Question

The area of an airplane wing is \(A=4\,m^{2}\). Air flows with velocity \(v_{1}=80\,m/s\) above the wing and \(v_{2}=60\,m/s\) below it. The density of air is \(1.2\,kg/m^{3}\). Find the pressure difference \((\Delta P)\) between the upper and lower surfaces of the wing.

• A. \(1200\;Pa\)
• B. \(1680\;Pa\)
• C. \(2400\;Pa\)
• D. \(3600\;Pa\)

Hint:

For airplane wing problems, remember: \[ \boxed{\Delta P=\frac{1}{2}\rho\left(v_{\text{fast}}^{2}-v_{\text{slow}}^{2}\right)} \] Greater fluid velocity corresponds to lower pressure according to Bernoulli's principle. If the lift force is asked, \[ \boxed{F=\Delta P\times A.} \] Always check whether the question asks for pressure difference or lift force.

Answer 1

The Correct Option is B

Concept: The pressure difference between the upper and lower surfaces of an airplane wing is explained using Bernoulli's Principle. According to Bernoulli's theorem, \[ P+\frac{1}{2}\rho v^{2}+\rho gh=\text{Constant} \] where

• \(P\) = Pressure of the fluid,

• \(\rho\) = Density of the fluid,

• \(v\) = Velocity of the fluid,

• \(h\) = Height above the reference level.
Since the upper and lower surfaces of the wing are nearly at the same height, \[ h_{1}=h_{2}, \] the gravitational potential energy terms cancel out. Hence, \[ P_{1}+\frac{1}{2}\rho v_{1}^{2} = P_{2}+\frac{1}{2}\rho v_{2}^{2}. \] Therefore, the pressure difference is \[ \boxed{\Delta P=P_{2}-P_{1} =\frac{1}{2}\rho\left(v_{1}^{2}-v_{2}^{2}\right)} \] where \[ v_{1}\gt v_{2}. \] Thus, the faster-moving air above the wing produces lower pressure.

Step 1: Write the given data.
Given, \[ A=4\,m^{2} \] \[ v_{1}=80\,m/s \] \[ v_{2}=60\,m/s \] \[ \rho=1.2\,kg/m^{3}. \] Although the area of the wing is provided, it is not required for calculating the pressure difference.

Step 2: Apply Bernoulli's equation.
Using, \[ \Delta P = \frac{1}{2}\rho \left(v_{1}^{2}-v_{2}^{2}\right), \] Substitute the given values, \[ \Delta P = \frac{1}{2}\times1.2 \left(80^{2}-60^{2}\right). \]

Step 3: Calculate the square of the velocities.
\[ 80^{2}=6400, \] \[ 60^{2}=3600. \] Hence, \[ 6400-3600=2800. \] Therefore, \[ \Delta P = 0.6\times2800. \]

Step 4: Evaluate the pressure difference.
\[ \Delta P = 1680\;Pa. \] Thus, \[ \boxed{\Delta P=1680\;Pa.} \] Hence, the correct answer is \[ \boxed{\textbf{Option (B)}}. \]

Additional Note: If the lift force acting on the wing is required, it can be calculated using \[ F=\Delta P\times A. \] Here, \[ F=1680\times4=6720\;N. \] However, since the question asks only for the pressure difference, the area is not used in the final answer.