Buoyancy and stability of floating bodies

Why it matters

Buoyancy explains why balloons and airships rise, how a seaplane float or a ship stays up, and why a hydrometer reads density. Stability decides whether a floating body that is tipped by a gust or a wave rights itself or capsizes. The metacentric height computed here is also the starting point for roll behaviour of ships, floats and flying boats.

Key ideas

Archimedes' principle. A body wholly or partly immersed in a fluid at rest feels an upward force equal to the weight of the fluid it displaces. It follows directly from hydrostatics: pressure on the lower surfaces is larger than on the upper surfaces, and the net vertical force equals ρ_f·g·V_displaced. The force acts through the centre of buoyancy B, which is the centroid of the displaced volume, not the centre of gravity of the body.

Floating and sinking.

• Floating body: weight W equals buoyant force, so the displaced volume adjusts. A body of uniform density ρ_b floats with a fraction ρ_b/ρ_f of its volume submerged.
• Fully submerged body: if W > F_B it sinks; if W < F_B it rises; if equal it is neutrally buoyant (submarines and fish control this).
• Lighter-than-air craft: the gross lift of a gas envelope is (ρ_air − ρ_gas)·g·V; payload is gross lift minus envelope and structure weight. Lift falls with altitude because air density falls.

Stability of a fully submerged body. B and the centre of gravity G are fixed relative to the body. It is stable if G lies below B (a tilt creates a restoring couple), neutral if they coincide, and unstable if G is above B.

Stability of a floating body. When a floating body heels through a small angle, the shape of the displaced volume changes and B moves sideways towards the side that went deeper. The vertical through the new B cuts the original centreline at the metacentre M. The distance GM is the metacentric height:

• GM > 0 (M above G): restoring couple, stable.
• GM = 0: neutral.
• GM < 0 (M below G): overturning couple, unstable. So a floating body can be stable even with G above B, provided the waterplane is wide enough. This is why wide, shallow hulls are stable and tall narrow ones are not.

Metacentric radius BM depends on the waterplane: BM = I/V, where I is the second moment of area of the waterplane about the tilt axis and V the displaced volume. Tilting about the axis with the smallest I is the critical case, so always use the minimum I.

Trade-off. A large GM gives a stiff body with a short, jerky roll period (uncomfortable, high structural loads); a small positive GM gives a gentle, slow roll but little reserve against capsizing. The roll period for small oscillations is T = 2πk/√(g·GM), where k is the radius of gyration about the roll axis.

Assumptions and limits. Results are for fluid at rest, small heel angles (the metacentre is fixed only for small angles), and no free liquid surfaces inside the body (sloshing liquid effectively raises G and reduces GM).

Formulas

F_B = ρ_f·g·V_d

• F_B: buoyant force (N); ρ_f: fluid density (kg/m³); V_d: displaced volume (m³). Acts at the centroid of V_d.

V_d / V = ρ_b / ρ_f

• Fraction submerged of a uniform floating body of density ρ_b.

L_gross = (ρ_air − ρ_gas)·g·V

• Gross lift of a gas envelope of volume V (N).

BM = I / V_d

• BM: metacentric radius (m); I: minimum second moment of area of the waterplane about the tilt axis (m⁴). Rectangle of length L along the tilt axis and beam b: I = L·b³/12. Circle of diameter D: I = π·D⁴/64.

GM = BM − BG (G above B) or GM = BM + GB (G below B)

GM = w·x / (W·tanθ)

• Experimental (inclining test): moving a mass of weight w through a distance x across the deck tilts the body of total weight W (including w) by angle θ.

T = 2π·k / √(g·GM)

• Roll period (s); k: radius of gyration about the roll axis (m).

Worked examples

Example 1 (standard): floating block. Given: a wooden block 2 m long, 1 m wide and 0.5 m high, relative density 0.6, floats in water with the 0.5 m side vertical. Check stability for roll about the long axis.

1. Draft: h = 0.6 × 0.5 = 0.30 m. Displaced volume V_d = 2 × 1 × 0.30 = 0.60 m³.
2. B is at h/2 = 0.15 m above the bottom; G is at 0.25 m. BG = 0.10 m.
3. Waterplane I about the long axis: I = L·b³/12 = 2 × 1³/12 = 0.1667 m⁴.
4. BM = I/V_d = 0.1667/0.60 = 0.2778 m.
5. GM = BM − BG = 0.2778 − 0.10 = 0.178 m > 0. Answer: GM ≈ 0.178 m, stable.

Example 2 (standard): helium balloon. Given: envelope volume 1000 m³, ρ_air = 1.225 kg/m³, ρ_He = 0.169 kg/m³.

1. L_gross = (ρ_air − ρ_gas)·g·V = (1.225 − 0.169) × 9.81 × 1000.
2. = 1.056 × 9810 = 10 359 N. Answer: about 10.4 kN (about 1056 kg) before subtracting envelope and payload.

Example 3 (GATE level): inclining experiment. Given: a vessel of total weight 2000 kN; a 10 kN weight moved 6 m across the deck tilts it by 2°.

1. GM = w·x/(W·tanθ) = 10 × 6/(2000 × tan 2°).
2. tan 2° = 0.03492, so GM = 60/69.84 = 0.859 m.
3. With k = 3 m, roll period T = 2π·k/√(g·GM) = 2π × 3/√(9.81 × 0.859) = 6.50 s. Answer: GM ≈ 0.86 m, T ≈ 6.5 s.

Example 4 (GATE level): floating cylinder. A solid cylinder of diameter D, height H and relative density s floats upright. Draft h = sH, BM = (πD⁴/64)/(πD²h/4) = D²/(16sH), BG = H(1 − s)/2. Stability needs D²/(16sH) > H(1 − s)/2, i.e. D/H > √(8s(1 − s)). For s = 0.8 this gives D/H > 1.131.

Common mistakes

• Putting the buoyant force through G instead of through B, the centroid of the displaced volume.
• Using the total volume of the body instead of the displaced volume in BM = I/V_d.
• Using I about the wrong axis. The critical axis is the one with the smallest waterplane I (usually roll, about the long axis).
• Using the base area instead of the waterplane area for I.
• Applying the submerged-body rule (G below B) to floating bodies. Floating bodies can be stable with G above B.
• Forgetting that a balloon's lift uses the difference of densities and decreases with altitude.

For GATE AE

Expect numericals on fraction submerged, apparent weight in a liquid, balloon lift at a given altitude (combine with the ISA density), metacentric height of rectangular or cylindrical floats, the inclining experiment, and conceptual questions on the conditions for stable, neutral and unstable equilibrium. Practise locating B and G from geometry and choosing the correct waterplane second moment.

Quick check

1. An iceberg (ρ = 917 kg/m³) floats in sea water (ρ = 1025 kg/m³). What fraction is above the water?
2. A body weighs 500 N in air and 300 N fully immersed in water. What is its volume?
3. A floating body has M below G. Is it stable?
4. What happens to a helium balloon's gross lift as it climbs?

Answers: 1. About 10.5 % 2. 200/9810 ≈ 0.0204 m³ 3. No, it is unstable 4. It decreases because air density decreases