Buoyancy and Flotation - Cheat Sheet
Your last-minute revision companion for Buoyancy and Flotation (Hydraulics & Fluid Mechanics). All formulas, stability criteria, and common pitfalls in one place.
Archimedes' Principle & Buoyant Force
Formulas
F_B = γ_fluid × V_displaced
Meaning
F_B = buoyant force (N); γ_fluid = specific weight of fluid (N/m³); V_displaced = volume of fluid displaced (m³)
Watch out
For floating bodies, V_displaced ≠ body volume; use only the SUBMERGED volume. For fully submerged, use the entire object volume.
When to use
Any submerged or floating body problem — this is your starting point.
V_displaced = W / γ_fluid
Meaning
W = weight of body (N); γ_fluid = specific weight of fluid (N/m³); V_displaced = submerged volume (m³)
Watch out
This comes from F_B = W at equilibrium. Only valid for FLOATING bodies in equilibrium, not submerged ones.
When to use
Finding the volume of fluid displaced when you know the body's weight (floating equilibrium).
Common values
Value
9.81 kN/m³ or 9810 N/m³
Symbol
γ_w
Quantity
Specific weight of water (fresh)
Value
10.05 kN/m³
Symbol
γ_sw
Quantity
Specific weight of seawater
Value
1.025 (approx. 1.03)
Symbol
s_seawater
Quantity
Specific gravity of seawater
Important facts
- Archimedes' principle applies to ANY fluid (water, seawater, oil, air).
- A body floats if its average specific gravity < fluid specific gravity.
- Apparent weight (in fluid) = actual weight − buoyant force.
- γ_water ≈ 9.81 kN/m³ (at 4°C, standard condition); γ_seawater ≈ 10.05 kN/m³ (SG ≈ 1.03).
Key definitions
Buoyant Force
Example
A 0.05 m³ object fully submerged in water experiences F_B = 9.81 × 0.05 = 0.4905 kN upward.
Upward force exerted by a fluid on a submerged or floating object, equal to the weight of the displaced fluid.
Displaced Volume
Example
A block floating at half-height displaces half its geometric volume.
For floating bodies: the volume of fluid pushed aside (= submerged portion only). For fully submerged: the entire object volume.
Diagrams to know
- Submerged object with buoyant force arrow pointing up, weight arrow pointing down.
- Floating block showing draft depth (d), waterline, displaced volume shaded.
Flotation & Draft Calculation
Formulas
d = V_displaced / A
Meaning
d = draft or depth submerged (m); V_displaced = submerged volume (m³); A = plan area (horizontal cross-section, m²)
Watch out
A must be the WATERLINE area (top horizontal area of the submerged part), NOT the vertical cross-section. For a rectangular barge, A = L × B.
When to use
Finding how deep a prismatic (uniform cross-section) floating body sits in water.
d = s × h (for homogeneous floating block)
Meaning
s = specific gravity of body; h = total height of body; d = draft (depth submerged)
Watch out
Only valid for homogeneous (uniform density) bodies. Does NOT work if weight is concentrated at one end.
When to use
Quick check: a uniform block of SG 0.6 floats with 60% submerged.
W = s × γ_fluid × V_total
Meaning
W = weight of floating body (N); s = specific gravity of body; V_total = total volume of body (m³)
Watch out
s must be less than 1 (or less than fluid SG) for the body to float. If s > 1, it sinks.
When to use
Finding the weight of a homogeneous floating body.
Important facts
- Floating equilibrium: Buoyant force = Weight (F_B = W).
- For a uniform body floating in water: submerged fraction = s (specific gravity).
- Draft is independent of water depth as long as the body floats freely.
- Adding weight increases draft proportionally (until sinking occurs).
Key definitions
Draft (d)
Example
A barge with draft 1.2 m sits 1.2 m deep in water.
Vertical distance from the bottom of a floating body to the waterline; the depth of immersion.
Waterline Area
Example
For a 4 m wide × 10 m long rectangular barge, waterline area A = 40 m².
The horizontal cross-sectional area of a body at the water surface; critical for draft and stability calculations.
Specific Gravity (s)
Example
Wood with s = 0.6 floats; steel with s = 7.85 sinks.
Ratio of a body's density to water's density; determines if an object floats (s < 1) or sinks (s > 1).
Diagrams to know
- Floating block with dimensions h (height), A (plan area), d (draft marked), center of gravity G, center of buoyancy B.
- Cross-section showing waterline, draft d, and how s × h = d.
