Buoyancy and stability

1. Stability &Stability &
BuoyancyBuoyancy

2. ObjectivesObjectives
•
Principles of StabilityPrinciples of Stability
•
Archimedes PrincipleArchimedes Principle
•
Terminology of ship’s hydrostaticsTerminology of ship’s hydrostatics
•
Stability & moments -> staying uprightStability & moments -> staying upright
•
Metacenter, Center of Gravity, Center ofMetacenter, Center of Gravity, Center of
Buoyancy, etc.Buoyancy, etc.
•
Stability curvesStability curves

3. Principles of StabilityPrinciples of Stability
•
Floating object is acted on by forces of gravityFloating object is acted on by forces of gravity
and forces of buoyancyand forces of buoyancy
• Static equilibriumStatic equilibrium ΣΣFFii = 0= 0
•
Three conditions of static equilibrium:Three conditions of static equilibrium:
• Stable:Stable: return to same position if tippedreturn to same position if tipped
•
Neutral:Neutral: when rotated, will come to rest in anywhen rotated, will come to rest in any
positionposition
• Unstable:Unstable: will come to rest in new position if forcewill come to rest in new position if force
acts on itacts on it

4. Archimedes PrincipleArchimedes Principle
•
Law: a body floating or submerged in aLaw: a body floating or submerged in a
fluid is buoyed up by a force equal to thefluid is buoyed up by a force equal to the
weight of the water it displacesweight of the water it displaces
•
Depth to which ship sinks depends onDepth to which ship sinks depends on
density of water (density of water (ρρ = 1 ton/35ft= 1 ton/35ft33
seawater)seawater)

5. Archimedes PrincipleArchimedes Principle
•
Ship sinks until weight of waterShip sinks until weight of water
displaced by the underwater volume isdisplaced by the underwater volume is
equal to the weight of the shipequal to the weight of the ship
• Forces of gravity:Forces of gravity: G = mG = mshipshipg =Wg =Wshipship
• Forces of buoyancy:Forces of buoyancy: B =B = ρρwaterwaterVVdisplaceddisplaced
WWshipship == ρρwaterwaterVVdisplaceddisplaced

6. Archimedes PrincipleArchimedes Principle
•
Forces act everywhere on ship -> tooForces act everywhere on ship -> too
tough to analyzetough to analyze
•
Center of Gravity (G):Center of Gravity (G): all gravity forcesall gravity forces
as one force acting downward throughas one force acting downward through
ship’s geometric centership’s geometric center
•
Center of Buoyancy (B):Center of Buoyancy (B): all buoyancyall buoyancy
forces as one force acting upwardforces as one force acting upward
through underwater geometric centerthrough underwater geometric center

7. Archimedes PrincipleArchimedes Principle
•
Center of Gravity (G):Center of Gravity (G):
•
Changes position only by change/shift inChanges position only by change/shift in
mass of shipmass of ship
•
Does not change positionDoes not change position with movement ofwith movement of
shipship
•
Center of Buoyancy (B):Center of Buoyancy (B):
• Changes positionChanges position with movement of ship ->with movement of ship ->
underwater geometric center movesunderwater geometric center moves
• Also affected by displacementAlso affected by displacement
G

8. Hydrostatics TerminologyHydrostatics Terminology
• Displacement:Displacement: total weight of ship = totaltotal weight of ship = total
submerged volume of ship (measured in tons)submerged volume of ship (measured in tons)
•
Draft:Draft: vertical distance from waterline to keel atvertical distance from waterline to keel at
deepest point (measured in feet)deepest point (measured in feet)
• Reserve Buoyancy:Reserve Buoyancy: volume of watertight portionvolume of watertight portion
of ship above waterline (important factor inof ship above waterline (important factor in
ship’s ability to survive flooding)ship’s ability to survive flooding)
•
Freeboard:Freeboard: vertical distance from waterline tovertical distance from waterline to
main deck (rough indication of reservemain deck (rough indication of reserve
buoyancy)buoyancy)

9. Hydrostatics TerminologyHydrostatics Terminology
• As draft & displacement increase,As draft & displacement increase,
freeboard and reserve buoyancy decreasefreeboard and reserve buoyancy decrease

10. MomentsMoments
•
Def’n: tendency of a force to produceDef’n: tendency of a force to produce
rotation or to move an object about anrotation or to move an object about an
axisaxis
• Distance between the force and axis ofDistance between the force and axis of
rotation is the moment armrotation is the moment arm
• Couple: two forces of equal magnitude inCouple: two forces of equal magnitude in
opposite and parallel directions,opposite and parallel directions,
separated by a perpendicular distanceseparated by a perpendicular distance
• G and B are a coupleG and B are a couple

11. MomentsMoments
• Depending on location of GDepending on location of G
and B, two types ofand B, two types of
moments:moments:
•
Righting moment:Righting moment: tends totends to
return ship to upright positionreturn ship to upright position
•
Upsetting moment:Upsetting moment: tends totends to
overturn shipoverturn ship
• Magnitude of rightingMagnitude of righting
moment:moment:
•
RM = W * GZRM = W * GZ (ft-tons)(ft-tons)
•
GZ: moment armGZ: moment arm (ft)(ft)

12. MetacenterMetacenter
•
Def’n: the intersectionDef’n: the intersection
of two successiveof two successive
lines of action of thelines of action of the
force of buoyancy asforce of buoyancy as
ship heels throughship heels through
small angles (M)small angles (M)
•
If angle too large, MIf angle too large, M
moves off centerlinemoves off centerline

13. MetacenterMetacenter
•
Metacentric HeightMetacentric Height
(GM)(GM)
•
Determines size ofDetermines size of
righting/upsetting armrighting/upsetting arm
(for angles < 7(for angles < 7oo
))
GZ = GM*sinGZ = GM*sinφφ
•
Large GM -> largeLarge GM -> large
righting arm (stiff)righting arm (stiff)
•
Small GM -> smallSmall GM -> small
righting arm (tender)righting arm (tender)

14. MetacenterMetacenter
•
Relationship between G and MRelationship between G and M
• G under M: ship is stableG under M: ship is stable
•
G = M: ship neutralG = M: ship neutral
• G over M: ship unstableG over M: ship unstable
STABLESTABLE UNSTABLEUNSTABLE

15. Metacenter v. Stability CurvesMetacenter v. Stability Curves
•
At this point, we could use lots ofAt this point, we could use lots of
trigonometry to determine exact valuestrigonometry to determine exact values
of forces, etc for all angles -> too muchof forces, etc for all angles -> too much
workwork
•
GM used as a measure of stability up toGM used as a measure of stability up to
7°, after that values of GZ are plotted at7°, after that values of GZ are plotted at
successive angles to create the stabilitysuccessive angles to create the stability
curvecurve

16. Stability CurveStability Curve

17. Stability CurveStability Curve
• Plot GZ (righting arm) vs. angle of heelPlot GZ (righting arm) vs. angle of heel
•
Ship’s G does not change as angle changesShip’s G does not change as angle changes
•
Ship’s B always at center of underwater portion ofShip’s B always at center of underwater portion of
hullhull
•
Ship’s underwater portion of hull changes as heelShip’s underwater portion of hull changes as heel
angle changesangle changes
• GZ changes as angle changesGZ changes as angle changes

18. Questions?Questions?