Vessel Stability Theory — Complete USCG Exam Guide
Vessel stability is one of the highest-weighted subjects on every USCG captain license exam, from the six-pack OUPV through Master Unlimited. This guide covers every stability concept the exam tests: Archimedes' principle, metacentric height GM, righting lever GZ, free surface effect, list vs loll, trim calculations, and load line marks. Worked examples and exam tips throughout.
Key Formulas at a Glance
Metacentric Height
GM = KM - KG
KM = KB + BM
BM = I / V
Righting Arm (small angles)
GZ = GM x sin(theta)
RM = GZ x Displacement
Weight Shift (G movement)
GG1 = (w x d) / W
w=weight moved, d=distance, W=displacement
Trim Change
dTrim = TM / MCT1in
TM = trimming moment (ft-tons)
1. Buoyancy and Archimedes' Principle
Archimedes' principle is the foundation of all vessel stability: a body immersed in a fluid is acted upon by an upward buoyant force equal to the weight of the fluid it displaces. For a steel vessel floating in seawater, the hull pushes aside a volume of water, and that water exerts an upward force equal to its weight.
When a vessel is in equilibrium (floating at rest), two forces are balanced: the downward force of gravity acting through the center of gravity (G), and the upward buoyant force acting through the center of buoyancy (B). These two forces must be equal in magnitude and act along the same vertical line for the vessel to float upright without listing.
Diagram: Equilibrium Forces on a Floating Vessel
Gravity (W) acts downward through G (center of gravity)
Buoyancy (B force) acts upward through B (center of buoyancy)
At equilibrium: W = buoyant force, G and B on same vertical line
Displacement (LT) = Underwater Volume (cu ft) / 35
The center of buoyancy (B) is the geometric centroid of the underwater volume of the hull. It is not a fixed point — it moves as the vessel heels or changes draft, because the shape of the underwater volume changes. This movement of B is the key to understanding why vessels can be stable or unstable.
Displacement in long tons equals underwater volume in cubic feet divided by 35, because one long ton of seawater occupies 35 cubic feet at standard salinity (specific gravity 1.025). Fresh water is less dense — one long ton occupies 36 cubic feet. This difference is why vessels float deeper in fresh water than in salt water and why fresh water allowance must be applied when loading near a load line.
Exam Tip: Buoyancy Units
USCG stability problems use long tons (1 LT = 2,240 lbs). Seawater is 64 lbs per cubic foot (35 cu ft / LT). Fresh water is 62.4 lbs per cubic foot (36 cu ft / LT). Always identify which water density applies before calculating displacement.
2. Center of Buoyancy (B), Metacenter (M), and the KB-BM-KM Chain
Three points on the vessel's centerline define the geometry of stability. All three are measured vertically from the keel (K), which gives us the familiar KB, BM, and KM distances.
KB
Height of center of buoyancy above the keel. For a box-shaped vessel, KB equals approximately half the draft. As draft increases, KB rises. KB is read from stability tables at the given displacement.
BM (Metacentric Radius)
BM equals I divided by V, where I is the second moment of area of the waterplane about the centerline and V is the displaced volume. Wider vessels have much larger I and therefore larger BM. BM decreases as draft increases.
KM
KM equals KB plus BM. It is the height of the metacenter above the keel. KM values are tabulated in stability booklets at each draft increment and are used directly in calculating GM once KG is known.
What Is the Metacenter?
The metacenter (M) is a theoretical point. When a vessel heels to a small angle, the center of buoyancy shifts to the low side as the underwater volume on that side increases. A vertical line drawn upward through the new position of B intersects the vessel's original centerline at a point called the metacenter. For small angles of heel (up to about 10 to 15 degrees), M stays at approximately the same height.
The metacenter is not fixed at large angles of heel — this is why initial stability (based on GM) gives a different picture than large-angle stability (based on the GZ curve). For exam purposes, remember that the small-angle assumption underlying GM applies only up to roughly 10 to 15 degrees. Beyond that, you need the full GZ curve to assess stability.
