Centripetal forces with tilts/banks

Question

A 220000 kg plane flying at 130 m/s tilts 30º above the horizontal to fly in a circle. What is the radius of the circle in which the plane flies?

Answer 1

This is a centripetal force problem.  Assume the aerodynamic lift L is perpendicular to the wings.  If the wings are tilted by θ above the horizontal, the lift is tilted by θ from the vertical.  That makes L sinθ the horizontal component of lift, which actually points in the radial direction.  The vertical component is L cosθ, and it must equal the weight of the airplane.  Set the horizontal (i.e. radial) component of lift equal to the centripetal force.

 

mv²/r = L sinθ ⇒

 

r = mv² / (L sinθ)

 

L cosθ = mg ⇒

 

L = mg/cosθ

 

r = mv² / [(mg/cosθ)sinθ]

 

r = v² / (g tanθ)

 

You know v, g, and θ.  Plug in the numbers and get r.