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KAZUNGULA FERRY : BOTSWANA – ZAMBIA BORDER. BUOYANCY. RECALL NEGATIVE WEIGHT ARCHIMEDES PRINCIPLE FLOATING BODY STABILITY METACENTRE STABILITY OF SHIPS. ADJUSTABLE BUOYANCY LIFE JACKET. NEGATIVE WEIGHT ON SURFACES.

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buoyancy

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

negative weight on surfaces
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

slide3

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

archimedes principle
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

slide5

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

floatation
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

slide7

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

stability
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

metacentre
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

determining metacentric radius
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

stability of ships
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

slide12

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