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STRATEGY This problem can be solved by substituting values into Bernoulli's equation to find the pressure difference between the air under the wing and the air over the wing, followed by applying Newton's second law to find the mass the airplane can lift. SOLUTION Apply Bernoulli's equation to the air P, + V½pv,² = P, + V½pv,² flowing under the wing (point 1) and over the wing (point 2). Gravitational potential energy terms are small compared with the other terms, and can be neglected. Solve this equation for the pressure AP = P, - P2 = V½pv,² - V2pv,? = V½p(v2² - v,²) %3D 1 %3D difference. Substitute the given speeds and Ap = ½(1.29 kg/m³)(2452 m²/s² - 2222 m2/s²) p = 1.29 kg/m3, the density of air. AP = 6.93 × 10° Pa Apply Newton's second law. To support the plane's weight, the sum of the lift and gravity forces must equal zero. 2AAP -mg = 0-m = 5.66 x 10 kg Solve for the mass m of the plane.

STRATEGY This problem can be solved by substituting values into Bernoulli's equation to find the pressure difference between the air under the wing and the air over the wing, followed by applying Newton's second law to find the mass the airplane can lift. SOLUTION Apply Bernoulli's equation to the air P, + V½pv,² = P, + V½pv,² flowing under the wing (point 1) and over the wing (point 2). Gravitational potential energy terms are small compared with the other terms, and can be neglected. Solve this equation for the pressure AP = P, - P2 = V½pv,² - V2pv,? = V½p(v2² - v,²) %3D 1 %3D difference. Substitute the given speeds and Ap = ½(1.29 kg/m³)(2452 m²/s² - 2222 m2/s²) p = 1.29 kg/m3, the density of air. AP = 6.93 × 10° Pa Apply Newton's second law. To support the plane's weight, the sum of the lift and gravity forces must equal zero. 2AAP -mg = 0-m = 5.66 x 10 kg Solve for the mass m of the plane.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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EXAMPLE 9.15 Lift on an Airfoil GOAL Use Bernoulli's equation to calculate the lift on an airplane wing. PROBLEM An airplane has wings, each with area 4.00 m², designed so that air flows over the top of the wing at 245 m/s and underneath the wing at 222 m/s. Find the mass of the airplane such that the lift on the plane will support its weight, assuming the force from the pressure difference across the wings is directed straight upwards. STRATEGY This problem can be solved by substituting values into Bernoulli's equation to find the pressure difference between the air under the wing and the air over the wing, followed by applying Newton's second law to find the mass the airplane can lift. SOLUTION Apply Bernoulli's equation to the air P,+ Vapv,² = P2 + V½pv,? flowing under the wing (point 1) and over the wing (point 2). Gravitational potential energy terms are small compared with the other terms, and can be neglected. Solve this equation for the pressure AP = P1 - P2 = V½pv,² - V½pv,? = V½p(v,² - v,²) difference. Substitute the given speeds and Ap = ½(1.29 kg/m³)(245² m²/s² - 2222 m2/s?) p = 1.29 kg/m³, the density of air. AP = 6.93 x 10' Pa Apply Newton's second law. To support 2AAP -mg = 0 - m = 5.66 x 10° kg the plane's weight, the sum of the lift and gravity forces must equal zero. Solve for the mass m of the plane. LEARN MORE REMARKS Note the factor of two in the last equation, needed because the airplane has two wings. The density of the atmosphere drops steadily with increasing height, reducing the lift. As a result, all aircraft have a maximum operating altitude.
Transcribed Image Text:EXAMPLE 9.15 Lift on an Airfoil GOAL Use Bernoulli's equation to calculate the lift on an airplane wing. PROBLEM An airplane has wings, each with area 4.00 m², designed so that air flows over the top of the wing at 245 m/s and underneath the wing at 222 m/s. Find the mass of the airplane such that the lift on the plane will support its weight, assuming the force from the pressure difference across the wings is directed straight upwards. STRATEGY This problem can be solved by substituting values into Bernoulli's equation to find the pressure difference between the air under the wing and the air over the wing, followed by applying Newton's second law to find the mass the airplane can lift. SOLUTION Apply Bernoulli's equation to the air P,+ Vapv,² = P2 + V½pv,? flowing under the wing (point 1) and over the wing (point 2). Gravitational potential energy terms are small compared with the other terms, and can be neglected. Solve this equation for the pressure AP = P1 - P2 = V½pv,² - V½pv,? = V½p(v,² - v,²) difference. Substitute the given speeds and Ap = ½(1.29 kg/m³)(245² m²/s² - 2222 m2/s?) p = 1.29 kg/m³, the density of air. AP = 6.93 x 10' Pa Apply Newton's second law. To support 2AAP -mg = 0 - m = 5.66 x 10° kg the plane's weight, the sum of the lift and gravity forces must equal zero. Solve for the mass m of the plane. LEARN MORE REMARKS Note the factor of two in the last equation, needed because the airplane has two wings. The density of the atmosphere drops steadily with increasing height, reducing the lift. As a result, all aircraft have a maximum operating altitude.
PRACTICE IT Use the worked example above to help vou solve this problem. An airplane has wings, each with area 4.10 m2, designed so that air flows over the top of the wing at 243 m/s and underneath the wing at 225 m/s. Find the mass of the airplane such that the VIlon the plane will support its weight, assuming the force from the pressure difference across the wings is directed straight upwards. kg EXERCISE HINTS: GETTING STARTED | I'M STUCK! Approximately what size wings would an aircraft need on Mars if its engine generates the same differences in speed as in the "Practice It" and the total mass of the craft is 450 kg? The density of air on the surface of Mars is approximately one percent Earth's density at sea level, and theacceleration of gravity on the surface of Mars is about 3.8 m/s. m2
Transcribed Image Text:PRACTICE IT Use the worked example above to help vou solve this problem. An airplane has wings, each with area 4.10 m2, designed so that air flows over the top of the wing at 243 m/s and underneath the wing at 225 m/s. Find the mass of the airplane such that the VIlon the plane will support its weight, assuming the force from the pressure difference across the wings is directed straight upwards. kg EXERCISE HINTS: GETTING STARTED | I'M STUCK! Approximately what size wings would an aircraft need on Mars if its engine generates the same differences in speed as in the "Practice It" and the total mass of the craft is 450 kg? The density of air on the surface of Mars is approximately one percent Earth's density at sea level, and theacceleration of gravity on the surface of Mars is about 3.8 m/s. m2
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