Question

In a test experiment on a model aeroplane in a wind tunnel, the flow speed on the upper and lower surfaces of the wings are ${v_1}$ and ${v_2}$ respectively. If $\rho$ is the density of air, then the upward lift is (Given: A = cross-sectional area of the wing)
A. $\dfrac{1}{2}\rho A\left( {v_1^2 - v_2^2} \right)$
B. $\dfrac{1}{2}\rho A\left( {v_2^2 - v_1^2} \right)$
C. $A{v_1} - A{v_2}$
D. ${v_1} - {v_2}$

Answer 1

This question is based on the lift of the aeroplane. We know that the lift of the aeroplane is due to Bernoulli’s theorem. Bernoulli’s theorem states that when the pressure of the fluid flowing inside the narrow section increases, the velocity decreases. It is the addition of all the energies present in the fluid under steady-state conditions. The pressure on the upper surface is higher than the pressure on the lower surface of the airplane.

Complete step by step answer:
We know that Bernoulli's principle is applicable here to lift the aeroplane easily. The lift of an aeroplane is due to the pressure difference between the upper and lower surface of an aeroplane. The velocity at the upper surface is more while the velocity at the lower surface is less.
So, by using Bernoulli’s theorem we get,
$\begin{array}{l} {P_1} + \dfrac{1}{2}\rho v_1^2 = {P_2} + \dfrac{1}{2}\rho v_2^2\\\ {P_2} - {P_1} = \dfrac{1}{2}\rho \left( {v_1^2 - v_2^2} \right) \end{array}$
Here, ${P_1}$ is the upper surface pressure, ${P_2}$ is the lower surface pressure, ${v_1}$ is the velocity of the upper surface, ${v_2}$ is the velocity of the upper surface and $\rho$ is the density.
So, the pressure difference between upward and lower surface is given as,
$\Delta P = \dfrac{1}{2}\rho \left( {v_1^2 - v_2^2} \right)$
Now, we have to calculate the lift of the aeroplane.
${\rm{lift}} = \Delta P \times A$
We can now substitute the known expressions in the above expression, such that the result is,
${\rm{lift}} = \dfrac{1}{2}\rho \left( {v_1^2 - v_2^2} \right) \times A$

Thus, the correct option from the given options is (A).

Note: In this question, students might have knowledge about Bernoulli's theorem. We have to know where the pressure is higher or lower. We have to know the expression of Bernoulli’s theorem to calculate the upward lift.