# Chapter 4: Buoyancy & Stability

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1 Chapter 4: Buoyancy & Stability

2 Learning outcomes By the end of this lesson students should be able to: Understand the concept of buoyancy hence determine the buoyant force exerted by a fluid to a body fully submerged or floating. Determine whether a floating body is stable or not. Understand the metacentre and identify the metacentric height of a body fully or partially immersed. UiTMKS/ FCE/ BCBidaun/ ECW211 2

3 Introduction Buoyancy (upthrust): The vertical force exerted by the fluid on an immersed body. Upthrust on body = Weight of fluid displaced by fluid F b gv d the body W object gv Centre of buoyancy: The point which buoyancy is acting & the centroid of the displaced volume of fluid. o UiTMKS/ FCE/ BCBidaun/ ECW211 3

4 F b1 Fluid of density ρ 1 cb 1 V 1 cb 2 V2 Fluid of density ρ 2 F b2 Upthrust on upper part, Upthrust on lower part, F F b1 b2 Totalupthrust gv gv gv 1 2 acting through cb acting through gv cb 2 UiTMKS/ FCE/ BCBidaun/ ECW211 4

5 Procedure for solving buoyancy problems 1. Draw FBD. 2. Write vertical equilibrium equation, 3. Solve for the required by applying buoyancy principles: i. Buoyant force, F V ii. Weight of object, W Object V Object iii. W < R object floats at the surface of fluid iv. W > R object sinks in the fluid v. W = R floats at a certain depth of immersion vi. Depth of immersion, b fluid Vd x A displaced FV 0 UiTMKS/ FCE/ BCBidaun/ ECW211 5

6 Example 4.1 A stone weighs 665 N in the air. When the stone is completely submerged in water, it weighs 420 N. Calculate the volume of the stone. The unit weight of the stone is Nm -3. UiTMKS/ FCE/ BCBidaun/ ECW211 6

7 Example 4.2 A 60 mm cube is made of rigid foam material and floats in water with a depth of 40 mm below the water surface as shown in Fig. 4.2 (a). Determine the magnitude and direction of the force required to hold the cube completely submerged in glycerine, which has a specific gravity of W F b F e W F b 40 mm Water Glycerine UiTMKS/ FCE/ BCBidaun/ ECW211 7

8 Example 4.3 (Douglas, 2006) A rectangular pontoon has a width B of 6 m, a length l of 12 m, and a draught D of 1.5 m in fresh water (density 1000 kgm -3 ). Calculate (a) the weight of the pontoon, (b)its draught in sea water (density 1025 kgm -3 ) and (c) the load (in kn) that can be supported by the pontoon in fresh water if the maximum draught permissible is 2 m. UiTMKS/ FCE/ BCBidaun/ ECW211 8

9 Example 4.4 (Bansal, 2003) A wooden log of 0.6 m diameter and 5 m length is floating in river water. Find the depth of the wooden log in water when the specific gravity of the log is 0.7. h A B 2θ O D C 0.6 m UiTMKS/ FCE/ BCBidaun/ ECW211 9

10 Example 4.5 (Bansal, 2003) Find the density of a metallic body which floats at the interface of mercury of specific gravity 13.6 and water such that 40% of its volume is submerged in mercury and 60% in water. UiTMKS/ FCE/ BCBidaun/ ECW211 10

11 Stability The equilibrium of a body may be stable, unstable or neutral, depending upon whether, when given a small displacement, it tends to return to the equilibrium position, move further from it or remain in the displaced position. 2 cases: Fully immersed body Floating body (partially immersed) UiTMKS/ FCE/ BCBidaun/ ECW211 11

12 Stability of a fully immersed body Small angular displacement of θ from equilibrium position will generate a moment W x BG x θ G below B : Righting moment, return to its equilibrium position G above B: Overturning moment, body unstable UiTMKS/ FCE/ BCBidaun/ ECW211 12

13 UiTMKS/ FCE/ BCBidaun/ ECW211 13

14 Stability of floating body When body is displaced through an angle θ, the shape of volume changes and cb moves from B to B. Turning moment (W x θ) is produced : righting moment as in (b) or turning moment as in (c) UiTMKS/ FCE/ BCBidaun/ ECW211 14

15 Metacentre, M : point at which the line of action of the upthrust R cuts the original vertical through G. x GM Where θ is small, so that sin θ = tan θ = θ in radians. Metacentric height: the distance GM UiTMKS/ FCE/ BCBidaun/ ECW211 15

16 If M lies above G, a righting moment W x GM x θ is produced, equilibrium is stable and GM is regarded as positive. If M lies below G, an overturning moment W x GM x θ is produced, equilibrium is unstable and GM is regarded as negative. If M coincides with G, the body is in neutral equilibrium. UiTMKS/ FCE/ BCBidaun/ ECW211 16

17 UiTMKS/ FCE/ BCBidaun/ ECW211 17

18 Determination of the metacentric height The metacentric height of a vessel can be determined if the angle of tilt θ caused by moving a load P a known distance x across the deck is measured. Overturning moment due to movement of load P Px If GM is the metacentric height and W = mg is the total weight of the vessel including P, Righting moment W GM UiTMKS/ FCE/ BCBidaun/ ECW211 18

