Figure P4.15 shows the cross-sectional view of a ship undergoing rolling motion. Archimedes’ principle states that the buoyancy force B acting on a floating object equals the weight of the liquid...

Figure P4.15 shows the cross-sectional view of a ship undergoing rolling motion. Archimedes’ principle states that the buoyancy force B acting on a floating object equals the weight of the liquid displaced by the object. The metacenter M is the intersection point of the line of action of the buoyancy force and the ship’s centerline. The distance h of M from the mass center G is called the metacentric height. (a) Obtain the equation of motion describing the ship’s rolling motion in terms of the angle θ. (b) The given parameters are the ship’s weight W, its metacentric height h, and its moment of inertia I about the center of gravity. Obtain an expression for the natural frequency of the rolling motion.