BUOYANCY

1. KAZUNGULA FERRY : BOTSWANA – ZAMBIA BORDER BUOYANCY RECALL NEGATIVE WEIGHT ARCHIMEDES PRINCIPLE FLOATING BODY STABILITY METACENTRE STABILITY OF SHIPS ADJUSTABLE BUOYANCY LIFE JACKET KK's FLM 221 - WK6: BUOYANCY

2. NEGATIVE WEIGHT ON SURFACES • A surface in contact with a fluid below it feels a force from the fluid with a vertical component pointing upwards • The vertical component is equal and opposite to the weight of the ‘imaginary’ fluid on the other side that would cause equilibrium • This vertical component is called the NEGATIVE WEIGHT of the fluid because: • It points upwards instead of downward • It is the weight of the imaginary fluid • If the surface is for an object immersed in the fluid, the horizontal forces from opposite sides of the object cancel out – leaving only the vertical negative weight! KK's FLM 221 - WK6: BUOYANCY

3. Imaginary fluid of weight ‘mg’ Fv -Fv=mg F -F G C C Fh -Fh Real fluid below surface FIGURE 6.1: Recall: How did we determine the force ‘F’ last week? Ans: Put imaginary fluid to same level on the other side of surface, then find the force ‘–F’ it exerts to balance the real ‘F’.Vertical component of –F is weight of the imaginary fluid, ‘mg’! KK's FLM 221 - WK6: BUOYANCY

4. ARCHIMEDES’ PRINCIPLE • Refers to the force exerted by a fluid NOT just on a surface BUT on an entire body immersed in the fluid. • States: “A body fully or partially immersed in a fluid experiences an upward force ‘Fu’ called BUOYANCY or UPTHRUST equal to the weight of fluid it displaces” • It means: all forces from the fluid onto all the surfaces of the body lead to a SINGLE upward vertical force equal to the weight of displaced fluid. • From our analysis of last week, this is due to: • Horizontal forces all round the body cancelling out • The imaginary fluid we have all along been putting on ‘other side’ is actually that one displaced by the object! • The negative weight we have been talking about is the BUOYANCY or UPTHRUST! The object feels a loss of some weight (not mass) because of this ‘negative weight’ KK's FLM 221 - WK6: BUOYANCY

5. FIGURE 6.2: Explaining Archimedes’ principle using pressures: HOW THE BODY weighs less when in fluid!! Downward Force from fluid at top of surface = ρgh1A Force from fluid here = 0 because of zero pressure h1 a) Body fully immersed b) Body partially immersed h h1+h Actual weight of body, ‘W’ h Upward Force from fluid here = ρhAg = mfluidg = Weight of displaced fluid = BUOYANCY Upward Force from fluid here = ρ(h1+h)Ag Net downward force on body = W+ρgh1A-ρ(h1+h)Ag = W-ρghA = W-mfluidg = True object weight - BUOYANCY KK's FLM 221 - WK6: BUOYANCY

6. FLOATATION • When an object floats on its own free will, two forces act on it and balance out: The true weight (due to gravity) and the Buoyancy force (see figs 6.2b and 6.3) • Archimedes’ principle then leads us to conclude: “For a floating object, the weight of the fluid displaced equals the body’s true weight” • This is the law of floatation Mathematically, we write: (6.1) Question 1: Show that if the object is floating then: (6.2) Question 2: Show that the fraction of the object volume below the water line is given by: (6.3) KK's FLM 221 - WK6: BUOYANCY

7. Water line for each object Fu1 Fu3 Fu2 d1 d3 d2 W1 W3 W2 Fu – up thrust or BUOYANCY; W – weight of floating object; d – draft – i.e. depth of object below the water line in the fluid Shaded volume is the water displaced and its weight is the BUOYANCY force Fu FIGURE 6.3: LAW OF FLOATATION: W = Fu = WEIGHT OF FLUID DISPLACED KK's FLM 221 - WK6: BUOYANCY

8. STABILITY • The true weight of the object acts through its centre of gravity ‘G’; • The BUOYANCY force acts through the centre of gravity of the displaced fluid. This is called the CENTRE of BUOYANCY, ‘B’ • If ‘B’ and ‘G’ are along the same vertical line, there is no resultant couple on the object • If ‘B’ and ‘G’ are NOT on same vertical line, they form a couple – which may or may not overturn the object into the fluid θ θ G G G G B B B B STABLE EQUILIBRIUM: Buoyancy couple opposes disturbance UNSTABLE EQUILIBRIUM: Buoyancy couple reinforces disturbance FIGURE 6.4: STABLE & UNSTABLE EQUILIBRIUM KK's FLM 221 - WK6: BUOYANCY

9. METACENTRE • The point at which a buoyancy force intersects an original vertical line through ‘G’ (and hence old ‘B’) after a disturbance of the object is the METACENTRE ‘M’ • The distance GM is the METACENTRIC HEIGHT • The distance BM for small disturbances ‘θ’ is the METACENTRIC RADIUS • If ‘M’ is ABOVE ‘G’, there is STABLE equilibrium. If BELOW ‘G’, equilibrium is UNSTABLE. If at ‘G’, equilibrium is NEUTRAL M FIGURE 6.5: POSITION OF METACENTRE & STABILITY M G B G B UNSTABLE: ‘M’ is BELOW ‘G’ STABLE: ‘M’ is ABOVE ‘G’ KK's FLM 221 - WK6: BUOYANCY

10. DETERMINING METACENTRIC RADIUS • For all floating objects, the metacentre ‘M’ for small disturbances ‘θ’ is a property of the shape of the water line. • It can be shown mathematically that the METACENTRIC RADIUS ‘BM’ is given by: (6.4) • Where IG is the 2nd moment of area of the water line; V is the volume of fluid displaced; θ is the angle of tilt or disturbance in RADIANS; B and B’ are the centres of BUOYANCY before and after the disturbance KK's FLM 221 - WK6: BUOYANCY

11. STABILITY OF SHIPS • Ships get many disturbances while at sea – including waves and internal shift of weights • Have symmetrical water lines about a longitudinal axis but the shapes formed are not of the common type as listed in table 4.1(refer to week 4 notes) • Water line approximated by a rectangular shape but adjusted with a FORM factor ‘k’ for determining IG • Then: (6.5) • Where ‘l’ and ‘b’ are length and width of ship water line respectively • When a disturbance ‘θ’ causes a shift of weight ‘ΔW’ through a distance ‘x’ across the ship, stability requires that the buoyancy force Fu (=W) sets up a moment in accordance with: (6.6) • Where θis in RADIANS; W is the total weight of the ship and its cargo, MG is the metacentric height KK's FLM 221 - WK6: BUOYANCY

12. θ Approximate rectangle M Actual shape of water line W l x ΔW G B b Fu = W FIGURE 6.6: SHIP STABILITY KK's FLM 221 - WK6: BUOYANCY