(Solved): Consider an airplane flying in a circle of radius \( R \). The circle is horizontal, i.e. the plan ...

Consider an airplane flying in a circle of radius \( R \). The circle is horizontal, i.e. the plane of the circle is parallel to the ground, so as the airplane circles around, its height never changes. The airplane's speed, \( v \), is constant. According to fluid dynamics, air pressing against the bottom of the airplane's wings exerts a "lift" force \( F_{L} \). The direction of this lift force is always perpendicular to the wings. For instance, consider the diagram below, in which the airplane is seen from behind. The airplane is "banking" at an angle \( \theta \), and the lift force is perpendicular to the wings, so it points upward and a little bit to the left as shown. The airplane has mass \( M \). The airplane's engines exert a forward "thrust" force on the plane \( F_{t} \), while air resistance exerts a backwards "drag" force \( F_{d} \) ("forwards" and "backwards" being with respect to the airplane's direction of motion). For each of the questions below, keep an organized record of your work and attach it at the end. BACK VIEW (i.e. the airplane is flying "into the page") a. Is the thrust force \( F_{t} \) bigger, smaller, or the same magnitude as the drag force \( F_{d} \) ? Explain your answer. b. At what angle \( \theta \) must the airplane be banked in order to continue flying in its present circular path? Express your answer in terms of \( M, v \), and \( R \). c. What "lift" force \( F_{L} \) must the plane generate in order to continue flying in a horizontal circle? Express your answer in terms of \( M, v \), and \( D \) (as much as possible, express your answer as an algebraic expression). Make sure to attach your work for full credit on your answer and work.