Centers of Gravity, Buoyancy & the Metacenter
Every stability calculation begins with understanding where these points are and how they move. Know all six — the exam will test them individually and in combination.
K — Keel
The baseline reference point at the bottom of the vessel. All vertical measurements begin here. KG, KB, KM are all measured from K.
B — Center of Buoyancy
The centroid of the underwater volume of the hull. Buoyant force acts upward through B. B shifts to the low side when the vessel heels, generating the righting couple.
G — Center of Gravity
The point through which the entire weight of the vessel acts downward. G changes when weight is added, removed, or shifted. Lower G = better stability.
M — Metacenter
The point where the vertical line through the shifted center of buoyancy intersects the vessel's centerline (for small angles). If M is above G, the vessel is stable. If G rises above M, instability results.
GM — Metacentric Height
The vertical distance from G to M. Positive GM = stable. Negative GM = unstable (loll or capsize). GM = KM − KG = KB + BM − KG. The primary initial stability indicator.
GZ — Righting Arm
The horizontal distance between the lines of action of buoyancy (upward through B) and gravity (downward through G) when heeled. GZ > 0 = restoring moment. GZ = 0 = equilibrium or capsizing. Plotted against heel angle to create the stability curve.
Formulas You Must Know
These are the formulas that appear directly on USCG Master stability exams. Understand the variables and when to apply each.
Basic GM Formula
Exam FormulaKM is read from the vessel's hydrostatic tables at the current draft. KG is calculated from the loading condition (total moments / total displacement).
Metacentric Radius
I = second moment of area (inertia) of the waterplane about the centerline. V = displacement volume. Wider vessels have larger I and therefore larger BM and better initial stability.
Free Surface Correction
Exam Formulai = inertia of the free surface, ρ_L = density of the liquid in the tank, V = displacement volume, ρ_SW = density of seawater. Subtract FSC from solid GM to get corrected GM.
Corrected GM
Exam FormulaAlways apply the free surface correction when tanks are partially filled. The corrected GM is the value that determines actual stability — use this for all stability calculations.
Weight Shift (G movement)
Exam Formulaw = weight shifted, d = distance moved, W = total displacement. G moves in the same direction as the weight. For loading: W becomes W + w. For discharging: W becomes W − w.
Full GM Derivation — Step by Step
- 1Determine the vessel's mean draft from the loading condition.
- 2From hydrostatic tables at that draft, read KM (also called KG₀ or metacentric height above keel).
- 3Calculate KG = (Sum of all weight moments) / (Total displacement). Each item: moment = weight × vertical center (VCG).
- 4Calculate GM (solid) = KM − KG.
- 5Apply Free Surface Correction (FSC) for all slack tanks. Sum all FSC values from tank tables.
- 6GM (corrected) = GM (solid) − Total FSC. If negative, vessel is unstable.
The GZ Curve — Static Stability Curve
The GZ curve plots righting arm against angle of heel. It is the most comprehensive picture of a vessel's stability. Know every region and every critical point.
Initial Stability Zone
GZ increases approximately linearly. The slope at 0° = GM × sin(1°). This is the region governed by metacentric height. Exam: steep slope = large positive GM = stiff vessel.
Range of Positive Stability
GZ continues increasing, reaching its maximum at approximately 30–45° for well-designed vessels. Beyond initial stability, vessel shape (form stability) increasingly dominates over GM.
Maximum Righting Arm
The highest point of the GZ curve. IMO criteria require this to occur at an angle ≥ 25°. The magnitude of peak GZ determines the vessel's resistance to large-angle heeling.
Decreasing Positive Stability
GZ decreases as the deck edge submerges and form stability diminishes. The vessel remains stable (positive GZ) but with less restoring force. Flooding risk increases.
GZ = 0 — Capsize Point
Beyond AVS, the righting arm becomes negative — the vessel will capsize and not self-right. Well-designed vessels have AVS > 90°. If heeled beyond AVS by a wave, capsize is certain without external restoring force.
Positive Stability
A vessel has positive stability when GZ > 0 — the righting arm pushes the vessel back to upright. Three conditions for positive initial stability:
- ▸GM is positive (M is above G)
- ▸GZ increases with heel angle (up to the peak)
- ▸The vessel self-rights from any angle below AVS
Neutral & Negative Stability
These are dangerous conditions — know the difference:
- ▸Neutral: GM = 0. M coincides with G. Vessel heels to any angle without restoring force but doesn't capsize (yet).
