3. The SS Eureka is a small ferryboat which may be modelled as a cuboid of length , width (beam) b and height h (in metres). The draught d of the boat is the submerged depth of the boat; that is, the height from the bottom (the keel) to the water line. The mass of the boat is M kg and the density of water is pkg/m3. (a) For a draught d calculate the weight of water displaced by the boat. (b) Archimedes' Principle states that the buoyant force acting on the boat is equal to the weight of the water it displaces. Draw a force diagram to justify why the draught d(t) must satisfy the equation d + Pod = g M (1) where o = bl, g is the gravitational constant, and the time is t seconds. What is the stable draught do of the boat for which the boat is stationary? (c) Find the general solution to (1). d) A group of partygoers begin to dance on the boat. Their dancing imparts a force upon the boat equal to f (t) = F cos It. Amend Equation (1) accordingly and show that the solution for the initial conditions d(0) = do and d(0) = 0 is given by d(t) = do + M (w2 - $2 ) ( cos Sit - coswt) . Here w = vopg/M and we shall assume that w # 2. (e) We now introduce numerical values for our variables 1 = 10 b =4 h =1 g ~ 10 p ~ 1000 M = 3373 F = 100 (N) The frequency of the dancing is given by ? = 27BPM/60 where BPM are the beats per minute of the song they are listening to. We shall assume that the boat sinks if at any point it is entirely submerged under water. Which of the following songs will sink the boat? i. "Rock the Boat", The Hues Corporation, 93BPM ii. "Sailing", Rod Stewart, 64BPM iii. "Stayin' Alive", the Bee Gees, 104BPM iv. "Yellow Submarine", the Beatles, 111BPM v. "Beyond the Sea", Charles Trenet, 67BPM vi. "I Will Go Sailing No More", Randy Newman, 53BPM