Center of gravity (KG)¶
Every stability calculation in CargoForge-C starts from one number: KG, the
height of the ship's combined centre of gravity above the keel. This lesson
explains what a centre of gravity is, how it is computed from first principles
using moments, why the vertical component (KG) is the one that dominates safety
decisions, and how perform_analysis in src/analysis.c translates those
principles into the code that runs on every voyage plan.
The mental model 🧠¶
Balance a loaded tray on one fingertip. The spot where it sits dead level — tipping toward neither the heavy teapot nor the empty cup — is the centre of gravity.
A ship's CG is that same balance point for thousands of tonnes. CargoForge-C finds it exactly the way you'd find the tray's: take each item's weight times its position, add them all up, and divide by the total weight — a weighted average (each weight × distance term is called a moment). perform_analysis does this for the lightship plus every placed cargo item, in all three axes at once.
KG is just the vertical coordinate of that balance point: how high it sits above the keel. And height is what bites. Slide one heavy crate from the hold up onto the deck and the teapot moves toward the rim of the tray — KG climbs, and the whole thing gets twitchy. Because stability is a tug-of-war between KG and the metacentre (Lesson 23), every metre you let KG rise is a metre of safety you spend.
What this actually means (plain English)¶
No jargon — here's what the ideas in this lesson actually mean, and why they matter.
- Centre of gravity = "the single point where the whole ship's weight effectively acts" — not a physical object you can touch, but a calculated balance point that moves every time you load or shift a cargo item.
- Moment = "weight multiplied by height (or distance)" — it captures how much a heavy object at a given position "pulls" the balance point; a 20 t container high on deck contributes far more to the vertical moment than the same container sitting on the keel, because its arm (
cz) is larger inperform_analysis's accumulation loop. - KG = "how high above the keel the ship's combined centre of gravity sits, in metres" — it is the single number that most directly controls whether the ship is safe, because it appears with a minus sign in
GM = KB + BM − KG; every extra metre of KG shrinks GM by exactly one metre. - GM (metacentric height) = "the gap between where gravity pulls and where the hull's buoyancy pushes back" —
perform_analysiscomputes it asr.gm = r.kb + r.bm - r.kg; when that gap reaches zero the ship has no self-righting ability and will capsize. cz=pos_z + dimensions[2] / 2= "the arm used for each cargo box's vertical moment" — because a box's centre of gravity is at its geometric middle, not its bottom face; placing a 2 m box at z = 0 gives an arm of 1 m, not 0.- Free-surface correction = "a penalty subtracted from GM to account for liquid sloshing in tanks" —
perform_analysisapplies it asr.gm_corrected = r.gm - r.free_surface_correction, and every downstream IMO check uses the corrected value, never the raw one.
Why it matters: if KG is computed incorrectly — wrong arm, skipped lightship seed, or an unplaced item sneaking into the loop — the GM the code reports will be wrong, and a ship that looks stable on screen may capsize at sea; the pos_x < 0 guard in perform_analysis exists precisely to prevent unplaced cargo from corrupting this number.
What is a centre of gravity?¶
The centre of gravity (CG) of a system of objects is the single point where all the system's weight can be considered to act. It is a weighted average: objects that are heavier pull the CG toward themselves; objects that are lighter have less influence.
For a ship, the system has two major contributors:
- The lightship — the hull, machinery, and fittings, whose weight and CG are measured once during a ship's inclining experiment and recorded.
- The cargo — every item placed aboard, each with its own weight and position.
Tanks of ballast water and fuel are a third contributor; we will touch on them at the end of this lesson.
Moments: the arithmetic of "pulling"¶
To find the CG of a collection of objects you cannot simply average their positions — a 1 000 t container at one end of the ship dominates a 10 t crate at the other end. The correct operation is the moment:
where the arm is the distance from the object's position to a chosen reference point (the origin).
The CG is then:
This formula says: sum all the individual moments, then divide by the total weight. The result is the arm of the equivalent single weight.
Why moments work
Imagine a see-saw balanced at its centre. If you place a 2 kg weight 1 m to the left, its moment is 2 N·m counterclockwise. A 1 kg weight 2 m to the right has moment 2 N·m clockwise — they balance. The pivot point is exactly where the formula above would put the combined CG.
Three axes, one focus¶
A ship's CG has three components:
| Symbol | Axis | Reference |
|---|---|---|
| LCG | Longitudinal (fore–aft) | Midship or aft perpendicular |
| TCG | Transverse (port–starboard) | Centreline |
| KG | Vertical | Keel (K) |
For intact-stability purposes, KG dominates. Here is why.
The ship's righting ability depends on the gap between G (centre of gravity) and M (the transverse metacentre, a geometric property of the hull). That gap is:
If KG rises — because heavy cargo is loaded high — GM shrinks. When GM reaches zero the ship has no initial restoring force. If GM goes negative the ship capsizes. Nothing in the formula involves LCG or TCG in the same direct way; transverse offset causes heel, which can be corrected, but a high KG erodes the fundamental righting lever.
How CargoForge-C computes KG¶
Open src/analysis.c. The computation unfolds inside perform_analysis in a
single pass over the cargo array.
Step 1 — seed with the lightship moment¶
Before looping over any cargo, the code seeds the vertical moment accumulator with the lightship's known values:
/* from src/analysis.c, perform_analysis() */
float vertical_moment = ship->lightship_weight * ship->lightship_kg;
ship->lightship_weight is in kilograms (converted from tonnes by the parser).
ship->lightship_kg is the height of the lightship CG above the keel, in
metres. Their product is the lightship's contribution to the total vertical
moment, in kg·m.
