Problem 91 - Master Resource Book in JEE Main Physics

Question

Air is streaming past a horizontal air plane wing such that its speed is \(120 \mathrm{~ms}^{-1}\) over the upper surface and \(90 \mathrm{~ms}^{-1}\) at the lower surface. If the density of air is \(1.3 \mathrm{kgm}^{-3}\), what will be the gross lift on the wing? If the wing is \(10 \mathrm{~m}\) long and has an average width of \(2 \mathrm{~m}\) (a) \(81.9 \mathrm{~N}\) (b) \(8.19 \mathrm{kN}\) (c) \(81.9 \mathrm{kN}\) (d) \(819 \mathrm{kN}\)

Step-by-Step Solution

Answer

The gross lift on the wing is 81.9 kN, option (c).

Step 1: Understanding Bernoulli's Principle

To find the lift on the wing, we use Bernoulli's equation which relates the pressure difference to speed for a streamline flow. Bernoulli's equation is given by: \[ P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \]where \(P\) is pressure, \(\rho\) is the fluid density, and \(v\) is the fluid velocity. The pressure difference \(\Delta P\) can be found as:\[ \Delta P = \frac{1}{2}\rho (v_2^2 - v_1^2) \]

Step 2: Substitute Values for Pressure Difference

Given the speeds over the top (\(v_2 = 120 \mathrm{~ms}^{-1}\)) and bottom (\(v_1 = 90 \mathrm{~ms}^{-1}\)) surfaces and air density (\(\rho = 1.3 \mathrm{kgm}^{-3}\)), substitute these into the equation for pressure difference:\[ \Delta P = \frac{1}{2} \times 1.3 \times \left(120^2 - 90^2\right) \]Compute this to find \(\Delta P\).

Step 3: Calculate Pressure Difference

Continuing from the substitution:\[ \Delta P = \frac{1}{2} \times 1.3 \times (14400 - 8100) \]\[ \Delta P = \frac{1}{2} \times 1.3 \times 6300 \]\[ \Delta P = 4095 \mathrm{~N/m}^2 \]

Step 4: Calculate Lift Force

The lift force \(F_L\) is given by multiplying the pressure difference by the wing area, \(A\). The wing area can be calculated as length \(\times\) width = \(10 \mathrm{~m} \times 2 \mathrm{~m} = 20 \mathrm{~m}^2\). Substituting the values:\[ F_L = 4095 \times 20 \]\[ F_L = 81900 \mathrm{~N} \]

Step 5: Determine the Correct Answer

Since \( F_L = 81900 \mathrm{~N} \) is equivalent to \(81.9 \mathrm{kN}\), the correct answer from the options given is (c) \(81.9 \mathrm{kN}\).

Key Concepts

Lift force calculation

Calculating lift force involves understanding how certain variables contribute to the overall force that enables an aircraft's wing to lift it. The lift force is a crucial aspect of flight and is directly related to the pressure difference above and below the wing.
In essence, lift force can be computed by multiplying the pressure difference (\( \Delta P \)) by the area (\( A \)) of the wing. This can be represented by the equation:

• \[ F_L = \Delta P \times A \]

The area is calculated by multiplying the length and width of the wing. Therefore, for this exercise, the area of the wing is \( 20 \text{ m}^2 \).Replacing the obtained pressure difference with the wing area, the lift force for this wing is determined to be \( 81900 \text{ N} \), which is equivalent to \( 81.9 \text{ kN} \). This shows that a larger area or a higher pressure difference will result in a greater lift force.

Pressure difference

Pressure difference is a key factor in generating lift according to Bernoulli's Principle. It describes the variance in pressure between the top and bottom surfaces of a wing.

• Higher pressure on the bottom surface compared to the top creates an upward force.
• To find this difference, use the equation:\[ \Delta P = \frac{1}{2}\rho (v_2^2 - v_1^2) \]where:
  • \( \rho \) is the air density.
  • \( v_2 \) and \( v_1 \) are the velocities over the upper and lower surfaces, respectively.

Substituting the given speeds and density:

• \( v_2 = 120 \text{ m/s},\, v_1 = 90 \text{ m/s},\, \rho = 1.3 \text{ kg/m}^3 \)

The pressure difference becomes \( 4095 \text{ N/m}^2 \), highlighting how varying airflow speed results in pressure changes due to Bernoulli's Principle.

Fluid dynamics

Fluid dynamics involves studying the movement of fluids, including air, and their interactions with surfaces, such as airplane wings. It's essential for understanding how forces, like lift, are generated.
Factors like speed, pressure, and viscosity of the fluid contribute to these interactions:

• Bernoulli's Principle tells us that in a flowing fluid, an increase in velocity leads to a decrease in pressure.
• This principle helps explain why faster airflow over a wing's top surface creates lower pressure compared to slower airflow beneath.

As fluid dynamics theory is applied, it provides insights into efficient airplane wing design and operation, ensuring balanced lift and minimal drag.

Airplane wing aerodynamics

Aerodynamics is crucial for understanding how airplane wings generate lift through their shape and design.

• The design of the wing, often called an airfoil, is specifically crafted to create differences in airflow speed above and below the wing.
• Wings are angled slightly upwards, known as the angle of attack, to enhance lift.

The Bernoulli's Principle magic comes into play here:

• Faster airflow across the curved top of the wing means lower pressure, complemented by slower airflow below generating higher pressure.
• This pressure difference results in a net upward force, known as lift, essential for flight.

Thorough understanding of aerodynamics allows engineers to design wings that optimize lift while reducing resistance or drag, resulting in more efficient and safer flight.