CHAPTER FIVE-bouyance and floatation.pptx

It describes about bouyance and floatation

1. WOLAITA SODO UNIVERSITY DEPARTMENT OF HYDRAULIC AND WATER RESOURCES ENGINEERING FLUID MECHANICS ( HENG-2111 ) CHAPTER FIVE BUOYANCY AND FLOATATION 2 ND year HWRE (2023 G.C.) By: Manamno B. (MSc.) [email protected] 3/22/2023 By Manamno B. 1

2. TOPIC OUTLINE Introduction Buoyancy and Stability of Floating bodies Stability of Submerged bodies 3/22/2023 By Manamno B. 2

3. Specific Objective of the chapter Define Buoyancy, Archimedes Principle Estimate Center of Buoyancy Discuss about Metacenter To estimate Metacentric height Methods to estimate metacentric height 3/22/2023 By Manamno B. 3

4. INTRODUCTION Why an object floats ?   3/22/2023 By Manamno B. 4

5. An object floats If: They are less dense than the fluid they are in They weigh less than the buoyant force pushing up on them They are shaped so their weight is spread out 3/22/2023 By Manamno B. 5

6. Buoyancy When a body is immersed wholly or partially in a fluid, it is subjected to an upward force which tends to lift (buoy)it up. The tendency of immersed body to be lifted up in the fluid due to an upward force opposite to action of gravity is known as buoyancy. The force tending to lift up the body under such conditions is known as buoyant force or force of buoyancy or up-thrust. The buoyant force always acts vertically upward 3/22/2023 By Manamno B. 6

7. Buoyancy…….. The magnitude of the buoyant force can be determined by Archimedes’ principle which states “ When a body is immersed in a fluid either wholly or partially, it is buoyed or lifted up by a force which is equal to the weight of fluid displaced by the body” Center of Buoyancy: The point of application of the force of buoyancy on the body is known as the center of buoyancy It is always the center of gravity of the volume of fluid displaced 3/22/2023 By Manamno B. 7

8. Buoyancy…….. Center of Buoyancy 3/22/2023 By Manamno B. 8

9. Buoyancy…….. Consider a cylindrical body submerged in water as shown in the figure with cross sectional area,dA And an upward pressure force (p 2 –p 1 ) dA acting on the cylindrical element due to the pressure on the exposed end of the cylinder 3/22/2023 By Manamno B. 9

10. Buoyancy…….. dF B = (p 2 – p 1 ) dA =  (h 2 -h 1 ) dA =  dv , Where dv = volume of the prism The entire force exerted on the body is obtained by integrating the differential force on the small strip  is assumed constant through out the volume. V= Volume of the bod   3/22/2023 By Manamno B. 10

11. Buoyancy…….. A body immersed in two different fluids If the body is immersed so that part of its volume V 1 is immersed in a fluid of density  1 and the rest of its volume V 2 in another immiscible fluid of mass density  2 , Up thrust on upper part, R 1 =  1 gV 1, acting through G 1 , the centroid of V1, Up thrust on lower part,R 2 =  2 gV 2,, acting through G 2 , the centroid of V 2 ,   3/22/2023 By Manamno B. 11

12. Stability of Submerged bodies Condition of equilibrium Stable equilibrium: When center of buoyancy(B) is above center of pressure(G) Unstable equilibrium: G > B Neutral equilibrium: B and G are at the same point 3/22/2023 By Manamno B. 12

13. Stability of Floating body For the body to be in equilibrium , the weight, W, must equal the buoyant force, Fb both acting along the same vertical line (Fig a) For small angle of heel, the intersection point the vertical through the new center of buoyancy, B’, and the line, BG, produced is known as the meta- centre , M , and the body thus disturbed tends to oscillate about, M (Fig b) The distance between G and M is metacentric height. 3/22/2023 By Manamno B. 13

14. Stability of Floating body………… Condition of equilibrium of floating bodies Stable equilibrium: I f M lies above G then a righting moment is produced, GM is regarded as Positive 3/22/2023 By Manamno B. 14

15. Unstable equilibrium: If M lies below G an overturning moment is produced, GM is regarded as negative Neutral equilibrium: If M and G coincides 3/22/2023 By Manamno B. 15 Stability of Floating body…………

16. Tilting Ships 3/22/2023 By Manamno B. 16 Stability of Floating body…………

17. Determination of Metacentric height (GM) Analytical Method Experimental Method 3/22/2023 By Manamno B. 17 Stability of Floating body…………

18. Analytical Methods In the figure below, AA is the water-line and when a body is given a small tilt θ two wedge forces formed, Due to the submergence and emergence of the wedge area AOA’ on either side of the axis of rolling, are imposed on the body forming a couple which tends to restore the body to undisturbed condition. The effect of this couple is the same as the moment caused by the shift of the total buoyant force Fb from B to B’, the new center of buoyancy. 3/22/2023 By Manamno B. 18 Stability of Floating body…………

19. The buoyant force acting through B’ By moment about B, = = The wedge force,   3/22/2023 By Manamno B. 19 Stability of Floating body…………

20. , Where; I is the second moment of plane area of the body at water level about its longitudinal axis   3/22/2023 By Manamno B. 20 Stability of Floating body…………

21. Time of Oscillation Consider a floating body, which is tilted through an angle by an overturning couple as shown below. Let the overturning couple is suddenly removed. The body will start oscillating. Thus, the body will be in a state of oscillation as if suspended at the meta- center M. This is similar to a case of a pendulum. The only force acting on the body is due to the restoring couple due to the weight w of the body force of buoyancy F B . 3/22/2023 By Manamno B. 21

22. Restoring couple = W GM sin  Angular acceleration of the body, Torque due to inertia = But Where W=weight of body, K=radius of gyration about Y-Y Inertia torque = - Equating the above equations W GM sin  = -   3/22/2023 By Manamno B. 22 Time of Oscillation Cont.…..

23. For small angle  , sin  =  GM  = - or GM  + = 0 + = 0 This is second-degree differential equation, the solution is   3/22/2023 By Manamno B. 23 Time of Oscillation Cont.…..

24. Where C 1 and C 2 are constants of integration. The values of C 1 and C 2 are obtained from boundary conditions, which are at t=0,  =0 at t=(T/2),  =0 Where T=time of one complete oscillation Substituting the first boundary condition, C 2 =0 3/22/2023 By Manamno B. 24 Time of Oscillation Cont.…..

25. Substituting the first boundary condition, C 2 =0 Substituting the second boundary condition, we get But C 1 cannot be equal to zero and so the other alternative is = 0 = = or This gives the time period of oscillation or rolling of a floating body   3/22/2023 By Manamno B. 25 Time of Oscillation Cont.…..

26. Example (Quiz 1) Why an object floats ? In stable equilibrium the center of gravity and center of buoyancy are coincides. When this incidents have been happen ? 3/22/2023 By Manamno B. 26