When a plane turns, it banks as shown in the figure to give the lifting force of the wings a horizontal?

4 Answers

• Fw is the aerodynamic lift force acting on the wings, perpendicular to their surface

  Fg is the gravitational force from the Earth.

  Newton’s 2nd law in the vertical direction:

  Fw*cos(theta) = Fg

  Newton’s 2nd law in the horizontal direction:

  Fw*sin(theta) = m*a

  Recall the origin of Fg, Fg = m*g:

  Fw*cos(theta) = m*g

  solve for Fw:

  Fw = m*g/cos(theta)

  Substitute:

  m*g*sin(theta)/cos(theta) = m*a

  Cancel m’s and combine trigonometry:

  g*tan(theta) = a

  acceleration is due to centripetal acceleration:

  a = v^2/r

  Substitute:

  g*tan(theta) = v^2/r

  Solve for r:

  r = v^2/(g*tan(theta))

  Data:

  v:=266.67 m/s; g:=9.8 N/kg; theta:=35 deg;

  Result:

  r = 10363 meters

  r = 10.363 km

• Plane Banking

• At 45 deg bank, the lateral forces on the airplane are Fg = mgsin(45 d) and Fcent = mv^2/r*cos(45 d). To avoid sideslip, these forces must be equal to keep the net force on the plane perpendicular to the wing. Note sin(45 d) = cos(45 d). Thus Fg = Fcent ==> r = v^2/g = (940/3.6)^2/9.8 = 6957 m.

• Acceleration v^2/r for the circular movement must be provided by the lifting force horizontal component. Hence v^2/r = F/m* sin(theta).

  Now balance the vertical components : F*cos(theta) = m*g

  So v^2/r = g*tan(theta) If you want theta < 35 deg then r > v^2/(g*tan(35)) = 10352 m