Question

An airplane is flying at constant speed \(v\) in a horizontal circle of radius \(r\). The lift force on the wings due to the air is perpendicular to the wings. At what angle to the vertical must the wings be banked to fly in this circle?

Short answer

The wings must be banked at an angle \( \theta = \arctan\left(\frac{v^2}{rg}\right) \) to the vertical.

Step-by-step solution

Identify the Forces Acting on the Airplane

The main forces acting on the airplane are gravity, which pulls it downward, and lift, which acts perpendicular to the wings. Additionally, since the airplane is flying in a circle, there is a centripetal force required to maintain this circular motion.

Express the Centripetal Force

The necessary centripetal force to keep the airplane moving in a circle is provided by the horizontal component of the lift force. This centripetal force is given by the formula: \[ F_c = \frac{mv^2}{r} \] where \( m \) is the mass of the airplane, \( v \) is the speed, and \( r \) is the radius of the circle.

Decompose the Lift Force

The lift force \( L \) can be decomposed into two components: a vertical component \( L \cos(\theta) \) balancing the gravity \( mg \), and a horizontal component \( L \sin(\theta) \) providing the centripetal force. The angle \( \theta \) is the bank angle between the lift vector and the vertical.

Equate Vertical Forces

The vertical component of the lift must balance the gravitational force: \[ L \cos(\theta) = mg \] This ensures that the airplane does not climb or descend.

Equate Horizontal Forces

The horizontal component of the lift must provide the centripetal force: \[ L \sin(\theta) = \frac{mv^2}{r} \] This component ensures that the airplane maintains its circular path.

Calculate the Bank Angle

Divide the equation for the horizontal force by the equation for the vertical force to eliminate \( L \):\[ \frac{L \sin(\theta)}{L \cos(\theta)} = \frac{\frac{mv^2}{r}}{mg} \]This simplifies to:\[ \tan(\theta) = \frac{v^2}{rg} \]Solve for \( \theta \):\[ \theta = \arctan\left(\frac{v^2}{rg}\right) \]

Key concepts

Centripetal Force

When an airplane navigates a circular path, a special force, known as the centripetal force, plays a crucial role. This force is responsible for pulling the airplane towards the center of the circle, allowing it to maintain the curved trajectory without flying off in a straight line.

In the case of an airplane flying in a horizontal circle, the centripetal force is not directly applied by any individual component of the aircraft itself; instead, it is derived from the horizontal component of the lift force acting against the airplane's wings.

The formula for calculating the necessary centripetal force is given by: \[ F_c = \frac{mv^2}{r} \] Where:

• \( F_c \) is the centripetal force required.
• \( m \) is the mass of the airplane.
• \( v \) is the velocity or speed of the airplane.
• \( r \) is the radius of the circular path.

This equation clearly shows us that the force increases with higher speeds and tighter turns (smaller radius).

Lift Force

The lift force is central to flight, as it's what keeps the airplane suspended in the air. The wings' shape and the angle at which they cut through the air create this force. For an aircraft flying level in a horizontal circle, the lift force acts perpendicular to the wings.

To fly steadily without ascending or descending, this lift must counteract the force of gravity. In circular motion, lift can be broken into two components:

• Vertical Component: \(L \cos(\theta)\), which balances gravity, \(mg\).
• Horizontal Component: \(L \sin(\theta)\), which provides the centripetal force.

These components help maintain a stable, consistent flight path within the horizontal circular motion.

The interaction between these components ensures that the airplane doesn't lose altitude while navigating the turn.

Circular Motion

Circular motion refers to the path of an object that is moving in a circle. For an airplane, maintaining a circular path means that all its directional adjustments result in a curved trajectory instead of a straight line.

In physics, this type of motion requires a constant force directed towards the center of the circle, known as the centripetal force. The airplane continuously changes direction to keep the curve, using the lift's horizontal component as the centripetal force.

This type of motion is governed by the tight relationship between speed, the circle's radius, and the necessary centripetal force, making precise force balance crucial for stable flight within circular paths.

Angle of Bank

The angle of bank \( \theta \) is the angle between the lift force and the vertical plane. In order to fly in a circular path without altering altitude, an airplane must achieve the right bank angle, which ensures the correct balance between vertical lift that opposes gravity and the horizontal component that provides centripetal force.

To find the angle of bank, we use the following relationship derived from balancing horizontal and vertical forces:\[ \tan(\theta) = \frac{v^2}{rg} \] Here,

• \( \theta \) is the bank angle.
• \( v \) is the speed of the airplane.
• \( r \) is the radius of the circular path.
• \( g \) is the gravitational acceleration.

Using this formula allows us to calculate the precise angle necessary for maintaining a stable circular flight path at given speeds and radius. The angle of bank is crucial for a smooth and controlled circular motion.