19 For equlibrium in the tilted position, the righting moment must equal the overturning moment so that, W GM Metacentric height, GM Px Px W UiTMKS/ FCE/ BCBidaun/ ECW211 19

20 Determination of the position of the metacentric relative to centre of buoyancy For a vessel of known shape and displacement, the position of the centre of buoyancy B is comparatively easily found and the position of M relative to B can be easily calculated. Moment due to movement of But, Therefore, F b ρgv BB' MB BB' MB Total moment due to altered displaceme nt ρgθi I V d UiTMKS/ FCE/ BCBidaun/ ECW211 20

21 Procedure for determining stability of floating bodies Determine the position of floating body using the buoyancy principles. Locate B/ cb and calculate the distance from the bottom of body to B/ cb y cb Locate G/ cg and calculate the distance from the bottom of body to G/ cg y cg UiTMKS/ FCE/ BCBidaun/ ECW211 21

22 Calculate second moment of area I of the plane cut by the water surface hence calculate MB/ GM. MB Calculate the distance from the bottom of the body to MB using I V d y mc y cb MB UiTMKS/ FCE/ BCBidaun/ ECW211 22

23 Example 4.6 A flatboat hull weighs 130 kn when fully loaded. Fig. 4.7 (a) and (b) show the front and side views of the boat. Determine whether the boat is stable in water and find out if the boat will float. The centre of gravity is at 0.7 m measured from the bottom of the boat. The length of the boat is 6.0 m, the width is 2.2 m and the height is 1.2 m. UiTMKS/ FCE/ BCBidaun/ ECW211 23

24 Example 4.7 A piece of cork has been cut to take the shape as shown in Fig. 4.8 and has a specific weight of 2.36 knm -3. The cork is then put into turpentine which has a specific gravity of 0.87 as shown in Fig. 4.8 (b). Determine whether the cork is stable. UiTMKS/ FCE/ BCBidaun/ ECW211 24

25 Example 4.8 A wooden cone floats in water in the position shown in Fig. 4.9 (a). Determine whether the cone is stable or not if the specific gravity of the wood is UiTMKS/ FCE/ BCBidaun/ ECW211 25

26 Example 4.9 (Douglas, 2006) A cylindrical buoy (Fig. 4.10) 1.8 m in diameter, 1.2 m high and weighing 10 kn floats in salt water of density 1025 kgm -3. Its centre of gravity is 0.45 m from the bottom. If a load of 2 kn is placed on the top, find the maximum height of centre of gravity of this load above the bottom if the buoy is to remain in stable equilibrium. UiTMKS/ FCE/ BCBidaun/ ECW211 26

27 Exercises UiTMKS/ FCE/ BCBidaun/ ECW211 27

28 Douglas 3.17 A buoy floating in sea water of density 1025 kgm -3 is conical in shape with a diameter across the top of 1.2 m and a vertex angle of Its mass is 300 kg and its centre of gravity is 750 mm from the vertex. A flashing beacon is to be fitted to the top of the buoy. If this unit has a mass of 55 kg, what is the maximum height of its centre of gravity above the top of the buoy if the whole assembly is not to be unstable? ( The centre of volume of a cone of height h is at a distance ¾ h from the vertex). (Ans: 1.25 m) UiTMKS/ FCE/ BCBidaun/ ECW211 28

29 Munson 2.98 A river barge, whose cross section is approximately rectangular, carries a load of grain. The barge is 8.5 m wide and 27.4 m long. When unloaded its draft ( depth of submergence) is 1.5 m, and with the load of grain the draft is 2.0 m. Determine: a. The unloaded weight of the barge b. The weight of the grain (Ans: 3.42 MN, MN) UiTMKS/ FCE/ BCBidaun/ ECW211 29

30 Munson When the Tucurui Dam was constructed in northern Brazil, the lake that was created covered a large forest of valuable hardwood trees. It was found that even after 15 years underwater the trees were perfectly preserved and underwater logging was started. During the logging process a tress is selected, trimmed, and anchored with ropes to prevent it from shooting to the surface like a missile when cut. Assume that a typical large tree can be approximated as a truncated cone with a base diameter of 2.4 m, a top diameter of 0.6 m, and a height of 30.0 m. determine the resultant vertical force that the ropes must resist when the completely submerged tree is cut. The specific gravity of the wood is approximately 0.6. (Ans:233 kn) UiTMKS/ FCE/ BCBidaun/ ECW211 30

31 Munson An inverted test tube partially filled with air floats in a plastic water-filled soft drink bottle as shown in VideoV2.7 and Fig. P The amount of air in the tube has been adjusted so that it just floats. The bottle cap is securely fastened. A slight squeezing of the plastic bottle will cause the test tube to sink to the bottom of the bottle. Explain this phenomenon. UiTMKS/ FCE/ BCBidaun/ ECW211 31

