A sailboat is equipped with a fin underneath the hull and a

A sailboat is equipped with a fin underneath the hull and a large sail. While heading across the wind, the propelling force is given by the lift of the sail, while the fin acts as a stabilizer. The fin and sail can be modeled as thin, symmetric airfoils with planform areas. S_f and S_s respectively. The angle of attack of the sail is a, and the angle of attack of the fin is beta, where 1 >> alpha >> beta > 0. By considering the force balance diagram given below, calculate (a) the angle beta and (b) the velocity of the boat, v_w. Use the following parameters: rho_alpha = 1.2 kg/m^3, rho_w = 1025 kg/m^3, v_alpha = 10 m/s, alpha = 15 degree, C_Dh = 0.05, c_s = 20 m, C_h = lm, c_f= 0.75 m, b_s = 50 m, b_f = Sm, and d = 3 m Where C_Dh is the drag coefficient of the hull, the c\'s are the respective effective chord lengths, the b\'s are the spans, d is the depth of the hull below the waterline, and the subscripts a, w, s, f and h correspond to air, water, sail, fin, and hull respectively. You may assume that all the forces are acting in the same horizontal plane, and that the drag forces from the sail and fin are negligible.

Solution

Velocity of boat is the difference between wind velocity relative to water and velocity of wind relative to boat