An airplane is flying in a horizontal circle at a speed of 480 km / h. If its wings are tilted

An airplane is flying in a horizontal circle at a speed of...

An airplane is flying in a horizontal circle at a speed of $480 \mathrm{~km} / \mathrm{h}$. If its wings are tilted $40^{\circ}$ to the horizontal, what is the radius of the circle in which the plane is flying? (See Fig. 6-69.) Assume that the required force is provided entirely by an "aerodynamic lift" that is perpendicular to the wing surface.

Key Concepts

Force Decomposition

Force decomposition involves breaking a force vector into its horizontal and vertical components. In the context of a banking airplane, the lift force is resolved into a vertical component that counteracts gravity and a horizontal component that supplies the centripetal force required for the circular motion.

Banking Flight Dynamics

Banking flight dynamics refers to the manner in which an airplane tilts its wings during a turn so that a component of the lift force acts horizontally to provide the needed centripetal acceleration. This tilt reduces the effective vertical lift and requires balancing the vertical and horizontal force components to maintain level, controlled flight.

Centripetal Force

In circular motion, an object must experience a net inward force—called the centripetal force—to maintain its curved trajectory. This force is related to the object's mass, velocity, and the radius of the circle, and is expressed by the equation F = mv²/r.

Aerodynamic Lift

Aerodynamic lift is the upward force generated by an aircraft's wings, created by the flow of air over and under the wing surfaces. In maneuvers such as banking turns, this lift force is not entirely vertical but is instead oriented perpendicular to the wing, contributing both to lifting the aircraft and providing the necessary horizontal force for the turn.

Transcript

00:01 Okay, so in this problem, we've got a airplane that is flying in a big circle, and the airplane is banked at 40 degrees to the horizontal.

00:12 And so the lift that the airplane is going to feel, the force of the lift is going to be perpendicular to the angle of the wing.

00:22 So there's going to be a horizontal and a vertical component of that lift.

00:27 And the vertical component is going to keep it in the air.

00:31 The horizontal component keeps it flying the circle.

00:34 We're told that the speed of the airplane is flying is 480 kilometers per hour, which if we convert that, it becomes 133 .3 meters per second.

00:46 And now we need to analyze the forces that are acting on this.

00:50 So the forces in the vertical direction, since this is a horizontal circle, that there is no net force in the vertical direction.

01:01 So net force is zero.

01:03 We've got the weight force of the airplane.

01:07 And then we have the vertical component of the lift force.

01:12 So the weight force down and the vertical component up is going to be the lift force times the cosine of this angle 40 degrees.

01:23 So the cosine is going to give us the vertical component.

01:31 And so we can set these two equal to each other.

01:33 And what we really want is we want this to be solved for the lift force.

01:38 So if we solve this equation for the lift force, we get this equation.

01:43 Force lift is equal to the weight force divided by cosine of that angle...

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