Question

The speeds of air-flow on the upper and lower surfaces of a wing of an aeroplane are $ v_1 $ and $ v_2 $, respectively. If $ A $ is the cross-sectional area of the wing and $ \rho $ is the density of air, then the upward lift is

• A. \( \frac{1}{2} \rho A (v_1 - v_2) \)
• B. \( \frac{1}{2} \rho A (v_1 + v_2) \)
• C. \( \frac{1}{2} \rho A (v_1^2 - v_2^2) \)
• D. \( \frac{1}{2} \rho A (v_1^2 + v_2^2) \)

Hint:

The upward lift on the wing can be derived from Bernoulli's equation, and it depends on the difference in the squares of the velocities on the upper and lower surfaces of the wing.

Answer 1

The Correct Option is C

The upward lift on a wing is given by Bernoulli's principle, which relates the difference in the velocities of air over the upper and lower surfaces of the wing to the lift force. 
The pressure difference \( \Delta P \) between the upper and lower surfaces is related to the velocities \( v_1 \) and \( v_2 \) by: \[ \Delta P = \frac{1}{2} \rho (v_1^2 - v_2^2) \] The upward lift \( L \) is the force exerted by this pressure difference on the cross-sectional area of the wing. Therefore, the lift force is: \[ L = \Delta P \cdot A = \frac{1}{2} \rho A (v_1^2 - v_2^2) \] 
Thus, the correct answer is: \[ \text{(3) } \frac{1}{2} \rho A (v_1^2 - v_2^2) \]

Answer 2

To determine the upward lift on the wing of an aeroplane due to different speeds of air-flow across its surfaces, we leverage Bernoulli's principle. According to this principle, the pressure exerted by a fluid in motion decreases as the speed of the fluid increases, which can be expressed mathematically using the Bernoulli's equation:

Bernoulli's Equation

\( P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \)

where \( P_1 \) and \( P_2 \) are the pressures on the upper and lower surfaces respectively, and \( v_1 \) and \( v_2 \) are the corresponding air speeds.

Lift Force Calculation

The lift force (\( F \)) is the result of pressure difference between the lower and upper surfaces of the wing and is given by:

\( F = (P_2 - P_1) \times A \)

From Bernoulli's equation, we can express the pressure difference \( P_2 - P_1 \) as:

\( P_2 - P_1 = \frac{1}{2} \rho v_1^2 - \frac{1}{2} \rho v_2^2 \)

Plugging this into the lift force equation, we get:

\( F = [\frac{1}{2} \rho v_1^2 - \frac{1}{2} \rho v_2^2] \times A \)

Simplifying, the formula for the lift force becomes:

\( F = \frac{1}{2} \rho A (v_1^2 - v_2^2) \)

Therefore, the correct expression for the upward lift on the wing is:

\( \frac{1}{2} \rho A (v_1^2 - v_2^2) \)