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Chapter 14: Problem 43

Lift on an Airplane. Air streams horizontally past a small airplane's wings such that the speed is 70.0 \(\mathrm{m} / \mathrm{s}\) over the top surface and 60.0 \(\mathrm{m} / \mathrm{s}\) past the bottom surface. If the plane has a wing area of 16.2 \(\mathrm{m}^{2}\) on the top and on the bottom, what is the net vertical force that the air exerts on the airplane? The density of the air is 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) .

Short Answer

Expert verified
The net vertical force is 12,636 N upwards.

Step by step solution

01

Understand Bernoulli's Principle

Bernoulli's Principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure. This principle helps us understand that the pressure on the top surface of the wing, where the air moves faster, will be lower than the pressure on the bottom surface of the wing, where the air moves slower.
02

Calculate Pressure Difference Using Bernoulli's Equation

Apply Bernoulli's equation to find the pressure difference between the top and bottom of the wing: \[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]Let the surface above be 1 and below be 2. Rearranging, the pressure difference \( \Delta P = P_2 - P_1 \) is:\[ \Delta P = \frac{1}{2} \rho (v_2^2 - v_1^2) \]Substitute \( v_1 = 70.0 \ \text{m/s} \), \( v_2 = 60.0 \ \text{m/s} \), and \( \rho = 1.20 \ \text{kg/m}^3 \).
03

Perform Calculation of Pressure Difference

Calculate \( \Delta P \) using the given values:\[ \Delta P = \frac{1}{2} \times 1.20 \times ((60.0)^2 - (70.0)^2) \]\[ \Delta P = \frac{1}{2} \times 1.20 \times (3600 - 4900) \]\[ \Delta P = \frac{1}{2} \times 1.20 \times (-1300) = -780 \ \text{Pa} \]This tells us that the pressure is lower on the top, as expected.
04

Calculate Net Vertical Force

The net vertical force \( F \) can be found by multiplying the pressure difference by the wing area \( A \):\[ F = \Delta P \times A \]Substitute \( \Delta P = -780 \ \text{Pa} \) and \( A = 16.2 \ \text{m}^2 \).
05

Perform Calculation of Net Force

Now, perform the calculation:\[ F = -780 \times 16.2 = -12636 \ \text{N} \]This negative sign indicates that the lift direction is upward.

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Key Concepts

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

Aerodynamics
Aerodynamics is the study of how gases interact with moving bodies. In aviation, it plays a crucial role in understanding how airplanes fly. The main focus is on the behavior of air as it flows over and under the various surfaces of an aircraft, particularly the wings. These wings are designed to create differences in airspeed and pressure above and below them, leading to lift.
The shape of an airplane wing, known as an airfoil, is curved on the top and flatter on the bottom. This shape allows air to travel faster over the top than beneath the wing due to the longer path it has to cover. Consequently, aerodynamics is not only about speed but also about how speed affects pressure on different parts of an aircraft. These concepts are vital for engineers to design efficient, safe, and reliable aircraft.
Pressure Difference
Understanding pressure differences is crucial in applying Bernoulli's Principle, which is key for lift in flight. Bernoulli's Principle explains that in a steady streamline flow of a fluid, regions where the fluid moves faster will have lower pressure, and slower-moving regions will have higher pressure.
In our airplane wing scenario, the air travels faster over the top surface than the bottom. This results in lower pressure on top and higher pressure on the bottom. The difference in these pressures is what generates lift, allowing the airplane to rise. We calculate this pressure difference using Bernoulli's equation, which considers airspeed and air density.

How It Works

- Faster air speed on top leads to lower pressure.- Slower air speed on the bottom creates higher pressure.- Pressure difference (\(\Delta P\)) is found by evaluating the change due to speeds over both surfaces.
This principle not only supports aircraft flight but is also applicable to various real-world scenarios, such as explaining why a shower curtain moves inward when a shower is running.
Net Vertical Force
Net vertical force is the total force acting on the aircraft due to pressure differences, impacting its lift. Once we know the pressure difference (\(\Delta P\)) and the area of the wings (\(A\)), we can find this force using the equation \(F = \Delta P \times A\).
This equation tells us how much lift or downward force the wings experience. In our case, the calculation revealed a value of \( -12636 \, N\), indicating an upward lift. The negative sign indicates direction, showing that the net vertical force is acting upwards against gravity.

More on Lift

- Lift is crucial for flight; it counters the weight of the airplane.- A greater net vertical force means more effective and efficient lift.- It's essential in designing aircraft to ensure they are capable of taking off, cruising, and landing safely.
Effective manipulation of net vertical force allows aircraft to perform various maneuvers, emphasizing the importance of understanding both pressure differences and aerodynamic principles in the field of aviation.

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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.

A hunk of aluminum is completely covered with a gold shell to form an ingot of weight 45.0 \(\mathrm{N}\) . When you suspend the ingot from a spring balance and submerge the ingot in water, the balance reads 39.0 \(\mathrm{N}\) . What is the weight of the gold in the shell?

An object of average density \(\rho\) floats at the surface of a fluid of density \(\rho_{\text { fluid }}\). (a) How must the two densities be related? (b) In view of the answer to part (a), how can steel ships float in water? (c) In terms of \(\rho\) and \(\rho_{\text { fluid }}\) what fraction of the object is submerged and what fraction is above the fluid? Check that your answers give the correct limiting behavior as \(\rho \rightarrow \rho_{\text { fluid }}\) and as \(\rho \rightarrow 0 .\) (d) While on board your yacht, your cousin Throckmorton cuts a rectangular piece (dimensions \(5.0 \times 4.0 \times 3.0 \mathrm{cm} )\) out of a life preserver and throws it into the ocean. The piece has a mass of 42 g. As it floats in the ocean, what percentage of its volume is above the surface?

A block of balsa wood placed in one scale pan of an equal-arm balance is exactly balanced by a \(0.0950-\mathrm{kg}\) brass mass in the other scale pan. Find the true mass of the balsa wood if its density is 150 \(\mathrm{kg} / \mathrm{m}^{3}\) . Explain why it is accurate to ignore the buoyancy in air of the brass but not the buoyancy in air of the balsa wood.

Water is flowing in a pipe with a varying cross-sectional area, and at all points the water completely fills the pipe. At point 1 the cross-sectional area of the pipe is \(0.070 \mathrm{m}^{2},\) and the magnitude of the fluid velocity is 3.50 \(\mathrm{m} / \mathrm{s}\) . (a) What is the fluid speed at points in the pipe where the cross-sectional area is (a) 0.105 \(\mathrm{m}^{2}\) and (b) 0.047 \(\mathrm{m}^{2} ?\) (c) Calculate the volume of water discharged from the open end of the pipe in 1.00 hour.
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