- ▸Negative: GM < 0. G is above M. Vessel will heel to a loll angle or capsize. The GZ curve begins below zero.
List vs. Loll — The Critical Distinction
Confusing list and loll is dangerous. The corrective action for one can capsize a vessel suffering from the other. This distinction is heavily tested on the Master exam.
List
Cause
Off-center weight — G has shifted to one side (port or starboard). Transverse G is not on the centerline.
GM Status
Positive GM — vessel is stable, just asymmetric.
Behavior
Vessel heels consistently to the heavy side. If you rock the vessel, it returns to the same list angle. It resists being pushed further.
Correction
- ▸Move weight from the low (heavy) side to the high side
- ▸Add weight to the high side
- ▸Discharge weight from the low side
Loll
Cause
Negative GM — G is above M. The vessel is inherently unstable and seeks a heel angle where GZ returns to zero (the loll angle).
GM Status
Negative GM — vessel is unstable. Emergency condition.
Behavior
Vessel may flop from side to side (port loll to starboard loll). Pushing toward upright increases instability. Loll angle shifts randomly.
Correction — CRITICAL SEQUENCE
- 1.Fill lowest available tanks (port and starboard equally)
- 2.Empty high tanks to reduce top-heaviness
- 3.Reduce free surface — pump tanks to full or empty
- !NEVER transfer weight to the high side first — this temporarily worsens negative GM and may capsize the vessel.
Free Surface Effect (FSE)
Free surface effect is the single most important stability hazard for vessels with tanks. Understand it conceptually, mathematically, and operationally.
How FSE Works
When a partially-filled tank is present and the vessel heels, the liquid shifts to the low side. This shift moves the effective center of gravity upward and toward the low side — even though G itself doesn't physically move.
This produces a virtual rise in G — the vessel behaves as if its center of gravity has risen by the FSC amount. The corrected GM is reduced, sometimes to zero or negative.
The worst case is a wide, long tank filled to approximately 50% capacity — maximum inertia of the free surface. Subdividing a tank with a centerline wash bulkhead dramatically reduces FSE (by up to 75%).
What Increases FSE
- ▸Wider tanks — inertia (I) increases with the cube of tank width (b³/12)
- ▸Longer tanks — I increases linearly with tank length
- ▸50% full — maximum free surface area for most tank shapes
- ▸Denser liquids — seawater ballast causes more FSE than diesel
- ▸Multiple slack tanks — FSC values are cumulative and additive
What Reduces FSE
- ▸Fill tanks completely (100%) or empty them (0%)
- ▸Centerline wash bulkheads — divide wide tanks
- ▸Use lower-density liquids in upper tanks
- ▸Minimize number of slack tanks at any one time
| Tank Condition | Free Surface | Impact on GM | Risk Level |
|---|---|---|---|
| Tank 100% full | None | No FSE — liquid cannot shift | None |
| Tank 95–99% full | Minimal | Small FSE — air space allows minor movement | Low |
| Tank 50–75% full | Maximum | Largest inertia of free surface — worst FSE | High |
| Tank 25–50% full | Significant | Large FSE, weight lower but surface large | High |
| Tank nearly empty (5–25%) | Moderate | FSE reduces as surface narrows at bottom | Moderate |
| Tank 0% (completely empty) | None | No liquid present — no FSE | None |
Rule of thumb: A tank is either full, empty, or a stability hazard. Keep the number of slack tanks to an absolute minimum.
Loading, Discharging & Shifting — Effects on Stability
Every weight change moves G. Know the direction G moves for every scenario — the exam will test your ability to predict the effect on GM without calculating.
| Action | Effect on G | Effect on GM | Net Effect |
|---|---|---|---|
| Add low weight (below G) | ↓ G moves down | GM increases | Improves |
| Add high weight (above G) | ↑ G moves up | GM decreases | Worsens |
| Remove low weight (below G) | ↑ G moves up | GM decreases | Worsens |
| Remove high weight (above G) | ↓ G moves down | GM increases | Improves |
| Shift weight upward | ↑ G moves up | GM decreases | Worsens |
| Shift weight downward | ↓ G moves down | GM increases | Improves |
| Shift weight athwartship | → G moves to that side | Causes list | Creates heel |
| Partially fill a tank | ↑ G rises (virtual) | GM decreases (FSE) | Worsens |
Displacement, Deadweight & Tonnage
These terms are consistently confused on the USCG exam. Know which are weights (tons) and which are volumes (tonnage).
