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Stability & Buoyancy Objectives Principles of Stability Archimedes Principle Terminology of ship’s hydrostatics Stability & moments -> staying upright Metacenter, Center of Gravity, Center of Buoyancy, etc. Stability curves Principles of Stability

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### Stability & Buoyancy

Objectives

- Principles of Stability
- Archimedes Principle
- Terminology of ship’s hydrostatics
- Stability & moments -> staying upright
- Metacenter, Center of Gravity, Center of Buoyancy, etc.
- Stability curves

Principles of Stability

- Floating object is acted on by forces of gravity and forces of buoyancy
- Static equilibrium SFi = 0

- Three conditions of static equilibrium:
- Stable: return to same position if tipped
- Neutral: when rotated, will come to rest in any position
- Unstable: will come to rest in new position if force acts on it

Archimedes Principle

- Law: a body floating or submerged in a fluid is buoyed up by a force equal to the weight of the water it displaces
- Depth to which ship sinks depends on density of water (r = 1 ton/35ft3 seawater)

Archimedes Principle

- Ship sinks until weight of water displaced by the underwater volume is equal to the weight of the ship
- Forces of gravity: G = mshipg =Wship
- Forces of buoyancy: B = rwaterVdisplaced
Wship = rwaterVdisplaced

Archimedes Principle

- Forces act everywhere on ship -> too tough to analyze
- Center of Gravity (G): all gravity forces as one force acting downward through ship’s geometric center
- Center of Buoyancy (B): all buoyancy forces as one force acting upward through underwater geometric center

Archimedes Principle

- Center of Gravity (G):
- Changes position only by change/shift in mass of ship
- Does not change position with movement of ship

- Center of Buoyancy (B):
- Changes position with movement of ship -> underwater geometric center moves
- Also affected by displacement

Hydrostatics Terminology

- Displacement: total weight of ship = total submerged volume of ship (measured in tons)
- Draft: vertical distance from waterline to keel at deepest point (measured in feet)
- Reserve Buoyancy: volume of watertight portion of ship above waterline (important factor in ship’s ability to survive flooding)
- Freeboard: vertical distance from waterline to main deck (rough indication of reserve buoyancy)

Hydrostatics Terminology

- As draft & displacement increase, freeboard and reserve buoyancy decrease

Moments

- Def’n: tendency of a force to produce rotation or to move an object about an axis
- Distance between the force and axis of rotation is the moment arm

- Couple: two forces of equal magnitude in opposite and parallel directions, separated by a perpendicular distance
- G and B are a couple

Moments

- Depending on location of G and B, two types of moments:
- Righting moment: tends to return ship to upright position
- Upsetting moment: tends to overturn ship

- Magnitude of righting moment:
- RM = W * GZ (ft-tons)
- GZ: moment arm (ft)

Metacenter

- Def’n: the intersection of two successive lines of action of the force of buoyancy as ship heels through small angles (M)
- If angle too large, M moves off centerline

Metacenter

- Metacentric Height (GM)
- Determines size of righting/upsetting arm (for angles < 7o)
GZ = GM*sinf

- Large GM -> large righting arm (stiff)
- Small GM -> small righting arm (tender)

- Determines size of righting/upsetting arm (for angles < 7o)

Metacenter

- Relationship between G and M
- G under M: ship is stable
- G = M: ship neutral
- G over M: ship unstable

STABLE

UNSTABLE

Metacenter v. Stability Curves

- At this point, we could use lots of trigonometry to determine exact values of forces, etc for all angles -> too much work
- GM used as a measure of stability up to 7°, after that values of GZ are plotted at successive angles to create the stability curve

Stability Curve

- Plot GZ (righting arm) vs. angle of heel
- Ship’s G does not change as angle changes
- Ship’s B always at center of underwater portion of hull
- Ship’s underwater portion of hull changes as heel angle changes
- GZ changes as angle changes

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