Modern airplane design calls for a lift, due to the net force of the moving air on the wing, of about 2000 N per square meter of wing area. Assume the air flows past the wing of an aircraft with streamline flow. If the speed of flow past the lower wing surface is 120m/s, what is the required speed over the upper surface to give a lift of 2000 N/m^2? The density of air is 1.20 kg/m^3

The net lift per wing area, using the Bernoulli equation, is

(1/2)(air density)[(Vtop)^2-(Vbottom)^2]
= 2000 N/m^2

In your case Vbottom = 120 m/s
Solve for Vtop.

ok I got 133.2 m/s..is that right?

Yes.

11

To determine the required speed over the upper surface of the wing, we can use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid flowing through a streamline.

The equation is as follows:

P + (1/2)ρv^2 + ρgh = constant

Where:
P is the pressure of the fluid
ρ is the density of the fluid
v is the velocity of the fluid
g is the acceleration due to gravity
h is the height of the fluid

At two different points along the streamline flow, we can equate the constants and solve for the velocity.

In this case, we'll consider two points on the wing: the lower surface and the upper surface. At both points, the height is the same, so the term involving h can be ignored.

Let's assume the pressure on the lower surface is P1, the pressure on the upper surface is P2, and the velocity on the lower surface is v1 (known as 120 m/s). We can then express Bernoulli's equation at each point:

Point 1 (lower surface):
P1 + (1/2)ρv1^2 = constant

Point 2 (upper surface):
P2 + (1/2)ρv2^2 = constant

Since the density of air (ρ) is given as 1.20 kg/m^3 and the force required is 2000 N/m^2, we can rewrite the equation as:

P1 + (1/2)(1.20 kg/m^3)(120 m/s)^2 = P2 + (1/2)(1.20 kg/m^3)(v2)^2

The pressure difference between the upper and lower surfaces (P2 - P1) can be related to the required lift force using the equation:

P2 - P1 = (lift force / wing area)

Therefore, the equation becomes:

(lift force / wing area) = (1/2)(1.20 kg/m^3)(v2)^2 - (1/2)(1.20 kg/m^3)(120 m/s)^2

Solving for v2:

(v2)^2 = ((2 * lift force) / (ρ * wing area)) + (120 m/s)^2

v2 = √(((2 * lift force) / (ρ * wing area)) + (120 m/s)^2)

Substituting the given values:
lift force = 2000 N/m^2,
ρ = 1.20 kg/m^3,
wing area = 1 m^2,
v2 = √(((2 * 2000 N/m^2) / (1.20 kg/m^3 * 1 m^2)) + (120 m/s)^2)

Calculating v2, we get:
v2 ≈ 238.06 m/s

Therefore, the required speed over the upper surface of the wing to give a lift of 2000 N/m^2 is approximately 238.06 m/s.