Worked Example: Finding KM
A vessel is loaded to a displacement of 800 LT. From the stability booklet:
KB at this draft = 4.2 ft
BM at this draft = 6.8 ft
KM = KB + BM = 4.2 + 6.8 = 11.0 ft
If the vessel's KG (from loading calculation) is 9.4 ft, then:
GM = KM - KG = 11.0 - 9.4 = 1.6 ft (positive, vessel stable)
3. Metacentric Height GM — Positive, Negative, and Critical Values
GM is the master indicator of initial stability. Its sign tells you immediately whether a vessel is stable or in danger.
Positive GM
M is above G. Any heel creates a righting moment. The vessel returns to upright. Larger positive GM means stiffer, faster rolling. A very large GM can cause a short, violent roll that strains crew and cargo.
Zero GM
G and M coincide. The vessel has neutral stability — no righting moment and no capsizing moment at small angles. It will remain at whatever angle it is heeled to. This is a dangerous borderline condition.
Negative GM
G is above M. Any heel creates a capsizing moment rather than a righting moment. The vessel will heel over and settle at a loll angle (or capsize if the loll angle exceeds the angle of vanishing stability).
What Affects GM in Practice?
Adding high weights raises G and reduces GM. Removing high weights or adding low ballast lowers G and increases GM. Taking on deck cargo that is heavy and mounted high is a classic way to reduce GM dangerously. Adding ballast water to low double-bottom tanks lowers G and increases GM — the preferred corrective action for a vessel with insufficient stability.
Draft also affects GM. As a vessel loads down, draft increases, KB rises (favorable), but BM decreases (unfavorable) because displacement volume V increases faster than the waterplane moment of inertia changes. In most vessel shapes, KM decreases as draft increases, so a deeply loaded vessel typically has less GM than the same vessel at a light draft — but the exact behavior depends on hull form.
Exam Tip: GM and Roll Period
A vessel with very large GM has a short, stiff roll period — it snaps back quickly. A vessel with small GM has a long, gentle roll. A vessel with negative GM lolls. If you see a vessel with a very slow, sluggish roll that just barely comes back, it likely has very small positive GM. If you see a vessel that lists to one side but rolls easily back through center, it likely has list with positive GM.
4. Righting Lever GZ and the Stability Curve
The righting lever GZ is the horizontal distance between the line of action of gravity (through G) and the line of action of buoyancy (through B), measured perpendicular to the vessel's heeled axis. When the vessel is heeled, B has shifted to the low side. The vertical line through B no longer passes through G — it passes to one side of G — and this separation is the righting lever.
At small angles of heel, GZ equals GM times sin(heel angle). This linear relationship holds only in the initial stability range (typically under 10 to 15 degrees). At larger angles, the actual GZ must be read from the vessel's GZ curve, which is calculated for the specific hull form and loading condition.
Diagram: Key Features of a GZ Stability Curve
Initial slope at 0 degrees equals GM times sin(1 degree). A steeper slope means larger GM and stiffer initial stability.
Peak GZ at the maximum righting arm — the tallest point on the curve. The angle at which this occurs (typically 30 to 40 degrees for well-designed vessels) is also tested.
Angle of vanishing stability (AVS) — where GZ returns to zero. Heeling beyond this angle causes capsize. USCG stability criteria specify minimum AVS values (typically 70 degrees or greater for offshore vessels).
Area under the GZ curve represents dynamic stability — the energy needed to capsize the vessel. USCG regulations specify minimum areas from 0 to 30 degrees and 0 to 40 degrees.
Reading GZ Curves on the USCG Exam
Exam questions frequently show a GZ curve and ask you to identify: the vessel's GM (from the initial slope), whether stability is adequate (positive GZ at 30 degrees), the angle of maximum GZ, or the angle of vanishing stability. Some questions show two curves for different loading conditions and ask which is safer.