Stability of Floating Bodies — Metacentric Height
Formulas
BM = I / V_displaced
Meaning
BM = distance from center of buoyancy to metacenter (m); I = second moment of inertia of waterline area about axis of tilt (m⁴); V_displaced = displaced volume (m³)
Watch out
I must be about the AXIS OF TILT. For rolling (side-to-side), use I about the longitudinal axis; for pitching (fore-aft), use I about the transverse axis. For a rectangle: I = LB³/12 (rolling) or BL³/12 (pitching).
When to use
Always the first step in stability analysis; determines how far M is above B.
GM = BM - BG
Meaning
GM = metacentric height (m); BM = distance from B to metacenter (m); BG = distance from center of buoyancy to center of gravity (m, positive if G is above B)
Watch out
BG is signed: if G is ABOVE B, BG > 0 (reduces GM, reduces stability). If G is BELOW B, treat as negative (increases stability). Always: GM = BM − BG.
When to use
Determining stability: GM > 0 → stable; GM < 0 → unstable; GM = 0 → neutral.
Righting Moment = W × GM × sin(θ)
Meaning
W = weight of body (N); GM = metacentric height (m); θ = heel angle (small, in radians); righting moment restores body to upright.
Watch out
This is valid only for SMALL angles (typically < 15°). For large angles, use exact integration. Neglecting GM → zero restoring moment (unstable).
When to use
Quantifying the restoring torque when a floating body is tilted.
Common values
Value
I = (L × B³) / 12
Symbol
I_rolling
Quantity
Moment of inertia of rectangle about centroid (axis parallel to length)
Value
I = (B × L³) / 12
Symbol
I_pitching
Quantity
Moment of inertia of rectangle about centroid (axis parallel to width)
Important facts
- Stability condition: M must be ABOVE G (GM > 0) for equilibrium to be stable.
- M always lies on the vertical centerline and above B for a floating body.
- Increasing weight (lowering G) improves stability; raising G reduces it.
- I depends on the shape of the waterline area, not the body's draft or submerged volume shape.
- For symmetrical bodies, I about longitudinal axis (rolling) ≠ I about transverse axis (pitching).
- Stability is a SECOND-ORDER criterion: even if F_B = W (equilibrium), the body may be unstable if GM < 0.
Key definitions
Center of Buoyancy (B)
Example
For a rectangular barge at draft d, B is at height d/2 above the keel.
Centroid of the displaced fluid volume; point through which the buoyant force acts (always acts vertically upward).
Center of Gravity (G)
Example
For a uniform barge, G is at geometric center; for loaded cargo, G shifts toward the load.
Centroid of the body's mass; point where all weight may be considered to act.
Metacenter (M)
Example
M is always above B for a floating body. If M above G → stable; if M below G → unstable.
Point where the vertical line of buoyant force intersects the centerline after a small tilt; determines stability.
Metacentric Height (GM)
Example
GM = 0.5 m indicates a stable floating body with a restoring moment when tilted.
Vertical distance from center of gravity to metacenter; positive means stable, negative means unstable.
Diagrams to know
- Floating body diagram: keel, waterline, B (center of buoyancy at d/2 from keel), G (center of gravity), M (metacenter above G), dimension BG, dimension BM, dimension GM.
- Heeled (tilted) body showing how buoyant force shifts, new line of action intersects original centerline at M.
- Stability comparison: three cases — stable (M above G), unstable (M below G), neutral (M = G).
Apparent Weight & Submerged Objects
Formulas
F_apparent = F_actual - F_B = W - γ_fluid × V_object
Meaning
F_apparent = apparent weight (N); F_actual = actual weight in air (N); F_B = buoyant force (N); V_object = object volume (m³)
Watch out
Use the TOTAL volume of the object, not displaced volume (they are the same for fully submerged solids). If F_B > W, object floats upward (F_apparent is negative, i.e., upward).
When to use
Finding the apparent weight of a fully submerged object (e.g., on a scale underwater).
s_object = F_apparent / F_actual (for fully submerged)
Meaning
Apparent weight / actual weight = (W − F_B) / W = (ρ_object − ρ_fluid) / ρ_object
Watch out
Only valid for fully submerged objects. The ratio tells you the excess density over the fluid.
When to use
Quick way to find specific gravity from measured apparent weight.
Important facts
- Apparent weight is ZERO at neutral buoyancy (F_B = W), i.e., when object density = fluid density.
