An airplane wing is designed so that the speed of the air across the top of the wing is 274 m/s when the speed of the air below the wing is 193 m/s. The density of the air is 1.29 kg/m3. What is the lifting force on a wing of area 25.0 m2?

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I know that it is the pressure difference times the area of the wing.

I know I need to use Bernoulli's equation but I do not know how incorporate area into it.

find the pressure difference between below and above with Bernoulli

multiply that pressure difference by 25 m^2 to get the force

P1 + 1/2 rho v1^2 = P2 + 1.2 rho v2^2

P1 -P2 = (1/2) rho (v2^2-v1^2)
= (1/2) (1.29)(274^2-193)^2

multiply that by 25

P1 -P2 = .5* (v1^2-v2^2)

difference in pressure multiplied by area

To incorporate the area into Bernoulli's equation, you need to consider the pressure difference over an area. Here's how you can use Bernoulli's equation to find the lifting force on the wing:

1. First, recall Bernoulli's equation in its simplest form, which relates the pressure, velocity, and height of a fluid:

P + 1/2ρv^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, and
h is the height above some reference point.

2. Determine the pressure difference across the wing. In this case, we need to find the pressure difference between the top and bottom surfaces of the wing. Since Bernoulli's equation gives the relation between points in a fluid flow, we need to choose two points on the wing.

3. Let's consider the top surface of the wing as point 1 and the bottom surface as point 2. The equation for Bernoulli's principle between these two points is:

P1 + 1/2ρv1^2 = P2 + 1/2ρv2^2

Where:
P1 and P2 are the pressures on the top and bottom surfaces respectively,
v1 is the velocity of the air across the top surface, and
v2 is the velocity of the air below the wing.

4. Rearrange the Bernoulli's equation to solve for the pressure difference between the top and bottom surfaces:

P2 - P1 = 1/2ρ(v1^2 - v2^2)

5. Now, we can calculate the pressure difference using the given velocities. Plug the values into the equation:

P2 - P1 = 1/2 * 1.29 kg/m^3 * (274 m/s)^2 - (193 m/s)^2)

6. Calculate the pressure difference, which should have units of pascals (Pa).

7. Finally, to find the lifting force, multiply the pressure difference by the area of the wing:

Lifting force = Pressure difference * Area of the wing

Lifting force = (Pressure Difference) * 25.0 m^2

8. Calculate the lifting force using the pressure difference obtained in step 6 and multiply it by the wing's area.

This will give you the lifting force on the wing in newtons.