Displacement
Long Tons (LT)Total weight of the vessel including all contents. Equals weight of water displaced. 1 LT = 2,240 lbs. This is a weight measure.
Exam tip: Used in all stability calculations
Deadweight Tonnage (DWT)
Long Tons (LT)Total weight the vessel can carry: cargo, fuel, fresh water, crew, provisions, and stores. DWT = Loaded displacement − Light ship displacement.
Exam tip: Frequently tested — know DWT = Loaded − Light ship
Light Ship Displacement
Long Tons (LT)Vessel with no cargo, fuel, water, or provisions, but including all permanent equipment, machinery, and structure.
Exam tip: Starting point for loading calculations
Gross Tonnage (GT)
Volume (100 cu ft = 1 GT)Total enclosed volume of the vessel. Regulatory measure used for fees, manning requirements, and documentation. Not a weight.
Exam tip: GT vs displacement is a common distractor
Net Tonnage (NT)
Volume (100 cu ft = 1 NT)Cargo-carrying volume only (gross tonnage minus crew spaces, machinery, navigation spaces). Used for port dues and canal fees.
Exam tip: NT < GT always — know the relationship
Specific Gravity & Tank Weight Quick Reference
Common Specific Gravities
Weight Calculation Formula
Example: A ballast tank contains 2,100 cu ft of seawater (SG 1.025):
Weight = 2,100 × 1.025 / 35 = 61.5 LT
Fuel example: 1,400 cu ft diesel (SG 0.85):
Weight = 1,400 × 0.85 / 35 = 34.0 LT
Worked Stability Example — Step by Step
A typical USCG Master exam stability problem requires calculating new KG after loading, then finding corrected GM. Follow this sequence every time.
Problem Setup
A vessel's light ship displacement is 450 LT with KG = 8.2 ft. The following items are loaded:
| Item | Weight (LT) | VCG (ft) | Moment (ft-LT) |
|---|---|---|---|
| Light ship | 450 | 8.2 | 3690 |
| Cargo (hold) | 180 | 5.5 | 990 |
| Cargo (deck) | 60 | 14 | 840 |
| Fuel (slack — 60%) | 35 | 2.8 | 98 |
| Fresh water | 18 | 3.5 | 63 |
| Crew & provisions | 7 | 9 | 63 |
| TOTALS | 750 LT | — | 5,744 |
Step 1 — KG (solid):
KG = 5,744 / 750 = 7.66 ft
Step 2 — Read KM from hydrostatic tables at draft for 750 LT displacement:
KM (from tables) = 9.80 ft
Step 3 — GM (solid):
GM = KM − KG = 9.80 − 7.66 = 2.14 ft
Step 4 — Free Surface Correction (from tank tables for the slack fuel tank):
FSC = 0.22 ft (read from tank FSC table at 60% full)
Step 5 — Corrected GM:
GM (corrected) = 2.14 − 0.22 = 1.92 ft
Positive GM — vessel is stable. Meets minimum IMO criteria (> 0.150 m / 0.49 ft).
IMO & USCG Minimum Stability Criteria
These criteria (from 46 CFR Subchapters S and T) define minimum acceptable stability for passenger and cargo vessels. The USCG Master exam tests whether a vessel's GZ curve meets these standards.
Area 0° to 30°
Minimum energy to reach 30° — dynamic stability
Area 0° to 40°
Or to angle of flooding if less than 40°
Area 30° to 40°
Additional reserve stability beyond 30°
GZ at 30°
Minimum righting arm at 30° heel
Angle of max GZ
Peak of GZ curve must occur at or after 25°
Initial GM
Minimum metacentric height — initial stability floor
Exam Tips — Stability Section
GM = KM − KG, always
Every stability calculation on the exam ends with GM = KM − KG. KM comes from hydrostatic tables at the loaded draft. KG comes from your moment calculation. Know this formula cold.
