NATARAJAThe weather criterion & heel in a turn
Learning outcomes & permitted supports
- Tell the weather criterion as its story: steady wind → φ0, windward roll-back, then the gust at 1.5× the lever
- Compute the wind heeling lever lw1 = P·A·z/(1000·g·Δ) and say where every symbol lives on the ship
- Read areas a and b as ENERGY, and judge b ≥ a on the curve — with the Code's φ0 ≤ 16° side-check
- Show why the downflooding angle can fail a ship whose GZ curve never changed: the openings decide
- Work the heel in a turn — tan θ = v²(KG − d/2)/(g·R·GM) — outward, and quadratic in speed
Supports in this lab: Calculator. Box-form exact GZ; the roll-back angle is an input (aboard it comes from the Code's k·X1·X2·√(r·s) table) — the AREAS are the physics, integrated twice by independent routes in golden. 961/961.
Glossary: weather criterion · wind heeling lever · heel in a turn · IMO criteria · dynamical stability
The storm, told as areas
The IS Code's severe wind and rolling criterion is a short story in three sentences. A steady beam wind (504 Pa on your windage) presses with lever lw1 and she settles at φ0, where the GZ curve meets it. A deep sea rolls her windward of that by the roll-back angle. Then the gust arrives — lever lw2 = 1.5·lw1 — and from that windward edge it owns the red energy a; everything green under her curve to the cap is the energy b she can answer with. She passes when b ≥ a.
Model stated: the box hull's exact GZ; the roll-back angle is an input here — aboard it comes from the Code's table of k·X1·X2·√(r·s). The AREAS are the physics, and the golden suite integrates them twice by independent routes.
The openings decide
Area b ends at the FIRST of: the second intercept, 50°, or the downflooding angle — where a hatch, vent or door would start taking the sea. Drag it down and watch a ship with honest GM fail the criterion without her curve changing at all: the reserve was there, but the openings gave it away. This is why weathertight closures are a stability item, not housekeeping.
Right now the cap IS the downflooding angle (40.0°) — area b stops where the sea would come in.
Heel in a turn — the wheel is a heeling moment
Rudder over: the hull carves a circle of radius R and the centripetal push acts low, at the underwater body, while the mass rides high at G. The couple heels her OUTWARD — tan θ = v²·(KG − d/2)/(g·R·GM). Note what sits in the denominator: a tender ship (small GM) heels hard, and speed enters squared — the same law as squat, and the same cure.
At 14.0 kn in a 450 m circle with GM 0.60 m she heels 4.4° outward. Halve the speed and the tangent quarters.
| Steady heel φ0 · Code side-check ≤ 16° | 5.0° · OK |
|---|---|
| Area a (gust, windward) | 49.6 mm·rad |
| Area b (reserve to the cap) | 239.2 mm·rad |
| Verdict — b ≥ a | PASSES the weather criterion |
| Turn: heel outward | 4.4° |