Air streams horizontally past a small airplane's wings such that the speed is 70.0m/s over the top surface and 60.0m/s past the bottom surface. If the plane has a wing area of 16.2m^2 on the top and on the bottom, what is the net vertical force that the air exerts on the airplane?

http://hyperphysics.phy-astr.gsu.edu/Hbase/pber.html

As a first approximation assume the density of the air is the same on the bottom side as the top.

Pbottom+d*1/2 *Vbottom^2=Pressuretop + d*1/2*Vtop^2
or pressuretop-pressurebottom=d*1/2(Vtop^2-Vbottom^2)

So, after you calculate that pressure difference, multipy it by the wing area, and you have the lift force.

To find the net vertical force that the air exerts on the airplane, we first need to calculate the lift force on each wing surface separately.

Step 1: Calculate the pressure difference between the top and bottom surfaces of the wings.
The pressure difference is given by the formula:

ΔP = 0.5 * ρ * V^2 * (Cp - Cn)

Where:
ΔP is the pressure difference
ρ is the density of air
V is the velocity of the air
Cp is the coefficient of pressure on the top surface
Cn is the coefficient of pressure on the bottom surface

Given:
V(top) = 70.0 m/s
V(bottom) = 60.0 m/s
Cp = 0 (assuming no pressure difference on the top surface)
Cn = 1 (assuming maximum pressure on the bottom surface)

Step 2: Calculate the lift force on each wing surface.
The lift force is given by the formula:

L = ΔP * A

Where:
L is the lift force
ΔP is the pressure difference
A is the wing area

Given:
A(top) = 16.2 m^2
A(bottom) = 16.2 m^2

Step 3: Calculate the net vertical force.
The net vertical force is the difference between the lift forces on the top and bottom surfaces:

F_net = L(top) - L(bottom)

Let's calculate it.

Step 1: Calculate the pressure difference between the top and bottom surfaces.
ΔP = 0.5 * ρ * V^2 * (Cp - Cn)
ΔP = 0.5 * ρ * (V(top)^2 - V(bottom)^2)
ΔP = 0.5 * ρ * ((70.0 m/s)^2 - (60.0 m/s)^2)

Step 2: Calculate the lift force on each wing surface.
L(top) = ΔP * A(top)
L(bottom) = ΔP * A(bottom)

Step 3: Calculate the net vertical force.
F_net = L(top) - L(bottom)

Now, we need the density of air (ρ) to complete the calculation.

To find the net vertical force that the air exerts on the airplane, we need to calculate the difference in pressure between the top and bottom surfaces of the wings.

The pressure difference can be calculated using Bernoulli's equation, which states that the total pressure at any point in a fluid flow system is the sum of the static pressure, dynamic pressure, and potential energy pressure.

In this case, since the air flows horizontally, the potential energy pressure can be disregarded as it remains constant. Therefore, the equation can be simplified to:

P + 1/2 * ρ * v^2 = constant

Where:
P is the static pressure,
ρ (rho) is the density of the fluid,
v is the velocity of the fluid.

Let's calculate the pressure difference, step by step:

First, we calculate the dynamic pressure at the top surface of the wings:

Dynamic pressure (P_dyn, top) = 1/2 * ρ * v^2
= 1/2 * ρ * (70.0 m/s)^2

Next, we calculate the dynamic pressure at the bottom surface of the wings:

Dynamic pressure (P_dyn, bottom) = 1/2 * ρ * v^2
= 1/2 * ρ * (60.0 m/s)^2

Since the density (ρ) of the air is the same on both surfaces, we can ignore it to find the pressure difference:

Pressure difference (ΔP) = P_dyn, top - P_dyn, bottom

Now that we have the pressure difference, we can calculate the net vertical force using the formula:

Net vertical force (F_net) = ΔP * wing area

Substituting the values:

F_net = ΔP * wing area
= (P_dyn, top - P_dyn, bottom) * (wing area)

Calculate the value of (P_dyn, top - P_dyn, bottom) using the given velocities, and substitute it into the formula along with the wing area to obtain the net vertical force on the airplane.