Question

An airplane flies on a level path. There is a pressure difference of 500 Pa between the lower and upper surfaces of the wings. The area of each wing surface is about \(100 \mathrm{m}^{2} .\) The air moves below the wings at a speed of \(80.5 \mathrm{m} / \mathrm{s} .\) Estimate (a) the weight of the plane and (b) the air speed above the wings.

Short answer

Based on the given information and calculations, estimate the weight of the airplane and the airspeed above the wings.

Step-by-step solution

Understand the given information and the questions to be answered

We are given the pressure difference between the upper and lower surfaces of the wings, the area of each wing surface, and the airspeed below the wings. We need to estimate the weight of the plane and the airspeed above the wings.

Use Bernoulli's Equation

Bernoulli's Equation states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume of a fluid remains constant along a streamline. Since the airplane is flying at a level path, we can disregard the potential energy term. Thus, the Bernoulli's Equation for this case simplifies to: \(P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2\) Where \(P_1\) and \(v_1\) are pressure and airspeed below the wings, \(P_2\) and \(v_2\) are pressure and airspeed above the wings, and \(\rho\) is the air density.

Calculate the airspeed above the wings

We know the pressure below the wings, but we need the pressure above the wings to find the airspeed. We can obtain the value for air density using standard atmospheric density: \(\rho = 1.225 kg/m^3\) at sea level. Rearranging Bernoulli's Equation and solving for \(v_2\), we get: \(v_2 = \sqrt{\frac{2(P_1 - P_2)}{\rho} + v_1^2}\) We are given the pressure difference as 500 Pa, which implies \(P_1 - P_2 = 500 \thinspace Pa\). Substituting the values, we get: \(v_2 = \sqrt{\frac{2(500)}{1.225} + (80.5)^2}\) \(v_2 \approx 82.24 m/s\) The airspeed above the wings is approximately 82.24 m/s.

Calculate the force exerted by the wings

The pressure difference between the lower and upper surfaces of the wings causes the force that lifts the airplane. The force exerted by the wings can be calculated by multiplying the pressure difference by the wing area: \(F = \Delta P \cdot A\) Where \(F\) is the force exerted by the wings, \(\Delta P\) is the pressure difference, and \(A\) is the area of the wings. Substituting the given values, we get: \(F = 500 \cdot 100\) \(F = 50,000 N\) The force exerted by the wings is 50,000 N.

Estimate the weight of the plane

Since the airplane is flying at a level path, the weight of the plane (W) is equal to the lift force exerted by the wings: \(W = F\) \(W = 50,000 N\) The weight of the airplane is estimated to be 50,000 N. To summarize, the estimated weight of the airplane is 50,000 N, and the airspeed above the wings is approximately 82.24 m/s.