- Buoyant force always opposes weight (acts upward regardless of object density).
- If F_B > W, object has zero or negative apparent weight (floats even if held down).
Key definitions
Apparent Weight
Example
A 100 N steel block underwater with F_B = 10 N reads 90 N on a scale.
The weight reading of a submerged object (difference between actual weight and buoyant force); what a scale reads underwater.
Diagrams to know
- Submerged object with weight arrow down, buoyant force arrow up, net force (apparent weight) shown as difference.
Prismatic Bodies & Common Shapes
Formulas
I = (B × L³) / 12 or I = (L × B³) / 12 (rectangular waterline)
Meaning
B = width (m), L = length (m); first formula for rotation about width axis (pitching); second for rotation about length axis (rolling).
Watch out
Order matters: (L × B³)/12 for rolling (narrower dimension cubed) gives SMALLER I than (B × L³)/12. Pitching is typically less stable than rolling for long barges.
When to use
Computing metacentric height for rectangular barges, pontoons, caissons.
I_circle = π × R⁴ / 4
Meaning
I = second moment of inertia of circular waterline area (m⁴); R = radius of waterline circle (m).
Watch out
Applies only if waterline is a CIRCLE (cylinder floating level); if tilted, waterline becomes elliptical and I changes.
When to use
Cylindrical floating bodies (e.g., oil drums, submarine pressure hulls).
Important facts
- For long, narrow barges (L >> B), rolling stability (I about long axis = L×B³/12) is much smaller than pitching stability.
- Circular cross-section waterlines (I = πR⁴/4) are symmetric; same stability in all horizontal directions.
- Elongated waterlines concentrates buoyancy at center; more susceptible to rolling than pitching.
Diagrams to know
- Rectangular barge plan view (top) showing L, B, and which axis is which.
- Circle showing radius R and moment of inertia formula.
Step-by-Step Stability Check Procedure
Important facts
- 1. Calculate DISPLACED VOLUME: V_disp = W / γ_fluid (from weight and flotation condition).
- 2. Identify DRAFT: d = V_disp / A_waterline.
- 3. Locate CENTER OF BUOYANCY: B = d/2 from keel (for regular cross-sections).
- 4. Identify CENTER OF GRAVITY: G = distance from keel (given or calculated from load distribution).
- 5. Calculate BG = |G − B| (distance between B and G).
- 6. Calculate MOMENT OF INERTIA: I of the waterline area about the axis of tilt (use L×B³/12 for rolling about long axis).
- 7. Calculate BM = I / V_disp.
- 8. Calculate GM = BM − BG.
- 9. Check sign: GM > 0 → STABLE; GM < 0 → UNSTABLE; GM = 0 → NEUTRAL.
Must remember
- BUOYANT FORCE = γ_fluid × V_displaced (always upward, always equals weight of displaced fluid).
- FLOATING EQUILIBRIUM: F_B = W ⟹ V_displaced = W / γ_fluid (this defines draft and stability basis).
- METACENTRIC HEIGHT: GM = BM − BG, where BM = I / V_displaced (I is of WATERLINE AREA about TILT AXIS).
- STABILITY RULE: M above G (GM > 0) ⟹ stable; M below G (GM < 0) ⟹ unstable. Do NOT confuse the rule.
- For homogeneous block floating: d = s × h (draft = specific gravity × total height); immediate quick check.
- Moment of inertia for rectangle: I = L × B³ / 12 (rolling, narrow dimension B cubed) ≠ I = B × L³ / 12 (pitching).
- APPARENT WEIGHT: W_app = W − F_B. At neutral buoyancy (W_app = 0), object neither sinks nor floats.
- Draft depends on weight & waterline area, NOT water depth (as long as water is deep enough).
- Center of buoyancy B = centroid of displaced volume; for regular cross-section, B = d/2 from keel.
- Righting moment = W × GM × sin(θ): large positive GM ⟹ strong restoring torque; GM < 0 ⟹ capsize risk.
Last minute tips
- Always DRAW the floating body diagram: mark keel, waterline, B (at d/2), G (from problem), and check if M is above or below G. This single diagram prevents 80% of errors.
- For stability problems, FIRST find V_displaced from weight, THEN draft from area, THEN locate B and G, THEN calculate BM and GM in that order. Skipping steps causes mistakes.
- Watch the AXIS: rolling uses I about the LONG axis (L × B³/12 for a rectangle); pitching uses I about SHORT axis (B × L³/12). Mixing them reverses stability conclusions.