A vessel with free surface effect will show a reduced initial slope on the GZ curve compared to the solid GM curve. The corrected GZ values are lower across all angles, shifting the AVS to a smaller angle and reducing the area under the curve — both unfavorable for safety.
Worked Example: Righting Moment Calculation
Vessel displacement: 1,200 LT. GM: 2.1 ft. What is the righting moment at 10 degrees?
GZ = GM x sin(10 degrees) = 2.1 x 0.1736 = 0.364 ft
Righting Moment = GZ x Displacement = 0.364 x 1,200 = 437 ft-LT
The vessel exerts 437 foot-long-tons of righting force at 10 degrees of heel.
5. Static vs Dynamic Stability
Static stability refers to the vessel's ability to return to upright from a given static angle of heel — the righting moment at that angle. The GZ curve is a static stability curve because it plots the righting arm at each fixed angle without accounting for the energy of motion.
Dynamic stability is the ability to resist capsizing when heeling forces are applied dynamically — by waves, wind gusts, or sudden weight shifts. A vessel in motion can carry kinetic energy that propels it past an angle where static GZ is positive, reaching an angle where GZ becomes negative (capsizing territory). Dynamic stability is measured by the area under the GZ curve between two angles — this area in foot-degrees represents work done against capsizing.
The wind heeling moment criterion used in stability regulations compares the dynamic energy of a wind gust against the area under the GZ curve. The area under the righting arm curve from the leeward angle (angle of static equilibrium under steady wind) to the downflooding angle must exceed the area representing the dynamic wind gust energy by a specified factor of safety.
Key Distinction for the Exam
Static stability = GZ at a fixed angle. Dynamic stability = area under the GZ curve = energy to capsize. A vessel can have adequate static stability but inadequate dynamic stability if the GZ curve drops off steeply after its peak. USCG criteria address both.
6. Reserve Buoyancy and Freeboard
Reserve buoyancy is the watertight volume of the vessel above the waterline. It represents the additional buoyant force available before the vessel submerges the main deck. A vessel with large reserve buoyancy can absorb flooding or wave ingress without immediately sinking and can support large angles of heel before the deck edge submerges.
Freeboard is the distance from the waterline to the main deck amidships. Greater freeboard means greater reserve buoyancy. Overloading a vessel reduces freeboard and reserve buoyancy simultaneously — the deck edge submerges at a smaller angle of heel, which causes the GZ curve to peak earlier and fall more steeply, reducing the angle of vanishing stability.
The relationship between freeboard and stability is one reason load line regulations exist. Minimum freeboard requirements ensure that vessels maintain adequate reserve buoyancy for the waters and seasons they operate in.
Deck Edge Submersion and the GZ Curve
When the deck edge submerges, water can enter through deck openings (downflooding), and the rate of growth of the waterplane area suddenly changes. This often appears as a kink or change in slope in the GZ curve. Below the deck edge submersion angle, the curve may rise normally. Once the deck edge submerges, the curve may flatten or begin to fall more steeply. The angle at which the deck edge submerges is called the angle of deck edge immersion and is a key design parameter.
7. Effects of Adding, Removing, and Shifting Weight
Every change in loading changes the position of G and therefore changes GM. The master formula for G movement is GG1 equals (w times d) divided by W, where w is the weight involved, d is the distance between the original G and the position of the new weight (or the distance moved), and W is the total displacement.
Adding Weight
G moves toward the added weight. The new displacement W plus w is used.
GG1 = (w x d) / (W + w)
Adding high weight raises G, reducing GM. Adding low weight lowers G, increasing GM.
Removing Weight
G moves away from the removed weight. Displacement decreases to W minus w.
GG1 = (w x d) / (W - w)
Removing high weight lowers G and improves stability. Removing ballast from low tanks raises G.
