CELE Hydraulics & Fluid Mechanics — Buoyancy and FlotationCheat Sheet
Buoyancy and Flotation cheat sheet — the reference card you wish you had on exam day. Condensed from the full study notes, this is the high-yield core of Buoyancy and Flotation for CELE Hydraulics & Fluid Mechanics. Download, print, revise.
Exam context
On the CELE 2026, the Hydraulics & Fluid Mechanics subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Buoyancy and Flotation lands at position 3rd out of 10 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Hydraulics & Fluid Mechanics on a typical CELE paper.
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.
Sections
Formulas
Formula
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.
Formula
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
Section Title
Archimedes' Principle & Buoyant Force
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
Term
Buoyant Force
Example
A 0.05 m³ object fully submerged in water experiences F_B = 9.81 × 0.05 = 0.4905 kN upward.
Definition
Upward force exerted by a fluid on a submerged or floating object, equal to the weight of the displaced fluid.
Term
Displaced Volume
Example
A block floating at half-height displaces half its geometric volume.
Definition
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.
Formulas
Formula
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.
Formula
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.
Formula
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.
Section Title
Flotation & Draft Calculation
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
Term
Draft (d)
Example
A barge with draft 1.2 m sits 1.2 m deep in water.
Definition
Vertical distance from the bottom of a floating body to the waterline; the depth of immersion.
Term
Waterline Area
Example
For a 4 m wide × 10 m long rectangular barge, waterline area A = 40 m².
Definition
The horizontal cross-sectional area of a body at the water surface; critical for draft and stability calculations.
Term
Specific Gravity (s)
Example
Wood with s = 0.6 floats; steel with s = 7.85 sinks.
Definition
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.
Formulas
Formula
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.
Formula
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.
Formula
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)
Section Title
Stability of Floating Bodies — Metacentric Height
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
Term
Center of Buoyancy (B)
Example
For a rectangular barge at draft d, B is at height d/2 above the keel.
Definition
Centroid of the displaced fluid volume; point through which the buoyant force acts (always acts vertically upward).
Term
Center of Gravity (G)
Example
For a uniform barge, G is at geometric center; for loaded cargo, G shifts toward the load.
Definition
Centroid of the body's mass; point where all weight may be considered to act.
Term
Metacenter (M)
Example
M is always above B for a floating body. If M above G → stable; if M below G → unstable.
Definition
Point where the vertical line of buoyant force intersects the centerline after a small tilt; determines stability.
Term
Metacentric Height (GM)
Example
GM = 0.5 m indicates a stable floating body with a restoring moment when tilted.
Definition
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).
Formulas
Formula
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).
Formula
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.
Section Title
Apparent Weight & Submerged Objects
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
Term
Apparent Weight
Example
A 100 N steel block underwater with F_B = 10 N reads 90 N on a scale.
Definition
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.
Formulas
Formula
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.
Formula
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).
Section Title
Prismatic Bodies & Common Shapes
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.
Section Title
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
Rows
Values
- Entire body volume (V_body)
- Only submerged volume (V_displaced < V_body)
Property
Volume used in F_B = γV
Values
- F_B may be ≠ W (can be held at any depth)
- F_B = W (sinks/rises until equilibrium)
Property
Equilibrium condition
Values
- W_app = W − F_B = W(1 − 1/SG)
- W_app = 0 (floats at equilibrium)
Property
Apparent weight
Values
- No simple criterion; depends on shape & weight distribution
- Yes; if GM > 0 (M above G)
Property
Can be stable?
Values
- Submarines, anchored buoys, objects on lake bottom
- Ships, barges, pontoons, driftwood
Property
Typical applications
Columns
- Criterion
- Fully Submerged
- Floating
Table Title
Fully Submerged vs. Floating Bodies
Rows
Values
- GM > 0 (M above G)
- Returns to upright when tilted
- W × GM × sin(θ) > 0 (positive)
Property
Stable
Values
- GM < 0 (M below G)
- Continues to tilt when disturbed
- W × GM × sin(θ) < 0 (negative, increases tilt)
Property
Unstable
Values
- GM = 0 (M = G)
- No restoring moment; stays tilted
- W × GM × sin(θ) = 0
Property
Neutral
Values
- GM >> 0 (large positive)
- Stiff, high resistance to tilt (strong restoring moment)
- Increases rapidly with θ
Property
Large GM
Columns
- Condition
- GM Value
- Behavior
- Restoring Moment
Table Title
Stability Classification by Metacentric Height
Rows
Values
- I = L × B³ / 12 (about B-axis)
- About long axis (rolling, side-to-side)
- I_rolling
Property
Rectangle (L × B)
Values
- I = B × L³ / 12 (about L-axis)
- About short axis (pitching, fore-aft)
- I_pitching
Property
Rectangle (L × B)
Values
- I = π × R⁴ / 4 (any diameter)
- Any horizontal axis (symmetric)
- I_circle
Property
Circle (radius R)
Values
- I = π × a × b³ / 4 (about minor axis)
- About major axis
- I_ellipse
Property
Ellipse (a, b semi-axes)
Columns
- Shape
- Moment of Inertia (I) Formula
- Axis of Rotation
- Symbol
Table Title
Key Geometry Parameters by Shape (Waterline Area)
Rows
Values
- Floating bodies only displace the submerged part
- Use V_disp = W / γ_fluid or measure only submerged volume
Property
Using total body volume for V_displaced (floating)
Values
- 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
Property
Wrong I in BM = I / V_disp
Values
- Center of buoyancy ≠ center of gravity unless body is uniform and floats level
- Locate G from weight distribution; calculate BG; then GM = BM − BG
Property
Forgetting G location or assuming G = B
Values
- Draft = submerged depth from bottom to waterline, not depth of keel below surface
- Draft d = V_displaced / A_waterline
Property
Confusing draft with total immersion depth
Values
- 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
Property
Using wrong specific weight γ
Columns
- Mistake
- Why Wrong
- Correct Approach
Table Title
Common Exam Mistakes & Corrections
Previous chapter
Hydrostatic Pressure and Forces on Surfaces
Next chapter
Relative Equilibrium of Liquids
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