List vs Loll: correction sequence
The USCG exam tests whether you apply the correct remedy. For loll (negative GM), always fill the lowest tanks equally on both sides first. Never move high-side ballast first — that worsens negative GM.
FSC is cumulative
Add up the free surface corrections from all slack tanks. If there are three slack tanks each with FSC = 0.3 ft, total FSC = 0.9 ft. This can turn a positive GM into a negative one.
Wide tanks are the worst
Tank inertia (I) varies with the cube of tank width. Doubling tank width multiplies FSE by 8. The exam tests whether you know that width matters far more than length or fill level.
Displacement vs. tonnage
Displacement and DWT are weights (long tons). GT and NT are volumes (100 cu ft = 1 ton). The exam uses both in the same question set. Never mix units.
AVS on the GZ curve
The angle of vanishing stability is where GZ returns to zero on the downward slope. Once past AVS, the vessel capsizes. Well-found vessels have AVS > 90°. The exam may ask you to read AVS directly from a curve diagram.
Frequently Asked Questions
What is metacentric height (GM) and why does it matter on the USCG exam?
Metacentric height (GM) is the vertical distance between the center of gravity (G) and the metacenter (M), measured in feet or meters. It is the primary indicator of initial stability. A positive GM (M above G) means the vessel is stable and will return to upright after heeling. A negative GM (G above M) means the vessel is unstable and will capsize or settle at a loll angle. The larger the positive GM, the stiffer the vessel — but excessive GM produces a short, violent roll period. The USCG exam frequently tests the formula: GM = KB + BM − KG, where K is the keel.
What is the difference between list and loll?
List is a permanent heel caused by an off-center weight — the center of gravity has shifted to one side. The vessel has positive GM but heels to the heavy side. Correcting list means moving weight back to the centerline or adding weight to the high side. Loll is caused by negative GM — the vessel's center of gravity is above the metacenter. The vessel will heel to a random side and remain there at a 'loll angle' where the righting arm (GZ) equals zero. Loll is dangerous and correcting it by moving weight to the high side can worsen the situation. The correct action for loll is to lower G by adding low ballast, removing high weights, or flooding lower tanks. These two conditions are tested together on USCG Master exams.
How does free surface effect reduce stability?
Free surface effect occurs when a tank or compartment is partially filled with liquid. When the vessel heels, the liquid shifts toward the low side, moving the effective center of gravity (G) upward and toward the low side — this is called a 'virtual rise of G.' The free surface correction (FSC) is subtracted from GM to give the corrected value: GM (corrected) = GM (solid) − FSC. The formula for FSC is: FSC = (i × density of liquid) / (V × density of seawater), where i is the second moment of area (inertia) of the free surface and V is the vessel's displacement volume. Free surface effect is most severe in wide, partially-filled tanks and can cause a stable vessel to become unstable. Slack tanks are the most dangerous — a tank is either full, empty, or a stability hazard.
What does the GZ curve show, and what are the key points tested on the Master exam?
The GZ curve (righting arm curve or static stability curve) plots the righting arm (GZ, in feet or meters) against the angle of heel. Key exam points: (1) The slope of the GZ curve at 0° equals sin(1°) × GM — a steeper initial slope means stiffer initial stability. (2) The angle of vanishing stability (AVS) is where GZ returns to zero — if heeled beyond this point, the vessel will capsize. (3) The area under the GZ curve represents the vessel's dynamic stability — energy required to capsize. (4) Maximum GZ (the peak of the curve) and the angle at which it occurs are both important. Exam questions ask you to identify positive, zero, and negative righting arm regions from a curve diagram, and to determine whether a vessel has adequate stability based on the curve shape.
How do you calculate the shift in the center of gravity when loading or discharging cargo?
When a weight is added, removed, or shifted, the vessel's center of gravity (G) moves. The formulas are: (1) Loading: GG1 = (w × d) / (W + w), where w is the added weight, W is the original displacement, and d is the distance between G and the center of the new weight. G moves toward the added weight. (2) Discharging: GG1 = (w × d) / (W − w). G moves away from the discharged weight. (3) Shifting: GG1 = (w × d) / W, where d is the distance the weight is moved. G moves in the same direction as the weight. These calculations appear on USCG Master stability exams in conjunction with finding the new KG and then the new GM using stability tables.