Step 2 — accumulate cargo moments¶
The loop visits every cargo item. Unplaced items (those with pos_x < 0) are
skipped — they have not been assigned a physical location and therefore cannot
contribute to the moment.
/* from src/analysis.c, perform_analysis() */
for (int i = 0; i < ship->cargo_count; i++) {
const Cargo *c = &ship->cargo[i];
if (c->pos_x < 0) continue;
r.placed_item_count++;
r.total_cargo_weight_kg += c->weight;
float cx = c->pos_x + c->dimensions[0] / 2.0f;
float cy = c->pos_y + c->dimensions[1] / 2.0f;
float cz = c->pos_z + c->dimensions[2] / 2.0f;
moment_x += c->weight * cx;
moment_y += c->weight * cy;
vertical_moment += c->weight * cz;
lcg_moment += c->weight * (cx - ship->length / 2.0f);
}
Notice cz: the arm for the vertical moment is not pos_z alone, but
pos_z + dimensions[2] / 2.0f. A cargo item is a box; its centre of gravity
is at its geometric centre, which sits half a height above its bottom face.
The bottom face is at pos_z, so the centre is at pos_z + height/2.
For a 2 m × 2 m × 2 m box placed with its bottom at z = 0, the centre is at
z = 1 m — and its vertical moment contribution is weight × 1.0, not
weight × 0.
A worked moment
Suppose a 20 t container (20 000 kg) with height 2.6 m is placed so that its base sits at z = 0 m (directly on the keel deck).
- Arm = 0 + 2.6 / 2 = 1.3 m
- Moment = 20 000 × 1.3 = 26 000 kg·m
Another identical container stacked on top of it has its base at z = 2.6 m:
- Arm = 2.6 + 1.3 = 3.9 m
- Moment = 20 000 × 3.9 = 78 000 kg·m
Stacking doubles the height of the upper box's CG but triples its moment. This is why high stowage rapidly erodes GM.
Step 3 — add tank moments¶
If tanks are configured, their weight and vertical moment are folded in after the cargo loop:
/* from src/analysis.c, perform_analysis() */
if (ship->tanks && ship->tanks->count > 0) {
tank_weight_t = calculate_tank_weight(ship->tanks);
displacement_kg += tank_weight_t * 1000.0f;
vertical_moment += calculate_tank_vertical_moment(ship->tanks) * 1000.0f;
}
calculate_tank_vertical_moment (in tanks.c) returns a moment in tonne·m;
multiplying by 1 000 converts it to kg·m for consistency.
Step 4 — divide to get KG¶
After the accumulation is complete and the overweight check passes, one line computes KG:
This is the formula \(KG = \Sigma(w_i \times z_i) / \Sigma w_i\) evaluated over lightship + cargo + tanks in a single division.
From KG to GM¶
With KG in hand, perform_analysis proceeds to hydrostatics to obtain KB and
BM (covered in Lesson 22), then applies the fundamental stability equation:
The free-surface correction (Lesson 23) is subtracted next:
All downstream criteria — the six IMO thresholds, trim, heel, GZ curve — use
gm_corrected, never the uncorrected r.gm. KG is the single value that
makes or breaks the chain.
Why a high KG is dangerous¶
Consider what happens as KG rises while KB and BM remain fixed (same draft, same hull):
Every extra metre of KG subtracts directly from GM. Below GM = 0 the ship is unstable at rest: any small disturbance produces a heeling moment that the ship cannot oppose, and it capsizes without external help.
Even before GM reaches zero, a low positive GM means: - GZ at 30° falls below the IMO minimum of 0.20 m. - The area under the GZ curve (energy stored as righting work) shrinks. - A breaking wave, a shifted cargo load, or a flooding compartment can push the ship past its range of positive stability.
The console report produced by print_loading_plan shows this explicitly:
The label "CRITICAL - Too tender" appears when gm_corrected < 0.3f — a
value chosen to warn operators well before the IMO hard floor of 0.15 m.
Recap¶
- The centre of gravity is a weighted average of positions, computed via moments (weight × arm).
- CargoForge-C seeds the vertical moment with the lightship contribution, then
accumulates
weight × (pos_z + height/2)for every placed cargo item. - KG is the vertical CG measured from the keel; it is the single most influential number in the stability chain because it appears directly in \(GM = KB + BM - KG\).
- A high KG reduces GM; when GM reaches zero the ship has no self-righting ability and capsizes.
perform_analysisinsrc/analysis.ccomputes KG in one line —r.kg = vertical_moment / displacement_kg— after a single loop that accumulates moments from lightship, cargo, and tanks.- The corrected GM (
gm_corrected = gm - free_surface_correction) is used for all IMO checks and downstream calculations; KG is its primary driver.
Check yourself¶
Why is a cargo item's vertical moment arm pos_z + dimensions[2]/2, not just pos_z?
A box's centre of gravity sits at its geometric middle, not its bottom face. The bottom is at pos_z; the centre — where the weight effectively acts — is half a height higher.
If you stack an identical container directly on top of another, does its vertical moment double?
More than double. The upper container's centre of gravity roughly triples in height relative to the keel (from 1.3 m to 3.9 m in the worked example), so its moment contribution triples too — which is exactly why high stowage erodes GM so quickly.