Final answer:
The banking angle of the airplane's wings from the horizontal can be calculated using the speed of the plane, the radius of the circle it is moving in, and the acceleration due to gravity. By equating the horizontal component of lift to the centripetal force needed and the vertical component of lift to the weight of the plane, the ratio gives the tangent of the banking angle. Applying the values to the tangent formula allows for solving the angle using the arctangent.
Step-by-step explanation:
To determine the angle at which the plane's wings are banked from the horizontal as it circles the airport, we can use principles of physics to equate the centripetal force necessary for circular motion to the horizontal component of the lift force provided by the wings. The lift force L acts perpendicular to the wings, and components of this force can be broken down into vertical Lv and horizontal Lh components. The vertical component Lv balances the weight of the plane W, and the horizontal component Lh provides the centripetal force required to keep the plane moving in a circle of radius r.
We can write this as:
For vertical components: Lv = W = mg, where m is the mass of the airplane and g is the acceleration due to gravity.
For horizontal components: Lh = mv2/r, where v is the speed of the airplane and r is the radius of the circle.
The angle of banking θ can then be found by taking the ratio of the horizontal to vertical components of the lift force:
tan(θ) = Lh/Lv
Thus, tan(θ) = mv2/r/mg = v2/rg
Plugging in the values v = 185 m/s, r = 1.99 × 104 m, and g = 9.8 m/s2, we can solve for θ:
tan(θ) = (185 m/s)2/(1.99 × 104 m × 9.8 m/s2)
θ = atan((1852)/(1.99 × 104 × 9.8))
The banking angle θ can then be calculated using a calculator to find the arctangent.