An air-plane has an effective wing surface area of 17.0 m² that is generating the lift force. In level flight the air speed over the top of the wings is 66.0 m/s, while the air speed beneath the wings is 40.0 m/s. What is the weight of the plane? (The density of air is 1.29 kg/m³)

Given that,

Surface area A = 17m²

The speed at the top v" = 66m/s

Speed beneath is v' = 40 m/s

The density of air p = 1.29kg/m³

Weight of plane?

Assuming that,

the height difference between the top and bottom of the wind is negligible and we can ignore any change in gravitational potential energy of the fluid.

Using Bernoulli equation

P' + ½pv'² + pgh' = P'' + ½pv''² + pgh''

Where

P' is pressure at the bottom in N/m²

P" is pressure at the top in N/m²

v' is velocity at the bottom in m/s

v" is velocity at the top in m/s

Then, Bernoulli equation becomes

P' + ½pv'² = P'' + ½pv''²

Rearranging

P' - P'' = ½pv"² - ½pv'²

P'-P" = ½p (v"² - v'²)

P'-P" = ½ * 1.29 * (66²-40²)

P'-P" = 1777.62 N/m²

Lift force can be found from

Pressure = force/Area

Force = ∆P * A

Force = (P' - P") * A

Since we already have (P'-P")

Then, F=W = (P' - P") * A

W = 1777.62 * 17

W = 30,219.54 N

The weight of the plane is 30.22 KN

Weight of plane; W = 30219.54 N

Explanation:

For us to determine the lift force of the system, let's multiply the pressure difference with the effective wing surface area given that the area is obtained by Bernoulli equation. Thus,

P_b + (1/2) ρ (v_b) ² + ρg (y_b) = P_t + (1/2) ρ (v_t) ² + ρg (y_t)

Now, since the flight is level, the height is constant.

Thus, (y_b) = (y_t)

So, we now have;

P_b + (1/2) ρ (v_b) ² = P_t + (1/2) ρ (v_t) ²

Rearranging, we have;

P_b - P_t = (1/2) ρ (v_t) ² - (1/2) ρ (v_b) ²

P_b - P_t = (1/2) ρ[ (v_t) ² - (v_b) ²]

Now, weight is given by the formula;

W = (P_b - P_t) •A

Thus,

W = (1/2) ρ[ (v_t) ² - (v_b) ²] •A

From the question,

Density; ρ = 1.29 kg/m³

Velocity over top of wings; v_t = 66 m/s

Velocity beneath the wings; v_b = 40 m/s

Surface Area; A = 17 m²

Thus;

W = (1/2) 1.29[ (66) ² - (40) ²] •17

W = (1/2) •1.29•17[2756]

W = 30219.54 N