Bhivarabai Sawant College of Engineering & Research Narhe, Pune-41Buoyancy and Floatation Prof. A. K. Mahale 1 INTRODUCTION In this chapter, the equilibrium of the floating and sub-merged bodies will be considered. Thus the chapter will include : 1. 2. 3. 4. 5. 6. 7. Buoyancy, Centre of buoyancy, Metacentre. Metacentric height, Analytical method for determining metacentric height, Conditions of equilibrium of a floating and sub-merged body, and Experimental method for metacentric height. 2 3 . the centre of buoyancy will be the centre of gravity of the fluid displaced. through which the force of buoyancy is supposed to act. an upward force is exerted by the fluid on the body.BUOYANCY When a body is immersed in a fluid. This upward force is equal to the weight of the fluid displaced by the body and is called the force of buoyancy or simply buoyancy CENTRE OF BUOYANCY It is defined as the point. As the force of buoyancy is a vertical force and is equal to the weight of the fluid displaced by the body. 0 m. when it floats horizontally in water. The density of wooden block is 650 kg/m3 and its length 6.Example 1 Find the volume of the water displaced and position of centre of buoyancy for a wooden block of width 2.5 m and of depth 1.5 m. 4 . Example 1 .solution 5 . Example 2 A stone weighs 392.2 N in water. Compute the volume of stone and its specific gravity 6 .4 N in air and 196. gr. 13.Example 3 Find the density of a metallic body which floats at the interface of mercury of sp. 7 .6 and water such that 40% of its volume is sub-merged in mercury and 60% in water. solution 8 .Example 3 . Example 3 .solution 9 . The length of OA is 20 cm. 0. gr.8 into a cistern.Example 4 Afloat valve regulates the flow of oil of sp. and a valve at the other end which closes the pipe through which oil flows into the cistern. and the distance between the centre of the float and the hinge is 50 cm. It was observed that the flow of oil is stopped when the free surface of oil in the cistern is 35 cm below the 10 . The spherical float is 15 cm in diameter.81 N to completely stop the flow of oil into the cistern. AOB is a weightless link carrying the float at one end. The link is mounted in a frictionless hinge at 0 and the angle AOB is 135°. When the flow is stopped AO will be vertical. The valve is to be pressed on to the seat with a force of 9. Example 4.solution 11 . Example 4 .solution 12 . 13 . Consider a body floating in a liquid as shown in Fig. Let the body is in equilibrium and G is the centre of gravity and B the centre of buoyancy. both the points lie on the normal axis.META-CENTRE It is defined as the point about which a body starts oscillating when the body is tilted by a small angle. which is vertical. For equilibrium. (a). The meta-centre may also be defined as the point at which the line of action of the force of buoyancy will meet the normal axis of the body when the body is given a small angular displacement. ANALYTICAL METHOD FOR META-CENTRE HEIGHT Fig. This is shown in Fig. (b).META-CENTRIC HEIGHT The distance MG. The location of centre of gravity and centre of buoyancy in this position is at G and B. i. The new centre of buoyancy is at 5). (a) shows the position of a floating body in equilibrium. Hence M is the meta-centre and GM is meta-centric height. the distance between the meta-centre of a floating body and the centre of gravity of the body is called meta-centric height. The vertical line through B1 cuts the normal axis at M.. The floating body is given a small angular displacement in the clockwise direction.e. 14 . ANALYTICAL METHOD FOR META-CENTRE HEIGHT 15 . 20 m high. 3 m wide and 1.Example 5 A rectangular pontoon is 5 m long. If the centre of gravity is 0. The density of sea water = 7025 kg/m3. 16 .6 m above the bottom of the pontoon. The depth of immersion of the pontoon is 0. determine the meta-centric height.80 m in sea water. solution 17 .Example 5 . Determine the metacenter height of the block if its size is 2 m x 1 m x 0.7 floats in water.Example 6 A block of wood of specific gravity 0.8 m. 18 . solution 19 .Example 6 . Example 6 .solution 20 . CONDITIONS OF EQUILIBRIUM OF A FLOATING AND SUBMERGED BODIES Stability of a Sub-merged Body 21 . If FB = W and B and G are at the same point. (b). the body is said to be in Neutral Equilibrium. gives the couple due to W and FB also in the clockwise direction. (b) Unstable Equilibrium. the body is in unstable equilibrium as shown in Fig. but the centre of buoyancy (B) is below centre of gravity (G). If W = FB. (c). Thus the body does not return to its original position and hence the body is in unstable equilibrium. as shown in Fig.CONDITIONS OF EQUILIBRIUM OF A FLOATING AND SUBMERGED BODIES Stability of a Sub-merged Body (a) Stable Equilibrium. in the clockwise direction. A slight displacement to the body. the body is said to be in stable equilibrium. (c) Neutral Equilibrium. 22 . When W = FB and point B is above G. CONDITIONS OF EQUILIBRIUM OF A FLOATING AND SUBMERGED BODIES Stability of Floating Body. 23 . (c) Neutral Equilibrium.7 CONDITIONS OF EQUILIBRIUM OF A FLOATING AND SUB-MERGED BODIES 4. If the point M is at the centre of gravity of the body. If the point M is above G. If the point M is below G.7. (b) Unstable Equilibrium.4. the floating body will be in neutral equilibrium. 24 . If . . .1 Stability of a Sub-merged Body (a) Stable Equilibrium. 6 and it is floating in water with its axis vertical.Example 7 A solid cylinder of diameter 4.0 m has a height of 4. State whether the equilibrium is stable or unstable.0 m. h = 4 m 25 . D = 4m . Find the metacentric height of the cylinder if the specific gravity of the material of cylinder = 0. Example 7 .solution 26 . 27 .