Problem 67 - Physics

Question

An airplane has an effective wing surface area of \(16 \mathrm{~m}^{2}\) that is generating the lift force. In level flight the air speed over the top of the wings is \(62.0 \mathrm{~m} / \mathrm{s},\) while the air speed beneath the wings is \(54.0 \mathrm{~m} / \mathrm{s}\). What is the weight of the plane?

Step-by-Step Solution

Answer

The weight of the plane is approximately 9087 N.

Step 1: Understand Bernoulli's Principle

In this problem, we will use Bernoulli's equation to calculate the lift force. Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure. This can be expressed as: \( P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \), where \( P \) is the pressure, \( \rho \) is the density of the fluid (air in this case), and \( v \) is the velocity of the fluid.

Step 2: Calculate Pressure Difference

According to Bernoulli's principle, the difference in pressure above and below the wings can be expressed as: \( \Delta P = \frac{1}{2} \rho (v_{ ext{top}}^2 - v_{ ext{bottom}}^2) \). Substitute the given airspeed values: \( \Delta P = \frac{1}{2} \rho (62.0^2 - 54.0^2) \).

Step 3: Determine Density of Air

Assuming standard conditions, the density of air \( \rho \) is approximately \( 1.225 \text{ kg/m}^3 \). This is a commonly used approximation for the density of air at sea level.

Step 4: Calculate Lift Force

The lift force can be calculated by multiplying the pressure difference (\( \Delta P \)) by the effective wing surface area. This is expressed by the formula: \( F_{ ext{lift}} = \Delta P \times A \). Substitute the values: \( F_{ ext{lift}} = \frac{1}{2} \times 1.225 \times (62.0^2 - 54.0^2) \times 16.0 \) m\(^2\).

Step 5: Simplify and Solve for Lift Force

Carry out the calculations: \( F_{ ext{lift}} = \frac{1}{2} \times 1.225 \times (3844 - 2916) \times 16.0 = \frac{1}{2} \times 1.225 \times 928 \times 16.0 \). This simplifies to \( F_{ ext{lift}} \approx 9087 \, \text{N}\).

Step 6: Relate Lift Force to Weight of Plane

For level flight, the lift force equals the weight of the plane. Therefore, the weight of the plane is: \( W = F_{ ext{lift}} \approx 9087 \, \text{N}\).

Key Concepts

Lift Force Calculation

Lift force is crucial for an airplane to stay airborne. It acts perpendicular to the flow of the air around the wings and opposes the weight of the airplane. To calculate the lift force, we use Bernoulli's Principle, which helps us understand how pressure differences arise due to changes in fluid speed. The lift is computed by multiplying the pressure difference across the wing by the wing's surface area. Mathematically, it is given by:

• \( F_{\text{lift}} = \Delta P \times A \)

Here, \( \Delta P \) is the pressure difference, and \( A \) is the wing surface area. In our exercise, this calculation allows us to find the lift force generated by the wings, ultimately equal to the airplane's weight during level flight.

Pressure Difference

The pressure difference between the top and bottom surfaces of the wing is essential in creating lift. Bernoulli's Principle explains that an increase in airspeed results in a decrease in pressure. The wings of an airplane are designed so that air moves faster over the top surface compared to the bottom. This creates a lower pressure on the top, producing lift. To calculate the pressure difference, we apply the formula derived from Bernoulli's Principle:

• \( \Delta P = \frac{1}{2} \rho (v_{\text{top}}^2 - v_{\text{bottom}}^2) \)

Substituting the airspeed values into this formula helps us find the pressure difference, which in turn, is used to compute the lift force.

Wing Surface Area

The wing surface area plays a vital role in lift generation. It represents the actual area over which the pressure difference acts to generate lift. The larger the wing area, the more air can be affected by the pressure difference, resulting in a greater lift force. This is why airplanes generally have large wings. In our exercise, the wing surface area is given as \(16 \text{ m}^2\). This value is crucial, as a larger or smaller area would significantly affect the lift calculation, showcasing the importance of optimally designed wing sizes for balance between lift and drag.

Density of Air

Density of air is a key factor in calculating both pressure difference and lift. It represents the mass of air per unit volume and can vary with altitude and temperature. For the purpose of our exercise, the standard air density is used, which is approximately \(1.225 \text{ kg/m}^3\). This standardization helps in understanding typical conditions at sea level.Using this density in calculations ensures consistency, but it's essential to remember that variations can significantly impact lift force. Pilots and engineers must account for these in operations, especially when flying at different altitudes or in varying weather conditions, to ensure safe and efficient flights.