An airplane is flying in a horizontal circle at a speed of 482 km⁄h. The wings of the plane are tilted at 38.2° with respect to the horizontal as shown in the figure below. Find the radius of the circle in which the plane is flying. Assume that the centripetal force is provided entirely by the lift force perpendicular to the wing surface.
To find the radius of the circle in which the plane is flying, we need to analyze the forces acting on the plane.
The centripetal force required to keep the airplane in a circular motion is provided entirely by the lift force generated by the tilted wings. The lift force acts perpendicular to the wing surface.
In this case, we can decompose the lift force into two components: one component perpendicular to the plane's wing surface, which provides the centripetal force, and another component parallel to the wing surface, which counters the force of gravity.
Let's call the perpendicular component of the lift force Fc (centripetal force) and the parallel component Fg (force of gravity).
By considering the forces acting on the airplane, we have the following relations:
Fc = Lift × sin(θ)
Fg = Lift × cos(θ)
where θ represents the angle of tilt of the wings, and Lift is the total lift force generated by the tilted wings.
Since the airplane is flying in a horizontal circle, Fc is equal to the centripetal force required for circular motion:
Fc = mv²/r
where m is the mass of the airplane, v is the speed of the airplane, and r is the radius of the circle.
Now, if we equate the two expressions for Fc, we have:
mv²/r = Lift × sin(θ)
Since we want to find the radius, we rearrange the equation as follows:
r = mv² / (Lift × sin(θ))
Finally, we substitute the given values:
v = 482 km/h = 482000 m/h
θ = 38.2°
Note that m and Lift are not provided in the question. We would need those values to calculate the radius of the circle the airplane is flying in.