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Chapter 6: Problem 51

An airplane is flying in a horizontal circle at a speed of \(480 \mathrm{~km} / \mathrm{h}\) (Fig. 6-41). If its wings are tilted at angle \(\theta=40^{\circ}\) to the horizontal, what is the radius of the circle in which the plane is flying? Assume that the required force is provided entirely by an "aerodynamic lift" that is perpendicular to the wing surface.

Short Answer

Expert verified
The radius of the circle is approximately 2161.35 meters.

Step by step solution

01

Convert Speed to Meters per Second

First, we need to convert the plane's speed from kilometers per hour to meters per second. Use the conversion factor: \[1 \text{ km/h} = \frac{1}{3.6} \text{ m/s}\]. Thus, the speed \(v = 480 \text{ km/h} \times \frac{1}{3.6} = 133.33 \text{ m/s}\).
02

Identify Relevant Forces and Components

The aerodynamic lift \(L\) acts perpendicular to the wings and provides the centripetal force needed for circular motion. Lift can be decomposed into a vertical component (\(L\cos\theta\)) balancing the plane鈥檚 weight and a horizontal component (\(L\sin\theta\)) providing the centripetal force.
03

Write Down Newton's Second Law for Circular Motion

For horizontal circular motion, the horizontal component of lift \(L\sin\theta\) provides the centripetal force \(mv^2/r\). Thus, \[L\sin\theta = \frac{mv^2}{r}\].
04

Relate Vertical Forces

The vertical component of lift \(L\cos\theta\) balances the weight \(mg\) of the airplane: \[L\cos\theta = mg\].
05

Solve for the Radius \(r\)

From Step 4, solve for \(L\): \[L = \frac{mg}{\cos\theta}\].Substitute \(L\) into the centripetal force equation from Step 3: \[\frac{mg}{\cos\theta} \cdot \sin\theta = \frac{mv^2}{r}\].Simplify and solve for \(r\): \[r = \frac{v^2}{g \cdot \tan\theta}\].
06

Calculate the Radius

Plug the known values into the equation \(r = \frac{v^2}{g \cdot \tan\theta}\), letting \(v = 133.33 \text{ m/s}\), \(g = 9.81 \text{ m/s}^2\), and \(\theta = 40^\circ\): \[r = \frac{(133.33)^2}{9.81 \cdot \tan 40^\circ}\]. Calculate \(\tan 40^\circ \approx 0.8391\), then finally find \[r \approx \frac{17777.78}{8.2269} \approx 2161.35 \text{ m}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Aerodynamic Lift
Aerodynamic lift is the force that keeps airplanes airborne, counteracting the force of gravity. It acts perpendicular to the wings and is crucial in guiding an aircraft's circular motion. This lift is generated by the difference in air pressure on the upper and lower surfaces of an airplane's wings. When an airplane's wings are tilted at an angle, such as 40 degrees in this context, the lift can be divided into components.
  • The vertical component helps balance the weight of the airplane.
  • The horizontal component provides the necessary centripetal force for circular motion.
Understanding how these components work together is essential for analyzing circular flight paths and calculating flight properties like radius and speed.
Newton's Second Law
Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration: \( F = ma \). In circular motion, this principle helps us determine the forces an object experiences. For an airplane flying in a horizontal circle, the centripetal force is crucial.
Newton's Second Law allows us to express the centripetal force required to maintain circular motion as: \( F_c = rac{mv^2}{r} \).
Here:
  • \( F_c \) is the centripetal force.
  • \( m \) is the mass of the airplane.
  • \( v \) is the velocity.
  • \( r \) is the radius of the circular path.
This equation shows the relationship between velocity, mass, and the circle's radius, which helps determine how much force is needed for an aircraft to sustain its flight path.
Circular Motion
Circular motion refers to the movement of an object along a circular path. In the context of an airplane, understanding circular motion involves recognizing the key forces in play.
The centripetal force is necessary to keep the airplane moving in its circular path.
This force must always point towards the center of the circle. For an airplane:
  • The aerodynamic lift's horizontal component provides the centripetal force.
  • A balance of forces is required to maintain stable flight.
In circular motion, the radius of the circle affects the speed and force requirements. Smaller circles require more centripetal force at higher speeds, making the understanding of these dynamics critical in piloting and designing aircraft for specific maneuvers.
Centripetal Force
Centripetal force is essential for any object moving in a circular path. For airplanes, this force keeps them from flying straight off their intended circular trajectory.
In our scenario, the horizontal component of aerodynamic lift provides this force. The magnitude of centripetal force depends on:
  • Speed of the airplane: More speed means more centripetal force is needed.
  • Mass of the airplane: Heavier planes require more force.
  • Radius of the circular path: Smaller radii need higher force amounts.
The relationship can be calculated using the equation: \( F_c = rac{mv^2}{r} \).
This helps to determine and design the flight path, ensuring the airplane stays safely in its circular motion. Understanding centripetal force is vital for both pilots and engineers when managing safe and efficient circular flights.

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Most popular questions from this chapter

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