An airplane is flying with a constant speed of 275 m/s along a horizontal circle with a radius of 1.23 x 104 m. If the lift force of the air on the wings is perpendicular to the wings, at what angle, relative to the horizontal, should the wings be banked?

Question

An airplane is flying with a constant speed of 275 m/s along a horizontal circle with a radius of 1.23 x 104 m. If the lift force of the air on the wings is perpendicular to the wings, at what angle, relative to the horizontal, should the wings be banked?

Answer 1

centripetal force: mv^2/r

gravity force: mg

angle banked: arctan v^2/rg

Answer 2

To find the angle at which the wings should be banked, we need to consider the forces acting on the airplane when it is in circular motion.

First, we can start by drawing a free-body diagram of the airplane at the banked angle. There are three forces acting on the airplane: the gravitational force (mg) acting vertically downwards, the lift force (L) acting perpendicular to the wings, and the centrifugal force (Fc) acting towards the center of the circular path.

We can break down the gravitational force into two components: one perpendicular to the wings (mg * cosθ) and one parallel to the wings (mg * sinθ), where θ is the angle of banking.

The lift force can be decomposed into two components as well: one perpendicular to the wings (L * cosθ) and one parallel to the wings (L * sinθ).

The centripetal force is given by the equation Fc = (mv^2) / r, where m is the mass of the airplane, v is its velocity, and r is the radius of the circular path.

Since the airplane is flying at a constant speed and the lift force is perpendicular to the wings, we can equate the vertical components of the forces:

mg * cosθ = L * cosθ

Simplifying the equation:

mg = L

This means that the gravitational force is equal to the lift force, and there is no net vertical force acting on the airplane. This is necessary for the airplane to maintain level flight.

Next, we equate the horizontal components of the forces:

mg * sinθ = Fc

Substituting Fc = (mv^2) / r:

mg * sinθ = (m * v^2) / r

Simplifying the equation:

g * sinθ = (v^2) / r

Now, we can solve for the angle θ:

sinθ = (v^2) / (g * r)

θ = arcsin((v^2) / (g * r))

Plugging in the values given in the question:
v = 275 m/s
r = 1.23 x 10^4 m
g = 9.81 m/s^2

θ = arcsin((275^2) / (9.81 * (1.23 x 10^4)))

Using a calculator, we can find the value of θ to be approximately 49.5 degrees.

Therefore, the wings should be banked at an angle of approximately 49.5 degrees relative to the horizontal.