Shifting Weight
G moves in the same direction as the weight. Displacement is unchanged.
GG1 = (w x d) / W
Moving weight off-center creates list. Moving weight vertically changes GM.
Loading a Vessel — The KG Calculation
The standard loading calculation builds up the vessel's KG from a loading table. Each item aboard has a weight (in LT) and a vertical center of gravity (KG in feet above the keel). The product of weight times KG gives the vertical moment. The sum of all moments divided by the total displacement gives the vessel's KG. Then GM equals KM (from stability tables at the calculated draft) minus KG.
Worked Example: KG Calculation
| Item | Weight (LT) | KG (ft) | Moment (ft-LT) |
|---|---|---|---|
| Light ship | 450 | 8.2 | 3,690 |
| Cargo (hold) | 280 | 5.5 | 1,540 |
| Fuel (DB tank) | 80 | 1.8 | 144 |
| Fresh water | 20 | 2.1 | 42 |
| Totals | 830 | 5,416 |
KG = Total Moment / Total Displacement = 5,416 / 830 = 6.53 ft
If KM at 830 LT draft = 10.8 ft, then GM = 10.8 - 6.53 = 4.27 ft (positive)
8. Free Surface Effect — Slack Tanks and Stability Loss
Free surface effect is one of the most dangerous and commonly tested stability hazards. When a tank is partially filled with liquid, that liquid can shift when the vessel heels. This shift moves the effective center of gravity upward — reducing GM — even though no actual weight has been added or removed.
The free surface correction (FSC) is subtracted from solid GM to give the effective (corrected) GM:
GM (corrected) = GM (solid) - FSC
FSC = (i x rho-liquid) / (V x rho-seawater)
Where i = second moment of area of the free surface (ft^4), V = displacement volume (ft^3), and rho represents the respective fluid densities.
The second moment of area i equals (length times width cubed) divided by 12 for a rectangular tank. Because the width appears cubed, doubling the tank's beam increases the free surface effect eightfold. This is why wide slack tanks are so destructive to stability and why longitudinal centerline bulkheads are used to subdivide tanks — each half-width tank has only one-eighth the free surface moment of the full-width tank, dramatically reducing FSC.
Rules of Thumb for Slack Tanks
- ●A tank is either full (zero free surface), empty (zero free surface), or a stability hazard (any partial fill).
- ●Free surface effect is independent of the amount of liquid in the tank — a 10% full tank has almost the same FSC as a 90% full tank (same surface area, same second moment).
- ●Fuel consumption during a voyage creates slack tanks where full tanks existed at departure — GM decreases as voyage proceeds, which is why stability must be checked at worst-case fuel load.
- ●Denser liquids (saltwater ballast at SG 1.025) cause more free surface effect than lighter liquids (diesel fuel at SG 0.85) in the same tank.
Worked Example: Free Surface Correction
A tank 30 ft long by 20 ft wide contains slack diesel fuel (SG 0.85). Vessel displacement volume = 14,000 cu ft.
i = (L x W^3) / 12 = (30 x 8,000) / 12 = 20,000 ft^4
FSC = (20,000 x 0.85) / (14,000 x 1.025) = 17,000 / 14,350 = 1.18 ft
If solid GM was 1.50 ft, corrected GM = 1.50 - 1.18 = 0.32 ft (dangerously small!)
Adding a centerline bulkhead splits the tank into two 10 ft wide sections:
i (each half) = (30 x 10^3) / 12 = 2,500 ft^4
Total i = 2 x 2,500 = 5,000 ft^4 (one-quarter of original)
FSC = (5,000 x 0.85) / 14,350 = 0.30 ft (much safer)
9. The Wall-Sided Formula and Angle of Loll
The simple small-angle formula GZ equals GM times sin(theta) applies only up to about 10 to 15 degrees. For wall-sided vessels (vessels with vertical sides between the waterline and deck edge), the wall-sided formula extends this to moderate angles:
GZ = sin(theta) x (GM + (BM / 2) x tan^2(theta))
This formula accounts for the change in position of B at moderate heel angles in vessels with parallel sides. It is accurate up to the angle of deck edge immersion.
