Question

You are flying to Chicago for a weekend away from the books. In your last physics class, you learned that the airflow over the wings of the plane creates a lift force, which acts perpendicular to the wings. When the plane is flying level, the upward lift force exactly balances the downward weight force. Since O'Hare is one of the busiest airports in the world, you are not surprised when the captain announces that the flight is in a holding pattern due to the heavy traffic. He informs the passengers that the plane will be flying in a circle of radius \(7.00 \mathrm{mi}\) at a speed of \(360 . \mathrm{mph}\) and an altitude of \(2.00 \cdot 10^{4} \mathrm{ft}\). From the safety information card, you know that the total length of the wingspan of the plane is \(275 \mathrm{ft}\). From this information, estimate the banking angle of the plane relative to the horizontal.

Short answer

Answer: The banking angle of the plane relative to the horizontal is approximately 13.98°.

Step-by-step solution

Convert speed and radius to SI units

First, we'll convert the speed from miles per hour (mph) to meters per second (m\s) and the radius from miles to meters. conversion factors: 1 mile = 1609.34 m 1 hour = 3600 seconds Speed in \(\frac{\text{m}}{\text{s}}\): \(v = 360 \frac{\text{mi}}{\text{h}} \times \frac{1609.34 \text{m}}{1 \text{mi}} \times \frac{1 \text{h}}{3600 \text{s}} = 161.0 \ \frac{\text{m}}{\text{s}}\) Radius in meters: \(r = 7.00 \ \text{mi} \times \frac{1609.34 \text{m}}{1 \text{mi}} = 11265.38 \ \text{m}\)

Calculate the tangent of the banking angle

Now, we'll calculate the tangent of the banking angle using the formula \(\displaystyle\tan \theta = \frac{v^2}{g \cdot r}\). \(\displaystyle\tan \theta = \frac{(161.0 \ \text{m/s})^2}{(9.81 \ \frac{\text{m}}{\text{s}^2})(11265.38 \ \text{m})} = 0.2490\)

Calculate the banking angle

Finally, we'll calculate the banking angle by taking the arctangent of the tangent value obtained in step 2. \(\theta = \arctan(0.2490) = 13.98^\circ\) The banking angle of the plane relative to the horizontal is approximately \(13.98^\circ\).

Key concepts

Lift Force

When considering the principles of flight, lift force is a crucial concept. It's the upward acting force that opposes the downward pull of gravity, allowing airplanes to fly. This force is generated as air flows over the wings of an aircraft, with variations in air pressure above and below the wings according to Bernoulli's principle.

In a situation where a plane is flying level, as mentioned in the exercise, the lift force is balanced perfectly with the gravitational force acting on the plane. This equilibrium keeps the aircraft in steady horizontal flight. However, when an airplane enters a turn, the lift force must be adjusted to maintain circular motion, which introduces us to the next key concept.

Circular Motion

Circular motion refers to the movement of an object along a circular path. When a plane flies in a circular pattern or is in a so-called holding pattern as described in the exercise, it must maintain a constant inward force to stay on that path. This inward force is known as 'centripetal force' and is necessary for executing the turn.

The plane achieves circular motion by banking its wings, which tilts the lift vector and provides the necessary centripetal force. The centripetal force in the context of circular motion is exactly what leads us to understand tangential speed and centripetal acceleration.

Tangential Speed

Tangential speed, or the speed of an object moving along a circular path, is crucial when examining circular motion. It's the speed at any given point along the circle and is always directed tangent to the path, hence the name. In the provided exercise, the plane's tangential speed would be its velocity while following the circular holding pattern above the airport.

Tangential speed is different from angular speed, which measures how fast an object rotates or revolves relative to another point. In this case, we are interested in the tangential speed since it directly relates to the force needed to maintain the circular path, which in turn influences our next concept – centripetal acceleration.

Centripetal Acceleration

Centripetal acceleration is the acceleration of an object moving in a circle at constant tangential speed. This acceleration is always directed inward towards the center of the circle. It's not a force but rather describes how quickly the velocity of an object is changing in direction, not speed, as it moves along a curved path.

For objects in circular motion such as the airplane in a holding pattern mentioned in the exercise, centripetal acceleration is provided by the horizontal component of the lift force when the plane banks. The formula for centripetal acceleration, necessary to calculate the banking angle, is given by \( a_c = \frac{v^2}{r} \) where \( v \) is the tangential speed, and \( r \) is the radius of the circle. This concept is essential in our understanding of how and why the airplane banks at a specific angle to maintain its circular path.