A model airplane of mass 0.800 kg flies with a speed of 35.0 m/s in a horizontal circle at the end of a 62.0-m control wire as shown in Figure (a). The forces exerted on the airplane are shown in Figure (b): the tension in the control wire, the gravitational force, and aerodynamic lift that acts at 𝜃 = 20.0° inward from the vertical. Compute the tension in the wire, assuming it makes a constant angle of 𝜃 = 20.0° with the horizontal. N

To solve this problem, we need to use the principles of circular motion and the forces acting on the airplane.

Step 1: Identify the forces acting on the airplane There are three forces acting on the airplane: the tension in the wire (T), the gravitational force (mg), and the aerodynamic lift (L). The tension in the wire and the aerodynamic lift both have components that contribute to the centripetal force that keeps the airplane moving in a circle.

Step 2: Set up the equations for the vertical and horizontal forces The vertical forces must balance out, as the airplane is not moving up or down. This gives us the equation: Lcos(θ) + Tsin(θ) = mg. The horizontal force provides the centripetal force for the circular motion, giving us the equation: Lsin(θ) = Tcos(θ) + mv²/r.

Step 3: Solve the equations We can solve these equations simultaneously to find the tension in the wire. However, we first need to find the lift force (L). We can rearrange the second equation to find L = (Tcos(θ) + mv²/r) / sin(θ).

Step 4: Substitute the lift force into the first equation Substituting the expression for L from step 3 into the first equation gives us: ((Tcos(θ) + mv²/r) / sin(θ))cos(θ) + Tsin(θ) = mg.

Step 5: Solve for T This equation can be rearranged and solved to find the tension in the wire (T).

Note: The actual numerical solution will depend on the specific values of the mass (m), speed (v), radius of the circle (r), and angle (θ). In this case, m = 0.800 kg, v = 35.0 m/s, r = 62.0 m, and θ = 20.0°.