Problem 53 - Master Resource Book in JEE Main Physics

Question

A plane is in level flight at a constant apeed and each wing has an area of \(25 \mathrm{~m}^{2}\). During flight the speed of the air is \(216 \mathrm{kmh}^{-1}\) over the lower wing surface and \(252 \mathrm{kmh}^{-1}\) over the upper wing surface of each wing of aeroplane. Take density of air \(=1 \mathrm{kgm}^{-3}\) and \(g=10 \mathrm{~ms}^{-2}\) Pressure difference on each wing of aeroplane is (a) \(5 \mathrm{Nm}^{-2}\) (b) \(50 \mathrm{Nm}^{-2}\) (c) \(350 \mathrm{Nm}^{-2}\) (d) \(650 \mathrm{Nm}^{-2}\)

Step-by-Step Solution

Answer

The pressure difference on each wing of the airplane is (d) 650 Nm⁻².

Step 1: Convert Speeds to m/s

First, we need to convert the given speeds from km/h to m/s since the pressure difference formula requires speeds in m/s.- Speed over the lower surface = 216 km/h = \(\frac{216 \times 1000}{3600} \approx 60 \ \text{m/s}\)- Speed over the upper surface = 252 km/h = \(\frac{252 \times 1000}{3600} \approx 70 \ \text{m/s}\)

Step 2: Apply Bernoulli's Equation

We will use Bernoulli's equation to find the pressure difference, which states:\[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]Where:- \(P_1\) is the pressure beneath the wing.- \(v_1 = 60 \ \text{m/s}\) is the speed beneath the wing.- \(P_2\) is the pressure above the wing.- \(v_2 = 70 \ \text{m/s}\) is the speed above the wing.- \(\rho = 1 \ \text{kg/m}^3\) is the air density.

Step 3: Calculate Pressure Difference

Rearranging Bernoulli's equation to find the pressure difference \( \Delta P = P_1 - P_2 \), we have:\[ \Delta P = \frac{1}{2} \rho (v_2^2 - v_1^2) \]Substitute \(\rho = 1\), \(v_1 = 60\ \text{m/s}\), \(v_2 = 70\ \text{m/s}\):\[ \Delta P = \frac{1}{2} \times 1 \left((70)^2 - (60)^2\right) \]\[ \Delta P = \frac{1}{2} \times (4900 - 3600) \]\[ \Delta P = \frac{1}{2} \times 1300 \]\[ \Delta P = 650 \ \text{N/m}^2 \]

Step 4: Select the Correct Option

The calculated pressure difference on each wing of the airplane is \(650 \ \text{Nm}^{-2}\), which corresponds to option (d).

Key Concepts

Pressure Difference

In the study of fluid dynamics, one of the key principles used to understand the lift generated by an airplane wing is Bernoulli's Principle. This principle helps us calculate the pressure difference between the upper and lower surfaces of a wing. The pressure difference, denoted as \( \Delta P \), is crucial because it influences the lift force that keeps an airplane flying. According to Bernoulli's equation, the pressure difference can be expressed as:

• \( \Delta P = \frac{1}{2} \rho (v_2^2 - v_1^2) \)

- \( \rho \) is the air density,- \( v_1 \) and \( v_2 \) are the speeds of air above and below the wing, respectively.
To solve for \( \Delta P \), simply plug in the air densities and air speeds into the equation. The pressure difference is essential for determining if the correct lift force is being generated to keep the aircraft in steady flight.

Airplane Wing Dynamics

The dynamics of an airplane's wing are fascinating and directly linked to how pressure differences are created during flight. A wing is designed with a particular shape, often referred to as an airfoil. This design is crucial in influencing how air flows over and under the wing surfaces.

• The airfoil shape causes the air to travel faster over the top surface compared to the bottom.
• This speed variation creates a pressure differential according to Bernoulli's Principle, with lower pressure on the upper surface and higher pressure on the lower surface.

This difference in pressure is what generates lift, making it possible for the airplane to ascend and glide through the skies. Wing dynamics are meticulously engineered, with considerations for optimal lift-to-drag ratios, sustainability of flight, and fuel efficiency.

Speed Conversion

Speed conversion is a practical aspect of solving problems associated with airplane dynamics and Bernoulli's principle. Often, speeds are given in units such as kilometers per hour (km/h), which need to be converted to meters per second (m/s) for calculations based on Bernoulli's equation.

• To convert from km/h to m/s, use the conversion factor: \(1 \ \text{km/h} = \frac{5}{18} \ \text{m/s}\).

Converting speeds correctly ensures that all the units in equations match, which is critical for obtaining a valid solution. For example, converting the speeds of 216 km/h and 252 km/h for flight problems yields approximately 60 m/s and 70 m/s respectively, which are essential inputs for calculating pressure differences.

Air Density

Air density \( \rho \) is an important parameter when analyzing the dynamics of an airplane. It is the mass of air per unit volume and is typically measured in kilograms per cubic meter (kg/m\(^3\)). Air density affects how molecules are spaced, impacting lift generation as per Bernoulli's Principle.

• Generally, standard air density at sea level is \( 1.225 \ \text{kg/m}^3 \), but for many simplified calculations, \( 1 \ \text{kg/m}^3 \) may be used.
• A denser air increases lift since more air particles are interacting with the wing surfaces.

Changes in altitude, temperature, and humidity can all affect air density, which in turn alters flight dynamics. Pilots and engineers must understand these variations to properly assess and manage the performance of aircraft in different environmental conditions.