FLUID MECHANICS – 1 First Semester 2011 - 2012 Week – 3 Class – 1 Buoyancy and Flotation Compiled and modified by Sharma, Adam and Idriss
OBJECTIVES• Concept of buoyancy Buoyancy force Archimedes’ principle• Stability of floating bodies
BUOYANCY AND STABILITY Buoyant force: The upward force a fluid exerts on a body immersed in it. The buoyant force is caused by the increase of pressure with depth in a fluid. The buoyant force acting on the plate is equal to the weight of the liquid displaced by the plate. For a fluid with constant density, the buoyant force is independent of the distance of the body from the free surface. It is also independent of the density of the solidA flat plate of uniform thickness h submerged body.in a liquid parallel to the free surface. 3
ARCHIMEDES’ PRINCIPLEThe buoyant force acting on a body immersed in afluid is equal to the weight of the fluid displaced bythe body, and it acts upward through the centroidof the displaced volume.
The buoyant forces acting on a solid body submerged in a fluid and on a fluid body of the same shape at the same depth are identical.The buoyant force FB acts upward through the centroid C of thedisplaced volume and is equal in magnitude to the weight W of thedisplaced fluid, but is opposite in direction.For a solid of uniform density, its weight Ws also acts through thecentroid, but its magnitude is not necessarily equal to that of the fluidit displaces. (Here Ws > W and thus Ws > FB; this solid body wouldsink.) 5
For floating bodies, the weight of the entire body must be equal tothe buoyant force, which is the weight of the fluid whose volume isequal to the volume of the submerged portion of the floating body: A solid body dropped into a fluid will sink, float, or remain at rest at any point in the fluid, depending on its average density relative to the density of the fluid. 6
The altitude of a hot air balloon iscontrolled by the temperaturedifference between the air insideand outside the balloon, sincewarm air is less dense than coldair. When the balloon is neitherrising nor falling, the upwardbuoyant force exactly balances thedownward weight. 7
STABILITY OF IMMERSED AND FLOATING BODIES Stability is easily understood by analyzing a ball on the floor.For floating bodies such asships, stability is an importantconsideration for safety. 9
A floating body possesses vertical stability, while an immersedneutrally buoyant body is neutrally stable since it does notreturn to its original position after a disturbance.An immersed neutrally buoyant body is(a)stable if the center of gravity G is directly below the centerof buoyancy B of the body,(b)neutrally stable if G and B are coincident, and(c)unstable if G is directly above B. 10
When the center of gravity G of an A ball in a trough betweenimmersed neutrally buoyant body is not two hills is stable for smallvertically aligned with the center of disturbances, butbuoyancy B of the body, it is not in an unstable for largeequilibrium state and would rotate to its disturbances.stable state, even without any disturbance. 11
A floating body is stable if the body is bottom-heavy and thusthe center of gravity G is below the centroid B of the body, or ifthe metacenter M is above point G. However, the body isunstable if point M is below point G.Metacentric height GM: The distance between the center ofgravity G and the metacenter M—the intersection point of thelines of action of the buoyant force through the body before andafter rotation.The length of the metacentric height GM above G is a measureof the stability: the larger it is, the more stable is the floating 12body.
Example 1A block of wood of SG 0.8 and size 100mmx40mmx30mmfloats at depth of 24mm in water. Determine its metacentricheight, for tilt about its longitudinal axis.
Example 2A solid cylinder of 3 m diameter has a height of 3m. it ismade of material which SG is 0.8, and is floating in waterwith its axis vertical.Determine its metacentric height and state whether itsequilibrium is stable or unstable.
Example 3A solid cylinder 36cm long, 8cm diameter has its base 1 cmthick and its SG=7. The remaining part of the cylinder SG is0.5. Determine, if it can float vertically in water.
Example 4A uniform wooden circular cylinder of 400mm diameter andSG 0.6 is required to float in an oil of SG 0.8. Find themaximum length of the cylinder, in order that cylinder tofloat vertically in oil.