Problem 54 - 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 \(g=10 \mathrm{~ms}^{-2}\) Percentage of velocity difference of the upper and lower surface of the wings of aeroplane is (a) \(14.39\) (b) \(15.4 \%\) (c) \(16.7 \%\) (d) \(17.49\)

Step-by-Step Solution

Answer

The percentage velocity difference is approximately 16.7%, corresponding to option (c).

Step 1: Convert Speeds to m/s

To find the percentage of velocity difference, we first convert the wings' speeds from km/h to m/s. \[ \text{Velocity over lower surface: } 216 \text{ km/h} = \frac{216 \times 1000}{3600} \text{ m/s} = 60 \text{ m/s} \] \[ \text{Velocity over upper surface: } 252 \text{ km/h} = \frac{252 \times 1000}{3600} \text{ m/s} = 70 \text{ m/s} \]

Step 2: Calculate Velocity Difference

Next, calculate the difference in velocities between the upper and lower surfaces: \[ \Delta v = v_{\text{upper}} - v_{\text{lower}} = 70 \text{ m/s} - 60 \text{ m/s} = 10 \text{ m/s} \]

Step 3: Compute Percentage Velocity Difference

The percentage velocity difference is given by comparing the velocity difference to the lower surface velocity and multiplying by 100: \[ \text{Percentage difference} = \left( \frac{\Delta v}{v_{\text{lower}}} \right) \times 100 = \left( \frac{10}{60} \right) \times 100 \] \[ = \frac{1}{6} \times 100 \approx 16.67\% \]

Step 4: Identify the Correct Option

Compare the calculated percentage difference with the provided options. The closest match to our calculation of approximately \(16.67\%\) is option (c) \(16.7\%\).

Key Concepts

Aerodynamics

Aerodynamics is the study of how gases interact with moving bodies, particularly focusing on how air moves around objects. In the context of airplanes, aerodynamics is crucial as it determines how lift is generated, allowing an airplane to stay aloft. Understanding aerodynamics can help in designing aircraft to move more efficiently through the air by reducing drag. Consider an airplane wing in flight. The shape of the wing, often referred to as an airfoil, is designed to manage airflow differently above and below the wing. This results in a pressure difference, leading to the generation of lift according to Bernoulli's Principle. The faster the air moves over the top of the wing compared to the bottom, the more lift is generated. Key concepts involved in aerodynamics:

• Lift: The upward force that counters the weight of the airplane.
• Drag: The resistance force opposing the aircraft's motion through the air.
• Thrust: The forward force produced, usually by engines, propelling the airplane forward.
• Weight: The downward force due to gravity acting on the airplane's mass.

Velocity Difference

Velocity difference is a critical factor in determining how lift is generated on an airplane wing. In the provided problem, we measure the difference in speed between the air moving over the top surface of the wing and the air moving below it. This difference in velocity directly influences the pressure difference between the upper and lower surfaces of the wing. Bernoulli's Principle tells us that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure. For an airplane in smooth flight, the velocity over the upper surface is higher compared to the lower surface, resulting in lower pressure on top and higher pressure below, thus creating lift. To convert velocities from kilometers per hour to meters per second:

• Divide the speed in km/h by 3.6 to get m/s.

In our example:

• Lower surface speed: 216 km/h = 60 m/s
• Upper surface speed: 252 km/h = 70 m/s

The percentage velocity difference is calculated by comparing the velocity difference to the lower wing surface speed, resulting in approximately 16.67% in this case.

Fluid Dynamics

Fluid dynamics is the branch of physics that studies the movement of liquids and gases. For airplanes, it helps understand airflow over the wings and the resulting forces produced on the aircraft. These principles are applied to predict how air will move in relation to complex objects like airplane wings. Central to fluid dynamics and aerodynamics is Bernoulli's Principle, which describes the behavior of fluid under varying conditions of flow and height. In aviation, this principle explains how variations in velocity and pressure can lead to lift. To further grasp fluid dynamics in aviation:

• The air density plays a significant role, as seen with the given air density of 1 kg/m³, which impacts lift and drag forces.
• Pressure differences created by velocity variations are essential for generating lift.
• Flow patterns: The streamlined flow over wings is preferred to minimize turbulence and maximize efficient lift.

When analyzing an airplane in flight, understanding fluid dynamics helps engineers and pilots maximize performance and safety.