Angle of Loll from the Wall-Sided Formula
For a vessel with negative GM, the wall-sided formula can be solved for the angle at which GZ equals zero beyond upright — the loll angle:
tan(loll angle) = sqrt((-2 x GM) / BM)
A vessel with GM = -0.4 ft and BM = 8 ft:
tan(loll angle) = sqrt((2 x 0.4) / 8) = sqrt(0.1) = 0.316
Loll angle = arctan(0.316) = about 17.5 degrees
This vessel will settle at about 17.5 degrees of heel to either side.
The loll angle is stable in the sense that the vessel will remain there — GZ is zero at that angle. But it is not truly stable — any additional heeling force or cargo shift can push the vessel past the loll angle into capsizing territory if the GZ curve beyond the loll angle does not develop sufficient positive values. A large negative GM produces a large loll angle, which is extremely dangerous.
10. List vs Loll — The Critical Distinction
List
- Cause: Off-center weight — G has moved transversely to one side.
- GM: Positive. The vessel has normal righting characteristics.
- Behavior: Heels steadily to the heavy side. If pushed to the upright position, it springs back to the list.
- Correction: Move weight to the high side (the light side). Remove weight from the low side. Add ballast on the high side.
- Roll: Rolls about the listed angle. Stiffness is normal.
Loll
- Cause: Negative GM — G is above M due to high weights or excessive free surface.
- GM: Negative. No righting moment at zero heel.
- Behavior: Vessel rolls to a random side and stays there. May lurch from one loll angle to the other (dangerous).
- Correction: Lower G by adding low ballast, pumping out high tanks, or removing high weights. NEVER move weight to the high side — this is dangerous with loll.
- Roll: Sluggish roll, slow to come back, may flop from side to side.
Critical Exam Point: Do Not Confuse the Correction Actions
The single most dangerous mistake with a lolling vessel is to move weight to the high side to "correct" the lean. This action, correct for a list, is catastrophically wrong for loll. Moving a weight to the high side of a lolling vessel increases the heeling moment at the loll angle, potentially causing rapid capsize. If you cannot determine whether you have list or loll, treat the condition as loll and correct by lowering G.
How to Diagnose List vs Loll
1. Check the roll: A listing vessel rolls normally but about the list angle. A lolling vessel rolls very sluggishly and may lurch from side to side if disturbed.
2. Check the loading record: Was heavy deck cargo recently loaded? Has fuel been burned from high tanks? Large free surface in wide slack tanks? Any of these indicate potential negative GM (loll).
3. Calculate GM: If the loading calculation shows GM negative or very close to zero after applying free surface corrections, you have loll. If GM is clearly positive, you have list.
11. Damage Stability and Watertight Integrity
Damage stability refers to a vessel's ability to remain afloat and upright after flooding of one or more compartments due to hull breach, collision, or grounding. It is governed by subdivision requirements and the concept of the floodable length — the maximum length of a compartment that can be flooded without sinking the vessel.
The Lost Buoyancy Method
When a compartment is flooded, it loses its contribution to buoyancy. The vessel sinks to a new waterline where the remaining intact buoyancy equals the vessel's weight. The center of buoyancy shifts as the underwater volume changes. If the flooded compartment is off-center, the vessel will also develop an angle of heel. Damage stability calculations determine the final waterline, the heel angle, and whether adequate GM and freeboard remain after flooding.
Watertight Integrity
Watertight integrity is the condition in which all closures — hatches, watertight doors, portlights, and other openings — are secured so that flooding cannot spread between compartments. Maintaining watertight integrity at sea limits the extent of flooding in a casualty and preserves damage stability margins.