- If a problem gives 'G is 1.0 m above keel' and 'draft is 1.2 m', B is at 0.6 m, so BG = 1.0 − 0.6 = 0.4 m (G is ABOVE B, which reduces stability).
- Final sanity check: if you get GM < 0 (unstable), ask: 'Is G way above B?' If yes, result makes sense. If no, recalculate BM and BG—you likely swapped a dimension or axis.
Comparison tables
Fully Submerged vs. Floating Bodies
| Criterion | Fully Submerged | Floating |
|---|---|---|
| Volume used in F_B = γV | Entire body volume (V_body) | Only submerged volume (V_displaced < V_body) |
| Equilibrium condition | F_B may be ≠ W (can be held at any depth) | F_B = W (sinks/rises until equilibrium) |
| Apparent weight | W_app = W − F_B = W(1 − 1/SG) | W_app = 0 (floats at equilibrium) |
| Can be stable? | No simple criterion; depends on shape & weight distribution | Yes; if GM > 0 (M above G) |
| Typical applications | Submarines, anchored buoys, objects on lake bottom | Ships, barges, pontoons, driftwood |
Stability Classification by Metacentric Height
| Condition | GM Value | Behavior | Restoring Moment |
|---|---|---|---|
| Stable | GM > 0 (M above G) | Returns to upright when tilted | W × GM × sin(θ) > 0 (positive) |
| Unstable | GM < 0 (M below G) | Continues to tilt when disturbed | W × GM × sin(θ) < 0 (negative, increases tilt) |
| Neutral | GM = 0 (M = G) | No restoring moment; stays tilted | W × GM × sin(θ) = 0 |
| Large GM | GM >> 0 (large positive) | Stiff, high resistance to tilt (strong restoring moment) | Increases rapidly with θ |
Key Geometry Parameters by Shape (Waterline Area)
| Shape | Moment of Inertia (I) Formula | Axis of Rotation | Symbol |
|---|---|---|---|
| Rectangle (L × B) | I = L × B³ / 12 (about B-axis) | About long axis (rolling, side-to-side) | I_rolling |
| Rectangle (L × B) | I = B × L³ / 12 (about L-axis) | About short axis (pitching, fore-aft) | I_pitching |
| Circle (radius R) | I = π × R⁴ / 4 (any diameter) | Any horizontal axis (symmetric) | I_circle |
| Ellipse (a, b semi-axes) | I = π × a × b³ / 4 (about minor axis) | About major axis | I_ellipse |
Common Exam Mistakes & Corrections
| Mistake | Why Wrong | Correct Approach |
|---|---|---|
| Using total body volume for V_displaced (floating) | Floating bodies only displace the submerged part | Use V_disp = W / γ_fluid or measure only submerged volume |
| Wrong I in BM = I / V_disp | Using second moment about wrong axis or about centroid instead of waterline area | Always use I of the WATERLINE AREA about the AXIS OF TILT |
| Forgetting G location or assuming G = B | Center of buoyancy ≠ center of gravity unless body is uniform and floats level | Locate G from weight distribution; calculate BG; then GM = BM − BG |
| Confusing draft with total immersion depth | Draft = submerged depth from bottom to waterline, not depth of keel below surface | Draft d = V_displaced / A_waterline |
| Using wrong specific weight γ | Forgetting to use seawater γ ≈ 10.05 kN/m³ when in marine problem | Check problem statement; use γ_water = 9.81 for fresh water, γ_seawater ≈ 10.05 kN/m³ |
Diagrams
Buoyancy & Flotation Concept Map
Hierarchical map showing how buoyancy concepts relate: Archimedes' principle leads to flotation equilibrium, which determines draft and stability via metacentric height. Metacentric height is the key criterion for assessing whether a floating body is stable.
Floating Body Stability Assessment Flowchart
Step-by-step flowchart for conducting a complete floating body stability analysis. Guides users from calculating displaced volume through determining draft, locating centers, computing metacentric height, and final stability verdict. Essential for exam problems.
Floating Body Stability States
State diagram showing how a floating body transitions between stable equilibrium, unstable equilibrium, and neutral states based on metacentric height. Illustrates the consequence of each state when disturbed.
Typical Floating Body Stability Distribution (Training Fleet Data)
Example distribution of floating bodies by stability classification. Most well-designed vessels maintain GM > 0.2 m; newer designs aim for 0.5 m+ for increased seaworthiness. Helps contextualize what constitutes good vs. marginal stability.