Problem 30 - Master Resource Book in JEE Main Physics

Question

An aeroplane of \(\operatorname{mass} 3 \times 10^{4} \mathrm{~kg}\) and total wing area of \(120 \mathrm{~m}^{2}\) is in a level flight at some height. The difference in pressure between the upper and lower surfaces of its wings in kilo pascal is \(\left(g=10 \mathrm{~ms}^{-2}\right.\) ) (a) \(2.5\) (b) \(5.0\) (c) \(10.0\) (d) \(12.5\)

Step-by-Step Solution

Answer

The pressure difference is 2.5 kPa, so the answer is (a) 2.5.

Step 1: Understand the Problem

We need to find the pressure difference between the upper and lower surfaces of the airplane's wings in kPa. Given: mass of the aeroplane is \( 3 \times 10^4 \, \text{kg} \), wing area is \( 120 \, \text{m}^2 \), and acceleration due to gravity is \( g = 10 \, \text{ms}^{-2} \).

Step 2: Use Lift Force Equation

The lift force \( F_L \) needed to keep the airplane in level flight equals the gravitational force acting on it, which is \( F_L = mg \, \). The lift force is also given by the pressure difference times the wing area, i.e., \( F_L = \Delta P \times A \).

Step 3: Calculate Gravitational Force

Calculate the gravitational force (Weight): \( F = mg = 3 \times 10^4 \, \text{kg} \times 10 \, \text{ms}^{-2} = 3 \times 10^5 \, \text{N} \).

Step 4: Relate Force and Pressure Difference

We know from the lift force equation: \( 3 \times 10^5 \text{ N} = \Delta P \times 120 \, \text{m}^2 \). Therefore, the pressure difference \( \Delta P \) is \( \Delta P = \frac{3 \times 10^5}{120} \).

Step 5: Calculate Pressure Difference

Solve for \( \Delta P \): \( \Delta P = \frac{3 \times 10^5}{120} = 2500 \, \text{Pa} \).

Step 6: Convert Pressure Difference to kPa

To convert \( \Delta P \) from pascals to kilopascals, divide by 1000: \( \Delta P = \frac{2500}{1000} = 2.5 \, \text{kPa} \).

Key Concepts

Lift force

The lift force is a crucial aerodynamics concept, especially in the context of aircraft. It is the force that directly opposes the weight of an airplane. Think of it as the upward push that keeps the plane in the air.
Lift is generated by the wings as air moves over and under them. The shape of the wings—known as an airfoil design—plays a significant role. This design causes the air pressure on the wing's upper surface to be lower than that on the lower surface.
Mathematically, the lift force can be represented as:

• \( F_L = \Delta P \times A \)
• Where \( F_L \) is the lift force, \( \Delta P \) is the pressure difference between the upper and lower wing surfaces, and \( A \) is the wing area.

In level flight, the lift force must equal the gravitational force pulling the airplane downwards. This balance allows the airplane to maintain altitude without climbing or descending.

Pressure difference

The pressure difference between the upper and lower surfaces of an airplane’s wings is what creates lift force. This difference is a result of air moving faster over the top surface of the wing than beneath it.
This scenario is explained by Bernoulli’s principle, which states that faster moving air results in lower pressure. When this happens, the higher pressure beneath the wings pushes the airplane upwards.
To calculate the required pressure difference, one can rearrange the lift equation:

• \( \Delta P = \frac{F_L}{A} \)
• Where \( F_L \) is the lift force and \( A \) is the area of the wings.

In this specific exercise, substituting the known values, you calculate that the pressure difference required to keep the airplane in level flight is 2.5 kPa.

Level flight

Level flight is a stable flight condition where an airplane maintains a constant altitude. To achieve level flight, the lift force must balance the gravitational force acting on the plane.
In practice, pilots adjust the aircraft's speed and angle of attack—how the wing meets the air—to maintain this balance. Aerodynamic lift exactly compensates for the weight of the aircraft.
For example, if the airplane has a mass of \(3 \times 10^4 \text{ kg}\), the gravitational force is calculated as:

• \( F = mg = 3 \times 10^4 \text{ kg} \times 10 \text{ ms}^{-2} = 3 \times 10^5 \text{ N} \)

To stay in level flight, the airplane must produce a lift force equal to this gravitational force. Comprehending this balance is key for pilots and aerodynamics enthusiasts to understand how planes fly smoothly through the sky.