Problem 59 - Physics

Question

An airplane wing is designed so that the speed of the air across the top of the wing is \(251 \mathrm{~m} / \mathrm{s}\) when the speed of the air below the wing is \(225 \mathrm{~m} / \mathrm{s}\). The density of the air is \(1.29 \mathrm{~kg} / \mathrm{m}^{3} .\) What is the lifting force on a wing of area \(24.0 \mathrm{~m}^{2} ?\)

Step-by-Step Solution

Answer

The lifting force on the wing is approximately 191,373 N upwards.

Step 1: Understand the Problem

We need to calculate the lifting force on the airplane wing caused by the difference in airspeed across the top and bottom of the wing. We will use Bernoulli's principle and the given wing area to find the solution.

Step 2: Apply Bernoulli's Equation

According to Bernoulli's principle, the pressure difference \( \Delta P \) between the top and bottom of the wing is given by: \[ \Delta P = \frac{1}{2} \rho (v_{bottom}^2 - v_{top}^2) \]where \( v_{bottom} = 225 \) m/s, \( v_{top} = 251 \) m/s, and \( \rho = 1.29 \) kg/m³.

Step 3: Calculate Pressure Difference

Substitute the values into the Bernoulli's equation:\[ \Delta P = \frac{1}{2} \times 1.29 \times (225^2 - 251^2) \]Solving the expression in the parenthesis first gives:\[ 225^2 = 50625; \quad 251^2 = 63001 \]\[ 225^2 - 251^2 = 50625 - 63001 = -12376 \]Then calculating the pressure difference:\[ \Delta P = \frac{1}{2} \times 1.29 \times (-12376) \approx -7973.88 \, \text{Pa} \]

Step 4: Calculate Lifting Force

Now, use the pressure difference to find the lifting force \( F_L \):\[ F_L = \Delta P \times A \]Substitute in the area and pressure difference:\[ F_L = -7973.88 \, \text{Pa} \times 24.0 \, \text{m}^2 \]\[ F_L \approx -191373.12 \, \text{N} \]The negative sign indicates that the direction of lift is upward, opposite to the defined positive pressure direction.

Key Concepts

Aerodynamics

Aerodynamics is a branch of physics concerned with the study of air movement around objects, often used in the context of vehicles like cars and airplanes. It explores how air interacts with solid surfaces, such as an airplane wing, and how these interactions can be used to influence motion.
In our problem, aerodynamics plays a crucial role. The wing of the airplane is designed specially to manage air flow differently over the top and the bottom surfaces. This difference creates variations in air speed, which are fundamental to generating lift.

• The faster-moving air over the top of the wing reduces pressure compared to the slower-moving air underneath.
• This imbalance in air speed and pressure causes the airplane to lift.

Overall, aerodynamics helps experts design wings that use these principles to improve lift efficiency, minimizing fuel consumption and increasing flight stability.

Pressure Difference

The concept of pressure difference is central to Bernoulli's principle and to understanding how lift is created on a wing. When air flows over a wing, a pressure differential arises between the upper and lower surfaces. Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure. Let's put it into easier words:

• If the air moves faster across the top than below the wing, the pressure above is lower than the pressure below.
• This lower pressure on top helps to create a pressure difference known as \( \Delta P \).

Mathematically, the pressure difference can be calculated using the equation:\[\Delta P = \frac{1}{2} \rho (v_{bottom}^2 - v_{top}^2)\]Where:

• \( \rho \) is the air density, \( v_{bottom} \) is the speed of the air below the wing, and \( v_{top} \) is the speed over the wing.
• The negative result is interpreted as the direction of the lift being upward due to lower pressure on the top.

This concept is fundamental not only in aviation but also in many engineering applications involving fluid dynamics.

Lifting Force

Lifting force is the push that keeps an airplane in the sky, counteracting gravity. It emerges from the difference in air pressure created by the speed of airflow over the wing, essentially utilizing Bernoulli’s principle. In the given exercise, once the pressure difference \( \Delta P \) is calculated using the speeds and densities provided, we can find the lifting force \( F_L \). The formula used is:\[F_L = \Delta P \times A\]Where:

• \( F_L \) is the lifting force.
• \( \Delta P \) is the pressure difference.
• \( A \) is the area of the wing's surface.

The resulting force provides the upward push needed to counteract the weight of the plane. Importantly, the negative sign in the pressure calculation indicates the lift's upward direction, contrary to the assumed pressure flow.
Through the combination of this pressure difference over a sufficient wing area, airplanes can fly and maneuver effectively by maximizing lifting force.