An airplane, approching an airport for landing, had been asked to wait until the runway is clear. The plane flies a "holding pattern," which basically means it will fly in a circle at a constant height and speed. Suppose the 4.900×104 kg airplane has a speed of 405 km/h and needs to fly in a circle with a radius of 6.00 km. To turn, it will bank (tilt) its wings at some angle; this makes the aerodynamic lift force [(L)\vec] point up and to the left as shown below. What does the banking angle need to be for the plane to travel in the circle?
Question
An airplane, approching an airport for landing, had been asked to wait until the runway is clear. The plane flies a "holding pattern," which basically means it will fly in a circle at a constant height and speed. Suppose the 4.900×104 kg airplane has a speed of 405 km/h and needs to fly in a circle with a radius of 6.00 km. To turn, it will bank (tilt) its wings at some angle; this makes the aerodynamic lift force [(L)\vec] point up and to the left as shown below. What does the banking angle need to be for the plane to travel in the circle?
Solution
To solve this problem, we need to use the principles of circular motion and the forces acting on the airplane.
-
First, convert the speed of the airplane from km/h to m/s because the other units in the problem are in the metric system.
Speed = 405 km/h = 405,000 m/3600 s = 112.5 m/s
-
The force that keeps the airplane moving in a circle is the horizontal component of the lift force. This force can be calculated using the formula for the centripetal force, which is mass times velocity squared divided by the radius of the circle.
F = mv²/r = (4.9 x 10^4 kg)(112.5 m/s)² / (6.00 x 10^3 m) = 9.41 x 10^5 N
-
The lift force is the result of two components: the weight of the airplane acting downwards and the centripetal force acting towards the center of the circle. The lift force can be calculated using Pythagoras' theorem.
L = sqrt(F² + W²) = sqrt((9.41 x 10^5 N)² + (4.9 x 10^4 kg x 9.8 m/s²)²) = 1.03 x 10^6 N
-
The banking angle can be found by taking the inverse tangent of the ratio of the horizontal and vertical components of the lift force.
tan(θ) = F/W = (9.41 x 10^5 N) / (4.9 x 10^4 kg x 9.8 m/s²)
θ = atan(tan(θ)) = 43.6 degrees
So, the airplane needs to bank its wings at an angle of approximately 43.6 degrees to maintain its circular path.
Similar Questions
A train runs along an unbanked circular bend of radius 30 m at a speed of 54 kmh–1. What is the angle of banking required to prevent wearing out of the rail ? (g = 10 ms–2)A 53° B 45° C 60° D 37°
An airplane is undergoing horizontal acceleration of 5m/s2. A block weighing 2kg is suspended from the airplane, initially at rest relative to the airplane. What is the angle θ that the string makes with the horizontal.
The "leans" occurs when a slight change in bank angle goes unnoticed by the pilot, resulting in a perceived angle of bank when the aircraft is rolled back to wings level. This is because*1 pointSufficient visual cues are not presentThe original bank angle goes is small enough that it goes undetected by the semicircular canals in the inner earThe pilot cannot "feel" the angle of bankThere isn't a suitable horizon
A model airplane of mass 0.800 kg flies with a speed of 35.0 m/s in a horizontal circle at the end of a 62.0-m control wire as shown in Figure (a). The forces exerted on the airplane are shown in Figure (b): the tension in the control wire, the gravitational force, and aerodynamic lift that acts at 𝜃 = 20.0° inward from the vertical. Compute the tension in the wire, assuming it makes a constant angle of 𝜃 = 20.0° with the horizontal. N
An aircraft execute a horizontal loop at a speed of 720 km h–1, with its wings banked at 15°. The radius of the loop will be [Given: tan15° = 0.268]A 5.71 km B 10.41 km C 15.23 km D 20.63 km
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.