32 Munson An irregular shaped piece of a solid material weighs 36 N in air and 23 N when completely submerged in water. Determine the density of the material. (Ans: 2.77 x 10 3 kg/m 3 ) UiTMKS/ FCE/ BCBidaun/ ECW211 32

33 Munson A 1-m diameter cylindrical mass, M, is connected to a 2-m wide rectangular gate as shown in Fig. P The gate is to open when the water level, h, drops below 2.5 m. Determine the required value for M. Neglect friction at the gate hinge and the pulley. (Ans: 2480 kg) UiTMKS/ FCE/ BCBidaun/ ECW211 33

34 Suhaimi 4.17 A flashing beacon of mass 15 kg is fitted at the top of a 150 kg conical shaped buoy. The diameter across the top of the buoy is 1.0 m with a 50 0 vertex angle. Check the stability of the whole assembly if the center of gravity is 600 mm from the vertex and density of ocean water is 1025 kg/m 3. (Ans: Stable) UiTMKS/ FCE/ BCBidaun/ ECW211 34

35 Review of past semesters final exam questions UiTMKS/ FCE/ BCBidaun/ ECW211 35

36 Figure Q3(b) shows the wooden block with S.G = 0.8, is floating in water with depth of immersion, h m. Compute the volume and height, h for the immersed wooden block. Classify and shows the stability of the wooden block. (14 marks) OCT 2010 UiTMKS/ FCE/ BCBidaun/ ECW211 36

37 A spherical buoy has a diameter of 1.5 m weighs 8.50 kn and is anchored to the sea floor with a cable as is shown in Figure Q3(a) below. Although the buoy normally floats on the surface at certain times the water depth increases so that the buoy is completely immersed as illustrated. Determine the load tension of the cable. ( specific weight of sea water =10.1 kn/m 3 ) (8 marks) APR 2010 UiTMKS/ FCE/ BCBidaun/ ECW211 37

38 APR 2010 Figure Q3(b) shows a cylindrical buoy with a 1.8 m in diameter, 1.2 m high and weighs 10 kn, floats vertically in salt water of density 1025 kg/m 3. If the centre of gravity of the buoy is 0.45 m from the bottom, identify its stability. UiTMKS/ FCE/ BCBidaun/ ECW211 38

39 OCT 2009 A metallic cube 300 mm side and weighing 450 N is immersed into a tank containing two fluid layers of water and mercury as shown in Figure Q3(a). Use the buoyancy principles to determine the position of the cube at mercury-water interface when it has reached equilibrium. (7 marks) UiTMKS/ FCE/ BCBidaun/ ECW211 39

40 APR 2009 Figure Q3 (b) shows a buoy having a diameter of 2.4 m and length of 1.95 m is floating with its axis vertically located in sea water with specific weight of 10 kn/m 3. The buoy's weight is 16.5 kn and a load of 1.65 kn is placed on top of the buoy. If the buoy is to remain in stable equilibrium, determine the permissible height of the centre of gravity of the load above the top of the buoy using the buoyancy principles. UiTMKS/ FCE/ BCBidaun/ ECW211 40

41 UiTMKS/ FCE/ BCBidaun/ ECW211 41

42 APR 2009 A rectangular pontoon has a width of 8m, a length of 14m, and a draught of 2.5m in freshwater (p=1000 kg/m3). Using buoyancy concept calculate the following: i) The weight of pontoon ii) Its draught in seawater (p=1025 kg/m3) iii) The load that can be supported by the pontoon in freshwater if the maximum draught permissible is 4m. (10 marks) UiTMKS/ FCE/ BCBidaun/ ECW211 42

43 APR 2009 A block of wood has been cut as shown in Figure Q3(b) has a specific weight of 4.38kN/m 3. The wood is floating on glycerine which has a specific gravity of Determine the following properties using the buoyancy concept: i) Volume of displacement ii) Depth of immersion iii) Centre of buoyancy UiTMKS/ FCE/ BCBidaun/ ECW211 43

44 OCT 2008 A solid cylinder 2 m in diameter and 2 m in height is floating in water with its vertical axis as shown in Figure Q3(a). If the specific gravity of the material of cylinder is 0.65, apply the buoyancy concept to determine its metacentric height. Identify whether the equilibrium is stable or unstable. UiTMKS/ FCE/ BCBidaun/ ECW211 44

45 OCT 2008 A wooden block of width 1.25m, depth 0.75m and length 3.0m is floating in water as shown in Figure Q3(b). Given the specific weight of the wood is 6.4kN/m 3. Find the volume of water displaced and the position of centre of buoyancy by applying the Buoyancy concept. (10 marks) UiTMKS/ FCE/ BCBidaun/ ECW211 45

46 APR 2008 Figure Q3(b) shows the side view of a cylinder of 0.4 m diameter and height of 0.5 m floating in fresh water. Determine the stability of the cylinder if the specific weight of the cylinder is 5500 N/m 3. (14 marks) UiTMKS/ FCE/ BCBidaun/ ECW211 46

47 Summary End UiTMKS/ FCE/ BCBidaun/ ECW211 47

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