Downflooding angle is the angle of heel at which water begins to enter the vessel through non-watertight openings (ventilators, non-weathertight hatches, open doors). USCG stability criteria require that the downflooding angle must not be exceeded before adequate righting arm is developed. Vessels with low freeboard or large openings on the weather deck have small downflooding angles.
Passenger Vessel Stability Requirements
Passenger vessels under USCG jurisdiction must meet subdivision standards that ensure the vessel can survive flooding of specified compartments (the one- or two-compartment standard). These requirements are more stringent than cargo vessels because of the number of persons aboard and the limitations on rapid evacuation. Exam questions on passenger vessel stability often test knowledge of these special requirements and the associated stability criteria.
12. Load Line Marks (Plimsoll Mark) — Every Mark and What It Means
The Plimsoll mark is a system of maximum load line marks stamped and painted on the hull amidships of regulated vessels. It was introduced by Samuel Plimsoll in the 1870s to prevent overloading, which was causing widespread ship losses. Today it is governed by the International Load Line Convention and 46 CFR for U.S. vessels.
| Mark | Stands For | When Applied | Draft Relative to S |
|---|---|---|---|
| TF | Tropical Fresh Water | Tropical zone, fresh water ports | Highest (deepest loading allowed) |
| F | Fresh Water | Summer zone, fresh water | Above S by fresh water allowance |
| T | Tropical Salt Water | Tropical zone, salt water | Above S by 1/48 of summer draft |
| S | Summer Salt Water | Summer zone, salt water (baseline) | Reference mark |
| W | Winter Salt Water | Winter zone, salt water | Below S by 1/48 of summer draft |
| WNA | Winter North Atlantic | North Atlantic winter (north of 36N) | Lowest (least loading allowed) |
Fresh Water Allowance (FWA)
Fresh water is less dense than salt water, so a vessel loading in fresh water will be deeper in the water for the same weight. The fresh water allowance accounts for this: the vessel can load to a deeper draft in fresh water, because when it sails to sea and enters salt water, it will rise to the correct salt water load line.
FWA (inches) = Displacement / (4 x TPI)
Where displacement is in long tons and TPI (Tons Per Inch) is from the hydrostatic curves at the summer load line draft.
Dock Water Allowance (DWA) = FWA x (1.025 - SG of dock water) / 0.025
13. Trim Calculations — Moment to Change Trim
Trim is the difference between the after and forward drafts. A vessel trimmed by the stern has greater draft aft than forward, which is the normal operating condition for most displacement vessels. Trim affects propeller immersion, maneuverability, speed, and the accuracy of draft surveys.
Moment to Change Trim One Inch (MCT1in)
MCT1in (or MCTC in metric) is a hydrostatic parameter found in the stability booklet at each draft. It represents the longitudinal moment (in foot-long-tons) required to change the vessel's trim by one inch. To calculate the change in trim from a weight shift or addition, divide the trimming moment by MCT1in.
Worked Example: Trim Calculation
A vessel with LBP 180 ft has a 40 LT weight shifted 60 ft aft. MCT1in at current draft = 48 ft-LT/in. Distance from center of flotation to forward mark = 85 ft. Distance from center of flotation to aft mark = 95 ft.
Trimming Moment = 40 x 60 = 2,400 ft-LT (aft, increases stern draft)
Total Change in Trim = 2,400 / 48 = 50 inches
Change at Aft Mark = 50 x (95 / 180) = 26.4 inches (increase)
Change at Fwd Mark = 50 x (85 / 180) = 23.6 inches (decrease)
Aft draft increases by 26 inches; forward draft decreases by 24 inches.
Tons Per Inch Immersion (TPI)
TPI is the number of long tons required to change mean draft by one inch. It depends on the waterplane area at that draft: TPI equals (Aw times 1.025) divided by 420, where Aw is the waterplane area in square feet. TPI is used to calculate how much weight to add or remove to achieve a desired draft change, and to calculate fresh water allowance.
