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 5 mi at a speed of 440 mph and an altitude of 24,000 ft. From the safety information card, you know that the total length of the wingspan of the plane is 275 ft. From this information, estimate the banking angle of the plane relative to the horizontal.

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 5 mi at a speed of 440 mph and an altitude of 24,000 ft. From the safety information card, you know that the total length of the wingspan of the plane is 275 ft. From this information, estimate the banking angle of the plane relative to the horizontal.

Answer 1

tan theta = [MV^2/R]/[Mg] = V^2/(R*g)

Solve for the "bank" agle theta. It is the angle that the winds must be tilted from the vertical to allow a circling pattern.

You don't need the wingspan or the altitude.

I really hate these longwinded questions that make the probem seem harder than it really is.

Answer 2

To estimate the banking angle of the plane relative to the horizontal, we can use the concept of centripetal force and gravitational force.

1. Firstly, let's calculate the horizontal velocity of the plane:
The speed given is 440 mph. However, we need to convert it to feet per second (fps) since the altitude is given in feet.
440 mph = 440 * 5280 ft / 3600 s = 646.22 fps

2. Next, let's calculate the centripetal force acting on the plane:
The centripetal force is provided by the lift force acting on the wings due to the circular motion.
The radius of the circle is 5 miles, which equals 5 * 5280 ft = 26,400 ft.
The centripetal force (Fc) can be calculated using the formula:
Fc = m * v^2 / r, where m is the mass of the plane (not given), v is the velocity, and r is the radius.
However, since we are only looking to estimate the banking angle, we can assume that the mass cancels out, and the force depends on velocity and radius alone.

3. Now, let's calculate the weight force acting on the plane:
The weight force (Fw) is equal to the mass (m) of the plane multiplied by the acceleration due to gravity (g).
The altitude given is 24,000 ft, so we need to convert it to miles before calculating the weight force.
24,000 ft = 24,000 / 5280 miles = 4.55 miles.
The weight force can be calculated using the formula:
Fw = m * g, where g is approximately 32.2 ft/s^2.

4. Lastly, let's find the banking angle (θ):
The banking angle is the angle at which the plane is tilted relative to the horizontal plane.
The banking angle can be calculated using the formula:
tan(θ) = Fc / Fw

Let's plug in the values we have calculated:
Fc = (646.22 fps)^2 / 26,400 ft
Fw = m * 32.2 ft/s^2

However, since we don't know the mass of the plane, we cannot calculate the actual value of θ. We can estimate the banking angle by assuming that the lift force is equal to the weight force in the circular motion:

tan(θ) = 1
θ ≈ 45 degrees

Therefore, based on the given information and the assumption of balanced forces, we can estimate the banking angle of the plane to be approximately 45 degrees relative to the horizontal.