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Stability & Buoyancy

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**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

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**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

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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)

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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

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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

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**Archimedes Principle Center of Gravity (G): Center of Buoyancy (B):**

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 G

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**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)

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**Hydrostatics Terminology**

As draft & displacement increase, freeboard and reserve buoyancy decrease

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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

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**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)

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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

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**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)

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**Metacenter Relationship between G and M G under M: ship is stable**

G = M: ship neutral G over M: ship unstable STABLE UNSTABLE

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**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

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Stability Curve

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**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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