Quantifying Lift

Consider… 767 refused take-off: PEfO1Oo&hl=en PEfO1Oo&hl=en F16 =jaWNj-ZkADY =jaWNj-ZkADY Emergency landing w/o nose gear M4_Ullo&hl=en M4_Ullo&hl=en Theory is nice; we need numbers!

Factors that affect lift Depends on size and shape of solid o Airfoil shape o Wing size o Ratio of wing span to wing area

Factors that affect lift Depends on velocity of solid through fluid o Lift increases proportional to the SQUARE of velocity 2 x as fast = 4 x the lift 3 x as fast = 9 x the lift 4 x as fast = 16 x the lift

Factors that affect lift Depends on inclination of the solid relative to the direction of travel o “angle of attack” NOTE: when the angle gets too steep, the airfoil stops generating lift; the coefficient of lift is undefined. “Stall angle”

Factors that affect lift Depends on fluid’s mass compressibility viscosity Can be modeled mathematically with higher level math Usually calculated experimentally “Coefficient of Lift”, C L

Lift Equation L = ½  v 2 A C L where L = lift (N)  = density of air (kg / m 3 ) v = velocity (m/s) A = area of wing (m 2 ) C L = coefficient of lift (N s / kg)

Think about it… Why do larger aircraft need larger wings? How do fighter jets get away with smaller wings? How does our model incorporate angle of attack? L = ½  v 2 A C L What would you need to do to get a plane to fly on Mars? Here’s Randall Munroe’s answer to that question.answer

Think about it… In the emergency landing video, which factor did the pilot modify to maintain lift for as long as possible? In the refused take-off video, which factor likely led to lack of lift? In the video of the F16 crash, which factor likely led to lack of lift? L = ½  v 2 A C L

Finding lift Measure lift outside of a lab is hard. Measuring changes in motion up or down is easier. Fortunately, there’s physics! Lift acts opposite of weight.

Lift = weight - something - = downward acceleration + = upward acceleration

Example Calculate the lift produced by plane with a wing area of 16 m 2 and a coefficient of lift of 1.0 traveling at a velocity of 25 m/s through air with a density of 1.0 kg / m 3. L = ½  v 2 A C L = ½ (1.0 kg / m 3 ) (25 m/s) 2 (16 m 2 )(1.0) L= 5000 N  = 1.0 kg / m 3 v = 25 m/s A = 16 m 2 C L = 1.0 L = ?

Example Suppose an airplane with a weight of 5500 N generates 5000 N of lift during its take-off roll. In what direction, if any, will it accelerate? It cannot accelerate downward because the ground is in the way. It will not accelerate upward until lift is greater than weight. Assuming that the engines are on full throttle, the airplane will accelerate forward.

Example Suppose an airplane with a weight of 4500 N (mass of ~450 kg) generates 5000 N of lift during its take- off roll. In what direction, if any, will it accelerate? Lift = weight + (mass x acceleration) So, acceleration = (lift – weight) / mass a = (5000 N – 4500 N) / 450 kg a = 1.1 (m/s)/s upward

Example Calculate the coefficient of lift for a plane with a weight of 10,000 N, with a wing area of 25 m 2 traveling at a velocity of 30 m/s through air with a density of 1.2 kg / m 3. L = ½  v 2 A C L so, C L = L / (½  v 2 A) = (10,000 N ) / (½ (1.2 kg / m 3 ) (30 m/s) 2 (25 m 2 ) C L = 0.74  = 1.2 kg / m 3 v = 30 m/s A = 25 m 2 L = 10,000 N C L = ?

Example Calculate the lift for a plane with a weight of 10,000 N, with a wing area of 25 m 2 and coefficient of lift of 0.74 traveling at a velocity of 30 m/s through air with a density of 0.6 kg / m 3. L = ½  v 2 A C L = (½ (0.6 kg / m 3 ) (30 m/s) 2 (25 m 2 ) (0.74) L = 4995 N  = 0.6 kg / m 3 v = 30 m/s A = 25 m 2 C L = 0.74 L = ?

Example Suppose an airplane with a weight of 10,00 N (mass of ~1,000 kg) generates 5000 N of lift during flight. In what direction, if any, will it accelerate? Lift = weight + (mass x acceleration) So, acceleration = (lift – weight) / mass a = (5,000 N – 10,000 N) / 1,000 kg a = -5 (m/s)/s, downward