-air balloon has a volume of 2200 \(\mathrm{m}^{3}\) . The balloon fabric (the envelope) weighs 900 \(\mathrm{N}\) . The basket with gear and full propane tanks weighs 1700 \(\mathrm{N}\) . If the balloon can barely lift an additional 3200 \(\mathrm{N}\) of passengers, breakfast, and champagne when the outside air density is \(1.23 \mathrm{kg} / \mathrm{m}^{3},\) what is the average density of the heated gases in the envelope?

Short Answer

Expert verified
The average density of the heated gases is approximately \(0.268 \ \mathrm{kg/m}^3\).

Step by step solution

01

Calculate Total Lift Force Needed

First, add up all the weights that need to be lifted by the balloon, including the envelope, basket, and cargo (passengers, breakfast, and champagne). This is the total lift force that the hot air inside the balloon needs to generate.\[\text{Total weight carried} = 900 + 1700 + 3200 = 5800\text{ N}\]
02

Calculate the Lift Force Provided by Buoyancy

The lift force from the balloon is equal to the weight of the air displaced by the balloon. To find the buoyant force, multiply the volume of the balloon by the outside air density and gravitational acceleration (\(g = 9.8 \ \mathrm{ m/s}^2\)).\[\text{Buoyant Force} = 2200 \ \mathrm{m}^3 \times 1.23 \ \mathrm{kg/m}^3 \times 9.8 \ \mathrm{m/s}^2\]
03

Calculate Required Density of Heated Gas

The upward lift provided by the balloon should equal the total weight carried; thus, use the buoyant force equation relating it to the gravitational force on the total weight of the contents (including heated gas inside the balloon):\[ \text{Lift Force} = V \times \text{density of heated gas} \times g \].Substitute the buoyant force and solve for the density of the heated gas:\[ 5800 \ = 2200 \times \text{density of heated gas} \times 9.8 \]Re-arranging gives:\[\text{density of heated gas} = \frac{5800}{2200 \times 9.8}\]
04

Solve for Average Density of Heated Gas

Compute the numerical value using the expression from the previous step:\[\text{density of heated gas} = \frac{5800}{2200 \times 9.8} \approx 0.268 \ \mathrm{kg/m}^3\]

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.

Density
Density is a fundamental concept in fluid dynamics that describes how much mass exists in a given volume of a substance. It's an essential parameter when analyzing buoyancy, as it helps to determine the lift force a fluid can exert on an object.
To calculate density, use the formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). For the hot-air balloon, understanding the density of the heated gas compared to the outside air is crucial because it affects how much lift the balloon can generate.
Generally, lower density fluids tend to rise when surrounded by higher density fluids, a principle that allows hot-air balloons to float.
Lift Force
Lift force is the upward force that occurs when there is a difference in density between two fluids or two regions of the same fluid. In the case of a hot-air balloon, the lift force is generated when the air inside the balloon is heated, causing it to expand and decrease in density as compared to the cooler, denser air outside.
Here's how the lift force of the hot-air balloon works:
  • The total weight the balloon needs to lift consists of the weight of the envelope, basket, and cargo.
  • The balloon fabric and basket weigh a combined 2600 N. The cargo adds another 3200 N, resulting in a total lift requirement of 5800 N.
The lift provided by the balloon must equal this total weight. If the heated air inside the balloon has a density low enough compared to the surrounding air, it creates the necessary lift for the balloon to rise.
Hot-Air Balloon
The operation of a hot-air balloon is a practical application of fluid dynamics principles. It consists of an envelope (the large balloon part), a basket for carrying passengers and gear, and a propane burner to heat the air inside the envelope.
Here鈥檚 how it functions:
  • The heated air becomes less dense than the cooler air outside, creating buoyant force.
  • The buoyant force needs to overcome the total downward force due to the weight of the entire system, which includes the envelope, basket, and any passengers or cargo.
The difference in air density inside and outside the balloon directly affects its ability to rise and stay aloft.
Fluid Dynamics
Fluid dynamics deals with the behavior of fluids (liquids and gases) in motion. It鈥檚 pivotal in understanding how forces such as lift are generated and utilized, especially in aviation and marine applications. For hot-air balloons, fluid dynamics explains how manipulation of air density can achieve buoyancy, allowing the balloon to float.
Key concepts in fluid dynamics related to hot-air balloons include:
  • **Buoyancy:** A result of differences in fluid density, leading to a rising motion of the less dense fluid.
  • **Archimedes' Principle:** It helps predict the buoyant force, stating that the upward buoyant force on a submerged object is equal to the weight of the fluid it displaces.
Understanding these principles helps in calculating how much lift is necessary for a hot-air balloon to lift off and maintain altitude.

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

At one point in a pipeline the water's speed is 3.00 \(\mathrm{m} / \mathrm{s}\) and the gauge pressure is \(5.00 \times 10^{4} \mathrm{Pa}\) . Find the gauge pressure at a second point in the line, 11.0 \(\mathrm{m}\) lower than the first, if the pipe diameter at the second point is twice that at the first.

Submarines on Europa. Some scientists are eager to send a remote-controlled submarine to Jupiter's moon Europa to search for life in its oceans below an icy crust. Europa's mass has been measured to be \(4.78 \times 10^{22} \mathrm{kg}\) , its diamcter is 3130 \(\mathrm{km}\) , and it has no appreciable atmosphere. Assume that the layer of ice at the surface is not thick enough to exert substantial force on the water. If the windows of the submarine you are designing are 25.0 \(\mathrm{cm}\) square and can stand a maximum inward force of 9750 \(\mathrm{N}\) per window, what is the greatest depth to which this submarine can safely dive?

A plastic ball has radius 12.0 \(\mathrm{cm}\) and floats in water with 16.0\(\%\) of its volume submerged. (a) What force must you apply to the ball to hold it at rest totally below the surface of the water? (b) If you let go of the ball, what is its acceleration the instant you release it?

SHM of a Floating Object. An object with height \(h\) , mass \(M,\) and a uniform cross-sectional area \(A\) floats upright in a liquid with density \(\rho\) . (a) Calculate the vertical distance from the surface of the liquid to the bottom of the floating object at equilibrium. (b) A downward force with magnitude \(F\) is applied to the top of the object. At the new equilibrium position, how much farther below the surface of the liquid is the bottom of the object than it was in part (a)? (Assume that some of the object remains above the surface of the liquid.) (c) Your result in part (b) shows that if the force is suddenly removed, the object will oscillate up and down in SHM. Calculate the period of this motion in terms of the density \(\rho\) of the liquid, the mass \(M,\) and cross-sectional area \(A\) of the object. You can ignore the damping due to fluid friction (see Section \(13.7 ) .\)

You purchase a rectangular piece of metal that has dimensions \(5.0 \times 15.0 \times 30.0 \mathrm{mm}\) and mass 0.0158 \(\mathrm{kg}\) . The seller tells you that the metal is gold. To check this, you compute the average density of the piece. What value do you get? Were you cheated?
See all solutions

Recommended explanations on Physics Textbooks

Waves Physics

Read Explanation

Medical Physics

Read Explanation

Scientific Method Physics

Read Explanation

Solid State Physics

Read Explanation

Circular Motion and Gravitation

Read Explanation

Geometrical and Physical Optics

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.