TPI = Aw x 1.025 / 420
Weight to add = TPI x desired draft change (in inches)
Example: TPI = 35. To increase mean draft by 3 inches, add 35 x 3 = 105 LT.
14. USCG Exam Question Patterns and Tips
The USCG stability exam tests both conceptual understanding and numerical calculation. Most stability questions fall into a handful of recurring patterns. Knowing these patterns lets you approach each question systematically.
Pattern 1: Identify Positive/Negative GM
Given KB, BM (or KM), and KG values, calculate GM and state whether the vessel is stable. Always: KM = KB + BM, then GM = KM - KG. If GM is negative, the vessel is in loll condition. Practice this calculation until it is automatic.
Pattern 2: Effect of Loading/Discharging on GM
A cargo is loaded or discharged; find the new KG and new GM. Build a loading table: existing displacement and moment, add or remove the item, find new total displacement and new total moment, calculate new KG, look up new KM at new draft, calculate new GM. Watch for free surface corrections if tank fill levels change.
Pattern 3: Reading the GZ Curve
Given a GZ curve diagram, identify: initial GM (from slope), angle of maximum GZ, angle of vanishing stability, and whether USCG criteria are met (GZ at least 0.2 ft at 30 degrees, positive through 40 degrees for most offshore vessels). Negative GZ region means capsize. Zero-crossing after initial rise is the AVS.
Pattern 4: List vs Loll Recognition and Correction
A scenario describes a vessel heeled to one side. The question asks the cause and correct action. Key differentiators: Was high weight loaded? (Loll likely.) Was weight loaded off-center? (List likely.) Is GM positive or negative? Always correct loll by lowering G; correct list by moving weight to the high side.
Pattern 5: Free Surface Effect Calculation
Given tank dimensions and liquid SG, calculate FSC and corrected GM. Remember: i = L x W^3 / 12 for rectangular tank. FSC = i x liquid density / (V x 1.025). Corrected GM = Solid GM - FSC. This pattern may also ask what happens if the tank is subdivided — recall that FSC is proportional to the cube of tank width.
Pattern 6: Trim Calculations
A weight is added, removed, or shifted longitudinally. Find the change in trim and the new drafts fore and aft. Steps: calculate trimming moment (weight times distance from F), divide by MCT1in, distribute the total trim change proportionally to the distance of each mark from the center of flotation F.
Pattern 7: Load Line Questions
Know the sequence of marks from highest (TF) to lowest (WNA). Know that FWA equals displacement divided by (4 x TPI). Know that the Tropical mark is higher than Summer by (Summer draft / 48). Load line violation occurs if the applicable mark is submerged when the vessel sails. These are typically straightforward recall questions.
Study Strategy for Stability Questions
- ●Master the formula chain: KM = KB + BM, GM = KM - KG. Every numerical stability problem begins here.
- ●Know the sign rule: positive GM means stable. Negative GM means loll. G above M is always dangerous.
- ●Memorize the load line sequence: TF, F, T, S, W, WNA from highest to lowest draft allowed.
- ●Free surface effect is always a reduction in stability. More slack tanks = worse stability. Wide tanks = much worse than narrow tanks.
- ●For loll, always lower G. Never move weight to the high side of a lolling vessel. This is a safety-critical exam answer.
Frequently Asked Questions
What is Archimedes' principle and how does it apply to vessel stability?
Archimedes' principle states that a floating body is buoyed up by a force equal to the weight of the fluid it displaces. For a vessel, buoyancy equals the weight of seawater in the volume of hull below the waterline. At equilibrium, buoyancy equals ship weight, and the center of buoyancy (B) and center of gravity (G) are on the same vertical line. Displacement in long tons equals underwater volume in cubic feet divided by 35 (seawater) or 36 (fresh water).
