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Chapter 11: Problem 63

An airplane wing is designed so that the speed of the air across the top of the wing is 251 \(\mathrm{m} / \mathrm{s}\) when the speed of the air below the wing is 225 \(\mathrm{m} / \mathrm{s}\) . The density of the air is 1.29 \(\mathrm{kg} / \mathrm{m}^{3} .\) What is the lifting force on a wing of area 24.0 \(\mathrm{m}^{2} ?\)

Short Answer

Expert verified
The lift force is 191525.76 N.

Step by step solution

01

Understand Bernoulli's Principle

Bernoulli's principle can be used to find the lift force. It states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure. We'll use the difference in airspeed above and below the wing to find this pressure difference.
02

Use Bernoulli's Equation

Bernoulli's equation is given by \( P + \frac{1}{2} \rho v^2 = \text{constant} \). To find the pressure difference, calculate: \[ \Delta P = \frac{1}{2} \rho (v_1^2 - v_2^2) \] where \( v_1 = 251 \, \text{m/s} \) (air over the wing) and \( v_2 = 225 \, \text{m/s} \) (air under the wing).
03

Calculate the Pressure Difference

Substitute the given values into the equation: \( \rho = 1.29 \, \text{kg/m}^3 \), \[ \Delta P = \frac{1}{2} \times 1.29 \times (251^2 - 225^2) \]. Calculate this to find \( \Delta P \).
04

Compute the Lift Force

The lift force \( F \) is given by \( F = \Delta P \times A \), where \( A = 24.0 \, \text{m}^2 \) is the area of the wing. Substitute the pressure difference from Step 3 into this formula to find the lift force.
05

Perform the Calculation

Calculate each part step-by-step: \( 251^2 = 63001 \), \( 225^2 = 50625 \); now find \( 63001 - 50625 = 12376 \); then \( \Delta P = \frac{1}{2} \times 1.29 \times 12376 = 7980.24 \, \text{Pa} \). Finally, calculate the lift force: \( F = 7980.24 \times 24.0 = 191525.76 \, \text{N} \).

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Key Concepts

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

Lift Force
Lift force is a fundamental concept in aerodynamics, playing a crucial role in allowing airplanes to fly. It is generated when air flows over and under the wings of an aircraft.

According to Bernoulli's Principle, faster-moving air over the top of the wing reduces the pressure above it compared to the slower-moving air below. This pressure difference creates an upward lift force, allowing the plane to rise and remain in the air.

In engineering, understanding and calculating lift force is essential for designing efficient aircraft. Lift not only determines how well a plane can fly but also helps in optimizing fuel efficiency.
Pressure Difference
Pressure difference between the top and bottom surfaces of an airplane wing is vital for producing lift. As air flows over the wing, the speed difference causes a difference in pressure, as described by Bernoulli's Principle.

To calculate the pressure difference, we use the equation \( \Delta P = \frac{1}{2} \rho (v_1^2 - v_2^2) \). Here, \(v_1\) and \(v_2\) are the speeds of air over and under the wing, and \(\rho\) is the air density.
  • Higher speed over the wing results in lower pressure.
  • Lower speed under the wing results in higher pressure.
This difference creates a pressure lift force which supports the aircraft in the air.
Airplane Wing Design
The design of an airplane wing is pivotal in managing airflow and generating lift. The shape of the wing, known as an airfoil, is crafted to maximize lift while minimizing drag.

The curvature on the upper surfaces allows air to travel faster, reducing pressure above the wing. Meanwhile, the flatter lower surface maintains higher pressure, contributing to lift.

Wing design considerations also include controlling factors like stall conditions, where the lift dramatically reduces if the airflow separates from the wing, and balancing stability with maneuverability. These factors make airplane wing design a complex and essential field of aerospace engineering.
Fluid Dynamics
Fluid dynamics is a branch of physics that describes the behavior of liquids and gases in motion, crucial for understanding how lift forces are generated on airplane wings.

It deals with how air, as a fluid, flows around objects and the forces involved in these interactions. In the context of an airplane wing, fluid dynamics helps in analyzing how airspeed and pressure differences create lift.

Engineers apply principles of fluid dynamics by using computational models and wind tunnel testing to evaluate and enhance wing performance. This ensures that aircraft can fly safely, efficiently, and with minimal environmental impact.

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Most popular questions from this chapter

A person who weighs 625 \(\mathrm{N}\) is riding a \(98-\mathrm{N}\) mountain bike. Suppose that the entire weight of the rider and bike is supported equally by the two tires. If the pressure in each tire is \(7.60 \times 10^{5} \mathrm{Pa},\) what is the area of contact between each tire and the ground?

Two circular holes, one larger than the other, are cut in the side of a large water tank whose top is open to the atmosphere. The center of one of these holes is located twice as far beneath the surface of the water as the other. The volume flow rate of the water coming out of the holes is the same. (a) Decide which hole is located nearest the surface of the water. (b) Calculate the ratio of the radius of the larger hole to the radius of the smaller hole.

(a) The mass and the radius of the sun are, respectively, \(1.99 \times 10^{30} \mathrm{kg}\) and \(6.96 \times 10^{8} \mathrm{m}\) . What is its density? (b) If a solid object is made from a material that has the same density as the sun, would it sink or float in water? Why? (c) Would a solid object sink or float in water if were made from a material whose density was the same as that of the planet Saturn (mass \(=5.7 \times 10^{26} \mathrm{kg},\) radius \(=6.0 \times 10^{7} \mathrm{m} ) ?\) Provide a reason for your answer.

In the process of changing a flat tire, a motorist uses a hydraulic jack. She begins by applying a force of 45 N to the input piston, which has a radius \(r_{1} .\) As a result, the output plunger, which has a radius \(r_{2},\) applies a force to the car. The ratio \(r_{2} / r_{1}\) has a value of \(8.3 .\) Ignore the height difference between the input piston and output plunger and determine the force that the output plunger applies to the car.

A solid concrete block weighs 169 \(\mathrm{N}\) and is resting on the ground. Its dimensions are 0.400 \(\mathrm{m} \times 0.200 \mathrm{m} \times 0.100 \mathrm{m} .\) A number of identical blocks are stacked on top of this one. What is the smallest number of whole blocks (including the one on the ground) that can be stacked so that their weight creates a pressure of at least two atmospheres on the ground beneath the first block?
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