An airplane is moving through the air at a velocity v=200m/s. Streamlines just over the top of the wing are compressed to 80% of their original area, and those under the wing are not compressed at all.
a) Determine the velocity of the air just over the wing.
b) Find the difference in the pressure between the air just over the wing,P and that under the wing, P'.
c) Find the net upward force on both wings due to the pressure difference, if the area of the wig is 40m^2 and the density of the air is 1.3kg/m^3.
a) To determine the velocity of the air just over the wing, we can use the principle of continuity, which states that the product of the cross-sectional area A, the fluid (air) density ρ, and the fluid velocity v must be constant in the streamline:
A * ρ * v = constant
Let's assume the original area (before the compression) is A₀, so the compressed area is A₁ = 0.8 * A₀. The original velocity is v₀ = 200 m/s.
A₀ * ρ * v₀ = A₁ * ρ * v₁
We can cancel ρ because it's constant in the streamline:
A₀ * v₀ = A₁ * v₁
Now we can solve for v₁:
v₁ = (A₀ / A₁) * v₀ = (1 / 0.8) * 200 = 200 / 0.8 = 250 m/s
The velocity of the air just over the wing is 250 m/s.
b) To find the difference in pressure between the air just over the wing (P) and under the wing (P'), we can use Bernoulli's equation, which relates the pressure, density, and velocity of a fluid in a streamline:
P + (1/2) * ρ * v^2 = constant
Applying this equation to the air over the wing (1) and under the wing (0):
P₀ + (1/2) * ρ * v₀^2 = P₁ + (1/2) * ρ * v₁^2
Solving for the pressure difference P - P':
∆P = P₁ - P₀ = (1/2) * ρ * (v₀^2 - v₁^2)
Using the given density of the air, ρ = 1.3 kg/m³, and the calculated velocities:
∆P = (1/2) * 1.3 * (200^2 - 250^2) = 0.65 * (-7500) = -4875 Pa
The pressure difference between the air just over the wing and under the wing is -4875 Pa (meaning the pressure under the wing is higher).
c) To find the net upward force on both wings due to the pressure difference, we can use the formula:
F = ∆P * A
Using the given area of each wing (40 m²), and since there are two wings, we multiply the force by 2:
F = 2 * (-4875) * 40 = -390000 N
The net upward force on both wings due to the pressure difference is -390000 N (which is a downward force, as expected from a negative pressure difference).
a) To determine the velocity of the air just over the wing, we can use the equation of continuity, which states that the product of the cross-sectional area and velocity of a fluid remains constant.
Given that the streamlines just over the top of the wing are compressed to 80% of their original area, we can denote the original area as A and the compressed area as 0.8A.
Let's denote the original velocity of the air as v, and the velocity of the air just over the wing as v'. We can now apply the equation of continuity:
A * v = 0.8A * v'
Simplifying the equation, we find:
v' = v / 0.8
Plugging in the given velocity of the airplane, v = 200 m/s, we can calculate the velocity of the air just over the wing, v':
v' = 200 m/s / 0.8 = 250 m/s
Therefore, the velocity of the air just over the wing is 250 m/s.
b) To find the difference in pressure between the air just over the wing (P) and that under the wing (P'), we can use Bernoulli's equation, which relates the pressure, velocity, density, and height of a fluid flow.
Bernoulli's equation can be expressed as:
P + 1/2 * ρ * v^2 = P' + 1/2 * ρ * v'^2
Where P denotes the pressure just over the wing, P' denotes the pressure under the wing, ρ is the density of the air, v is the velocity of the airplane, and v' is the velocity of the air just over the wing.
We can rearrange the equation and plug in the known values:
P - P' = 1/2 * ρ * (v'^2 - v^2)
= 1/2 * 1.3 kg/m^3 * [(250 m/s)^2 - (200 m/s)^2]
Calculating the difference in pressure, we find:
P - P' ≈ 1/2 * 1.3 kg/m^3 * (62,500 m^2/s^2 - 40,000 m^2/s^2)
≈ 1/2 * 1.3 kg/m^3 * 22,500 m^2/s^2
≈ 14,625 kg/m·s^2
Therefore, the difference in pressure between the air just over the wing and that under the wing is approximately 14,625 N/m^2.
c) To find the net upward force on both wings due to the pressure difference (ΔP), we can multiply the pressure difference by the area of the wing and the density of the air.
The net upward force (F) can be calculated using the formula:
F = ΔP * A
Plugging in the given values:
F = 14,625 N/m^2 * 40 m^2
= 585,000 N
Therefore, the net upward force on both wings due to the pressure difference is 585,000 N.
To find the answers to these questions, we will use the equations of continuity and Bernoulli's principle. Now, let's go through each part step by step:
a) To determine the velocity of the air just over the wing, we can use the equation of continuity. According to this principle, the product of the velocity and the cross-sectional area of the streamlines remains constant along the streamline.
Mathematically, we can express this principle as:
A_1 * v_1 = A_2 * v_2
Where A_1 is the initial cross-sectional area of the streamlines, v_1 is the initial velocity of the air, A_2 is the reduced cross-sectional area just over the wing, and v_2 is the velocity of the air just over the wing.
Given that A_2 is 80% of A_1 and knowing A_1*v_1 = A_2*v_2, we can solve for v_2 as follows:
80% * A_1 * v_1 = A_2 * v_2
0.8 * A_1 * v_1 = 0.8 * A_1 * v_2
Since 0.8 * A_1 cancels out on both sides, we can conclude that v_2 = v_1.
Therefore, the velocity of the air just over the wing is equal to 200 m/s.
b) To find the difference in pressure between the air just over the wing (P) and that under the wing (P'), we can use Bernoulli's principle. According to Bernoulli's principle, an increase in the velocity of a fluid is associated with a decrease in its pressure and vice versa.
Mathematically, Bernoulli's equation is written as:
P + 1/2 * ρ * v^2 + ρ * g * h = constant
In this equation, P is the pressure of the fluid, ρ is the density of the air, v is the velocity of the air, g is the acceleration due to gravity, and h is the height above a reference point.
Now, since the velocity of the air, v, is the same above and below the wing (as established in part a), the pressure difference, ΔP = P - P', can be determined only by considering the height difference, h, above and below the wing.
Since there is no height difference, h = 0, the pressure difference, ΔP = 0.
Therefore, the difference in pressure between the air just over the wing, P, and that under the wing, P', is zero.
c) Since the pressure difference, ΔP, is zero (as calculated in part b), the net upward force on the wings due to the pressure difference can be found using the equation:
Force = Pressure difference * Area
Given that the area of the wing is 40 m^2, the density of the air is 1.3 kg/m^3, and the pressure difference, ΔP, is zero, we can calculate the net upward force as follows:
Force = ΔP * A = 0 * 40 = 0 N
Therefore, the net upward force on both wings due to the pressure difference is zero.