A plug inthe shape of the frustum of a cone (f.e.. the part of a truncated cone which includes the base) seals a 32 em dameter circular hole at the bottom of a tank holding fresh water. The top dameter of the plug 8 56 cm, and the top surface of the plug1s 15 cm above the base of the tank. The horzontal distance between the outer edge of the top surface of the plug and the circle of contact between the plug and the tank is 12 em. The depth of water in the tank 1sh = 0.5 m, The overall thickness of the plug 5 20 m and the pressure below the plug, external to the tank, is atmospheric pressure. See Figure 1. SDem A2 em Figure 1. Plug sealing a circular hole in the bottom of a tank holding water, Complete the following: {a) In no more than two sentences, clearly explain why the buoyaney foree arises [2 marks) (k) Determine the resultant force acting on the plug due only to the water [9 marks] {c) What depth of water in the tank (h)will result in no resultant water foree on the plug? [4 marks] (d) For this question, the depth of water in the tank is again h=0.5 m, If the plug is made of al um inum with a density of p,; = 2800 kg/m? and the volume of the plug (a frustrum) is given and is Volug = 2647 x 10 m?, what is the force per meter (N/m) around the rim of the 32 cm diameter hole exerted by the plug? [3 marks] [Hirts: Assumed knowledge the theorem of Papus states that the volume of a body of revolution 15 equal to the area of the 2D plan cross-section bemng rotated times the distance travelled by the centrowd of the 2D shape as it 15 rotated through one revolution. For example, the volume of a doughnut with circular cross-section of radius r and radius of the doughnut being B from the center of rotation to the centrod of the circle being rotated 8 W= (rr?)(2mR). When evaluating the force of the water on the plug, the water pressure acts at right angles to the surfaces of the plug whenever the water 18 in contact with the plug. Careful consideration of this will guide you as how you may best approach this problem. |