Question

Use Bernoulli's equation to estimate the upward force on an airplane's wing if the average flow speed of air is \(190 \mathrm{m} / \mathrm{s}\) above the wing and \(160 \mathrm{m} / \mathrm{s}\) below the wing. The density of the air is \(1.3 \mathrm{kg} / \mathrm{m}^{3}\) and the area of each wing surface is $28 \mathrm{m}^{2}$

Short answer

Question: Using Bernoulli's equation, estimate the upward force on an airplane's wing if the average flow speeds of air above and below the wing are 190 m/s and 160 m/s, respectively, the density of the air is 1.3 kg/m³, and the area of each wing surface is 28 m². Answer: The estimated upward force on the airplane's wing is 166,600 N.

Step-by-step solution

Understand Bernoulli's Equation

Bernoulli's equation states that for an incompressible, non-viscous fluid flow, the sum of pressure energy, kinetic energy, and potential energy is constant along any streamline. Mathematically, it can be expressed as: \(P + \dfrac{1}{2} \rho v^2 + \rho gh = constant\) where \(P\) is the fluid pressure, \(\rho\) is the fluid density, \(v\) is the flow velocity, \(g\) is the acceleration due to gravity, and \(h\) is the height above a reference point. For this problem, since the height difference above and below the wing is negligible, we can apply Bernoulli's equation as follows: \(P_1 + \dfrac{1}{2} \rho v_1^2 = P_2 + \dfrac{1}{2} \rho v_2^2\)

Find the Pressure Difference

To find the pressure difference, plug in the given values of density and flow speeds into Bernoulli's equation: \(P_1 + \dfrac{1}{2} (1.3) (190)^2 = P_2 + \dfrac{1}{2} (1.3) (160)^2\) Rearrange the equation to find the pressure difference: \(\Delta P = P_1 - P_2 = \dfrac{1}{2} (1.3) ((190)^2 - (160)^2)\) Calculate the pressure difference: \(\Delta P = 5950 \, \mathrm{Pa}\)

Calculate the Upward Force

Finally, to determine the upward force on the airplane's wing, we will use the following equation: \(F = \Delta P \cdot A\) where \(F\) is the upward force, \(\Delta P\) is the pressure difference, and \(A\) is the wing surface area. Using the calculated pressure difference and the given wing area, we can find the upward force: \(F = 5950 \, \mathrm{Pa} \cdot 28 \, \mathrm{m}^2\) Calculate the upward force: \(F = 166600 \, \mathrm{N}\) The estimated upward force on the airplane's wing is \(166600 \, \mathrm{N}\).