What is metacentric height GM and why is positive GM critical for safety?
GM is the vertical distance from the center of gravity (G) to the metacenter (M). Positive GM means M is above G — the vessel generates a righting moment when heeled and returns to upright. Negative GM means G is above M — the vessel has a capsizing moment and will settle at a loll angle or capsize. GM is calculated as KM minus KG, where KM equals KB plus BM (the metacentric radius, equal to waterplane moment of inertia divided by displacement volume). USCG exams provide KM tables; the student must calculate KG from the loading condition.
What is the difference between list and loll, and why does it matter on the USCG exam?
List is caused by an off-center weight with positive GM — the vessel has good righting ability but heels to the heavy side. Correction: move weight to the high side. Loll is caused by negative GM — G is above M. The vessel settles at a loll angle to one side without positive righting ability at zero heel. Correction: lower G by adding low ballast or removing high weights. Never move weight to the high side of a lolling vessel — this is dangerous and the opposite of the correct action for loll.
How does free surface effect reduce stability, and what causes it?
Free surface effect occurs when a partially filled tank allows liquid to shift when the vessel heels, effectively raising G and reducing GM. The free surface correction FSC equals (i times liquid SG) divided by (V times 1.025), where i is the second moment of area of the free surface (length times width cubed, divided by 12 for a rectangular tank). Because width is cubed, wide tanks cause dramatically more free surface effect than narrow tanks. Subdivision with centerline bulkheads reduces FSC to one-quarter per half (one-eighth the original per full width). Slack tanks are stability hazards — keep tanks either completely full or completely empty when possible.
What does the GZ stability curve show and what are the critical points on it?
The GZ curve plots the righting arm (horizontal distance between G and the vertical through B) against angle of heel. Initial slope equals GM times sin(1 degree) — steeper means larger GM and stiffer vessel. Peak GZ is the maximum righting arm and the angle at which it occurs. The angle of vanishing stability (AVS) is where GZ returns to zero — beyond this point the vessel capsizes. Area under the curve is dynamic stability. USCG regulations require minimum GZ of 0.2 ft at 30 degrees and positive GZ through 40 degrees (or beyond) for offshore vessels.
What is the wall-sided formula and when does the USCG exam test it?
The wall-sided formula gives GZ for vessels with vertical sides at moderate heel angles: GZ equals sin(theta) times (GM plus BM times tan-squared(theta) divided by 2). At small angles this reduces to GZ equals GM times sin(theta). For a lolling vessel with negative GM, the loll angle is found by setting GZ to zero and solving: tan(loll angle) equals the square root of (-2 times GM divided by BM). Exam questions may provide GM and BM values and ask you to calculate the loll angle, or to find GZ at a specified angle using the full wall-sided formula.
What are load line marks and what does each zone mean for loading?
Load lines (Plimsoll marks) are maximum loading draft marks on regulated vessels. From highest allowed draft to lowest: TF (Tropical Fresh), F (Fresh), T (Tropical Salt), S (Summer Salt — the reference mark), W (Winter Salt), WNA (Winter North Atlantic). Fresh water marks allow a deeper draft than salt water because fresh water is less dense — the vessel will rise to the correct salt water draft when it reaches sea. Fresh water allowance FWA equals displacement divided by (4 times TPI). Tropical marks allow slightly deeper loading than Summer by (Summer draft / 48 inches).
How do you calculate trim and what is the significance of trim in USCG exam problems?
Trim equals aft draft minus forward draft. Change in trim from a weight shift or addition equals trimming moment divided by MCT1in (Moment to Change Trim one inch, from the stability booklet). Trimming moment equals weight times the longitudinal distance from the center of flotation (F). The trim change is distributed to each draft mark proportionally: change at mark equals total trim change times (distance from F to mark divided by LBP). Shifts aft increase stern draft; shifts forward increase bow draft. Trim affects propeller immersion, speed, and draft survey accuracy.
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