Problem 51 - College Physics

Question

Lift on an airplane. Air streams horizontally past a small airplane's wings such that the speed is 70.0 \(\mathrm{m} / \mathrm{s}\) over the top surface and 60.0 \(\mathrm{m} / \mathrm{s}\) past the bottom surface. If the plane has a mass of 1340 \(\mathrm{kg}\) and a wing area of \(16.2 \mathrm{m}^{2},\) what is the net vertical force (including the effects of gravity) on the airplane? The density of the air is 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) .

Step-by-Step Solution

Answer

The net vertical force on the airplane is -10604.8 N (downward).

Step 1: Calculate Dynamic Pressure Difference

According to Bernoulli's Principle, the dynamic pressure difference between the top and bottom surfaces of the wing can be calculated using:\[\Delta P = \frac{1}{2} \rho \left(v_{\text{top}}^2 - v_{\text{bottom}}^2\right)\]Substituting known values: \(\rho = 1.20 \, \text{kg/m}^3\), \(v_{\text{top}} = 70.0 \, \text{m/s}\), \(v_{\text{bottom}} = 60.0 \, \text{m/s}\), the pressure difference is:\[\Delta P = \frac{1}{2} \times 1.20 \times (70.0^2 - 60.0^2) = 156 \, \text{N/m}^2\]

Step 2: Calculate Lift Force

The lift force on the wing is equal to the dynamic pressure difference times the wing area:\[F_{\text{lift}} = \Delta P \times A\]Where \(A = 16.2 \, \text{m}^2\). Therefore:\[ F_{\text{lift}} = 156 \times 16.2 = 2527.2 \, \text{N} \]

Step 3: Calculate Gravitational Force

The weight of the airplane, which acts downward, is given by:\[F_{\text{gravity}} = m \times g\]Where \(m = 1340 \, \text{kg}\) and \(g = 9.8 \, \text{m/s}^2\). This gives:\[F_{\text{gravity}} = 1340 \times 9.8 = 13132 \, \text{N}\]

Step 4: Calculate Net Vertical Force

The net vertical force on the airplane is the difference between the lift force and the gravity force:\[F_{\text{net}} = F_{\text{lift}} - F_{\text{gravity}}\]Substituting the calculated values:\[F_{\text{net}} = 2527.2 - 13132 = -10604.8 \, \text{N}\]This indicates that the net force is downward.

Key Concepts

Lift Force Calculation

To understand the concept of lift force, we need to delve into how it is generated on an airplane's wings. Lift is primarily created by the pressure difference between two surfaces of the wings, as explained by Bernoulli's Principle. The speed of the air above and below the wings is different, typically faster over the top and slower underneath. This speed difference results in a lower pressure on the top surface relative to the bottom. Bernoulli's Principle quantitatively describes this, stating that faster moving fluid leads to lower pressure.

The lift force can be calculated using the dynamic pressure difference and the wing area, with the formula:

• Dynamic pressure difference, \( \Delta P = \frac{1}{2} \rho \left(v_{\text{top}}^2 - v_{\text{bottom}}^2\right) \)
• Lift force, \( F_{\text{lift}} = \Delta P \times A \)

The area "A" refers to the total area of the wings that is exposed to the airflow, and the lift force is directed upward, opposing the force of gravity.

Dynamic Pressure Difference

The dynamic pressure difference is a fundamental concept in understanding how lift is generated on airplane wings. It arises due to the different speeds at which air flows over and under the wings. According to Bernoulli's Principle, a higher airspeed results in lower pressure. Thus, with the airplane wings, the higher airspeed over the top surface of the wing results in a lower pressure compared to the bottom surface.

For instance, if the speed is 70 m/s over the top surface and 60 m/s over the bottom, the pressure difference can influence how much lift is generated. This difference is calculated by:

• Substituting values into \( \Delta P = \frac{1}{2} \rho (v_{\text{top}}^2 - v_{\text{bottom}}^2) \)
• \[ \Delta P = \frac{1}{2} \times 1.20 \times (70.0^2 - 60.0^2) = 156 \text{ N/m}^2 \]

This pressure difference directly impacts how effectively the plane can overcome the gravitational pull and achieve lift.

Gravitational Force on Airplane

Gravitational force is the natural force that pulls the airplane towards the Earth, opposite to the lift force. It is determined by the mass of the airplane and the acceleration due to gravity. On Earth, this acceleration is approximately 9.8 m/s².

The gravitational force exerted on the airplane, often referred to as the plane's weight, can be calculated using the equation:

• \(F_{\text{gravity}} = m \times g\)
• \[ F_{\text{gravity}} = 1340 \times 9.8 = 13132 \text{ N} \]

This force acts downward, and for an airplane to ascend or maintain its altitude, the lift force must counterbalance the gravitational force. When these two forces are in equilibrium, a plane can maintain level flight. Understanding the interplay between lift and gravity is essential for controlling the aircraft.