Find the pressure difference on an airplane wing where air flows over the upper surface with a speed of 119 m/s, and along the bottom surface with a speed of 105 m/s.
(b) If the area of the wing is 40 m2, what is the net upward force exerted on the wing?
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To find the pressure difference on an airplane wing, we need to first calculate the difference in air speeds over the upper and lower surfaces.
Given:
Speed of air over the upper surface (v_upper) = 119 m/s
Speed of air along the bottom surface (v_bottom) = 105 m/s
The pressure difference (ΔP) can be calculated using Bernoulli's equation, which states that the total pressure at a point in a fluid flow (P_total) is equal to the sum of the static pressure (P_static) and the dynamic pressure (P_dynamic).
P_total = P_static + P_dynamic
Where:
P_static = static pressure (due to weight of air)
P_dynamic = dynamic pressure (due to motion of air)
The dynamic pressure (P_dynamic) is given by the equation:
P_dynamic = 0.5 * ρ * v^2
Where:
ρ = density of air (approximately 1.225 kg/m³ at sea level)
v = velocity of air
First, calculate the dynamic pressure over the upper surface (P_dynamic_upper):
P_dynamic_upper = 0.5 * ρ * v_upper^2
Next, calculate the dynamic pressure along the bottom surface (P_dynamic_bottom):
P_dynamic_bottom = 0.5 * ρ * v_bottom^2
Finally, calculate the pressure difference (ΔP) as:
ΔP = P_dynamic_upper - P_dynamic_bottom
Now we can calculate the pressure difference on the airplane wing.
Note: The values used are approximations based on the given information. For more accurate results, precise values of air density and speeds should be used.
Let's calculate ΔP:
ρ ≈ 1.225 kg/m³ (density of air at sea level)
P_dynamic_upper = 0.5 * 1.225 kg/m³ * (119 m/s)^2
P_dynamic_bottom = 0.5 * 1.225 kg/m³ * (105 m/s)^2
Now, subtract P_dynamic_bottom from P_dynamic_upper to find ΔP:
ΔP = P_dynamic_upper - P_dynamic_bottom
Once you have the pressure difference (ΔP), you can proceed to calculate the net upward force exerted on the wing.
To find the net upward force, we can use Bernoulli's principle, which states that the pressure difference across the wing surface will result in a net force perpendicular to the wing. This force is called the lift force.
The lift force (F_lift) is given by the equation:
F_lift = ΔP * A
Where:
ΔP = pressure difference
A = area of the wing
Given:
Area of the wing (A) = 40 m²
Now, we can calculate the net upward force exerted on the wing:
F_lift = ΔP * A
Substitute the calculated value of ΔP and the given area of the wing into the equation to find F_lift.