GATE Naval Architecture: five hydrostatics numerical patterns worth memorising

Hydrostatics is the most reliable scoring ground in the GATE Naval Architecture paper, because the questions repeat in a small number of shapes. If you can classify a question in the first ten seconds, you have converted a four-minute problem into a ninety-second one — and bought that time back for the questions that genuinely need thought.

Here are the five patterns worth drilling until the setup is automatic.

Pattern 1 — Change of draught from added or removed weight (TPC)

How to spot it: a small weight is loaded or discharged and you are asked for the new draught or the sinkage.

The tool: tonnes per centimetre immersion.

TPC = A_w · ρ / 100    (A_w in m², ρ in t/m³, TPC in t/cm)

Then sinkage in centimetres is simply w / TPC.

Where marks are lost: forgetting that TPC assumes the waterplane area does not change appreciably, which is only valid for small weights. If the weight is large, the question wants you to work from the hydrostatic curves instead. Also watch the density — TPC in dock water is not TPC in salt water.

Pattern 2 — Change of trim (MCT 1 cm)

How to spot it: a weight is shifted longitudinally, or loaded away from the centre of flotation, and you are asked for the new draughts forward and aft.

The tools:

Trimming moment = w · d   →   Change of trim = w · d / MCT₁_cm

Then split that trim between the ends about the centre of flotation, not about midships:

δT_aft = (change of trim) × L_{F, aft} / L     δT_fwd = (change of trim) × L_{F, fwd} / L

Where marks are lost: almost always in two places — dividing the trim equally between the ends when F is not amidships, and forgetting that a weight loaded somewhere other than at F causes both parallel sinkage (Pattern 1) and trim (Pattern 2). Do them as two separate effects and add.

Pattern 3 — Free surface effect

How to spot it: a slack tank, a partially filled tank, or a question about why GM fell after ballasting.

The tool: the virtual rise of G.

FSC = i · ρ_liquid / Δ    where i = l·b³/12 for a rectangular tank

Subtract it: GM_fluid = GM_solid − FSC.

The insight the examiner is testing: the loss depends on the cube of the tank breadth, and it does not depend on how much liquid is in the tank — only on the free surface. That is why one wide slack tank is far worse than several narrow ones of the same total volume, and why longitudinal subdivision is the fix. A question that gives you a tank subdivided into n equal-width compartments is asking you to notice the loss falls by a factor of n².

Pattern 4 — Angle of list, and GZ at small angles

How to spot it: a weight is shifted transversely, or loaded off the centreline, and you are asked for the resulting list.

The tool: for small angles,

tan θ = (w · d) / (Δ · GM)

and the righting lever is

GZ = GM · sin θ    (small angles only)

Where marks are lost: using GZ = GM · sin θ beyond about 10–15°, where it stops being valid because the metacentre moves. Beyond that you need the cross curves of stability, or the wall-sided formula:

GZ = sin θ · (GM + ½ · BM · tan² θ)

Also: if the question mentions a slack tank and asks for list, apply the free surface correction to GM first. Questions combining Patterns 3 and 4 are extremely common precisely because candidates handle each one correctly in isolation and forget to chain them.

Pattern 5 — Change of water density (dock water allowance)

How to spot it: a vessel moves from salt water to river or dock water, or vice versa, and you are asked for the change in draught.

The principle: displacement mass is unchanged, so the volume of displacement must change inversely with density.

Δ = ρ · ∇   →   ∇₂ = ∇₁ · ρ₁ / ρ₂

The standard fresh water allowance is

FWA (mm) = Δ / (4 · TPC)

and for an intermediate density, the dock water allowance scales it:

DWA = FWA × (1025 − ρ_DW) / 25    (ρ in kg/m³)

Where marks are lost: sign errors. Moving into less dense water means the vessel sinks deeper. Say that out loud before you compute, and it will catch a wrong sign every time.

A checklist for the exam itself

1. Classify first. Read the question and name the pattern before writing anything. Ten seconds spent here saves minutes.
2. Write the given values down with units. Density in t/m³ or kg/m³, and stay consistent. More marks are lost in this paper to unit slips than to conceptual errors.
3. Check whether effects need chaining. Sinkage plus trim. Free surface plus list. The compound questions are where the paper separates candidates.
4. Sanity-check the direction. Does the vessel sink or rise? Trim by the head or the stern? List to port or starboard? A quick physical check catches sign errors that the algebra will not.
5. Do not chase one question. Flag it, move on, come back. This is a pacing exam as much as a knowledge exam.

How to practise these

Not by reading worked examples — by producing them. Take one pattern, do eight or ten problems on it in a single sitting until the setup is automatic, then move to the next. Once all five are individually fluent, practise them mixed and timed, because classification speed is a separate skill from solving speed and it only develops under randomised conditions.

Half a minute a question is the pace the paper actually demands. Practise at that pace, and exam day feels like a repeat rather than a first attempt.