What is the relationship between KB, BM, and KM?
These three points all lie on the vessel's centerline and define stability geometry. K is the keel. B is the center of buoyancy (centroid of the underwater volume). M is the metacenter — the point where the vertical line through the shifted center of buoyancy intersects the vessel's centerline when heeled at a small angle. KB is the vertical distance from keel to center of buoyancy — roughly half the draft for a box-shaped vessel. BM is the metacentric radius, calculated as BM = I / V, where I is the second moment of area of the waterplane and V is the displacement volume. BM is larger for wider vessels. KM = KB + BM. From KM you subtract KG (keel to center of gravity) to get GM. On stability tables, KM is given at each draft, and KG is calculated from loading. GM = KM − KG.
What is deadweight, and how does it differ from displacement?
Displacement is the total weight of the vessel and everything aboard — it equals the weight of water displaced by the hull. It is expressed in long tons (LT) in U.S. practice, where 1 LT = 2,240 lbs. Deadweight tonnage (DWT) is the total weight a vessel can carry: cargo, fuel, water, provisions, crew, and stores. DWT = Loaded displacement − Light ship displacement. Light ship is the vessel with no cargo, fuel, water, or provisions but with all permanent equipment. Net tonnage is a regulatory measure of cargo-carrying capacity used for port fees. Gross tonnage measures total enclosed volume (1 GT = 100 cubic feet). These are commonly confused on the USCG exam — displacement and DWT are weights; gross and net tonnage are volumes.
How do you use tank sounding tables and specific gravity to calculate tank weights?
Tank sounding (or ullage) tables give the volume of liquid in a tank at a given sounding depth. To find weight: Weight (LT) = Volume (cu ft) × Specific Gravity (SG) / 35. Seawater has SG of 1.025 (35 cu ft per LT). Fresh water SG = 1.000 (36 cu ft per LT). Diesel fuel SG ≈ 0.85. Lube oil SG ≈ 0.90. Example: A fuel tank contains 1,400 cu ft of diesel (SG 0.85). Weight = 1,400 × 0.85 / 35 = 34 LT. Always confirm whether the problem uses long tons (2,240 lbs) or short tons (2,000 lbs) — USCG stability problems use long tons unless otherwise stated. The vertical center of the tank contents (vcg) from the sounding table is added to KG calculations.
What is the angle of loll and how should it be corrected?
The angle of loll is the angle at which a vessel with negative GM settles on one side. At this angle, the righting arm GZ is exactly zero — there is no net restoring force. If heeled further from the loll angle, GZ becomes negative and the vessel will capsize. Correcting loll: (1) Lower G by adding ballast to the lowest accessible tanks — both port and starboard equally to avoid worsening the heel. (2) Never move ballast or weights to the high side first — this temporarily worsens the negative GM condition and may capsize the vessel. (3) Empty high tanks and fill lower tanks. (4) Reduce free surface by pumping tanks either full or completely empty. The USCG exam tests the correct sequence: fill low tanks equally on both sides, never transfer high-side ballast first.
What are the IMO and USCG minimum stability criteria for small passenger vessels?
Under 46 CFR Subchapter T and S, small passenger vessels must meet minimum stability standards. Key criteria tested: (1) The area under the GZ curve from 0° to 30° shall not be less than 0.055 meter-radians. (2) The area under the GZ curve from 0° to 40° (or to the angle of flooding, whichever is less) shall not be less than 0.090 meter-radians. (3) The area under the GZ curve between 30° and 40° shall not be less than 0.030 meter-radians. (4) GZ at 30° shall be at least 0.200 meters. (5) The maximum GZ shall occur at an angle of not less than 25°. (6) The initial GM shall not be less than 0.150 meters. These criteria ensure the vessel has adequate range of stability and dynamic stability to survive wind gusts and wave action.
Related Study Guides
Deck General Safety
Fire classes, PFDs, EPIRB, MARPOL, and distress signals for the USCG exam.
Navigation Rules (COLREGS)
Lights, shapes, sound signals, and rules of the road — required for all license levels.
Man Overboard Procedures
Williamson Turn, Anderson Turn, Pan-Pan, hypothermia, and victim recovery.
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