CELE Hydraulics & Fluid Mechanics — Relative Equilibrium of LiquidsCheat Sheet
A printable cheat sheet for Relative Equilibrium of Liquids, built for CELE reviewers who want one go-to reference in the final stretch. Covers formulas, key definitions, common question types, and the Professional Regulation Commission (PRC) — Board of Civil Engineering-specific twists you will see on CELE day.
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. Relative Equilibrium of Liquids lands at position 4th 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.
Relative Equilibrium of Liquids - Cheat Sheet
Your final 30-minute reference for relative equilibrium problems: horizontal acceleration, vertical acceleration, rotation, and pressure distribution. Master the three core scenarios and their formulas.
Sections
Formulas
Formula
tan θ = a/g
Meaning
θ = angle of free surface tilt from horizontal; a = horizontal acceleration (m/s²); g = 9.81 m/s²
Watch Out
Surface slopes DOWN in the direction of acceleration. If a/g > 1, θ > 45° (steep slope). Never use sin or cos—only tangent.
When To Use
Tank or container accelerating horizontally; find surface slope angle
Formula
p = γ h_v
Meaning
p = pressure (kPa); γ = unit weight of water ≈ 9.81 kN/m³; h_v = vertical depth below tilted surface
Watch Out
Pressure depends on VERTICAL depth below the tilted surface, not normal to surface. Misidentifying h_v causes major errors.
When To Use
Calculate pressure at any point after finding the tilted surface orientation
Formula
h_rise = (a/g) × L_horizontal
Meaning
Rise of surface at one end of tank; L = horizontal distance traveled along tank length
Watch Out
This is a derived relation—use tan θ and geometry. Not all problems ask for this directly.
When To Use
Find how much higher the surface is at one end vs. the other in a rectangular tank
Common Values
Value
9.81 m/s²
Symbol
g
Quantity
Gravitational acceleration
Value
9.81 kN/m³
Symbol
γ
Quantity
Unit weight of water (fresh)
Value
10.05 kN/m³
Symbol
γ_salt
Quantity
Unit weight of water (salt, approximate)
Section Title
Horizontal Acceleration (Linear Motion)
Important Facts
- Free surface remains a plane in horizontal acceleration (no curvature).
- Pressure is hydrostatic relative to the tilted surface: p = γ × (vertical depth below surface).
- The lowest point of the surface is in the direction OPPOSITE to acceleration.
- In a closed tank, pressure increases in the direction of acceleration.
- The depth at one end rises by h_rise = (a/g) × L while the other end drops by the same amount (if tank is symmetric).
- Maximum acceleration before spillage: a_max occurs when surface reaches tank rim at steepest corner.
- For a = 0 (no acceleration), θ = 0—surface is horizontal, normal hydrostatic pressure applies.
Key Definitions
Term
Relative Equilibrium
Example
Water in an accelerating truck bed—no slosh, just tilted surface.
Definition
Liquid moving as a rigid body with zero relative motion between particles; no shear stress; free surface adjusts to new shape.
Term
Free Surface Tilt Angle (θ)
Example
a = 2 m/s², g = 9.81 m/s² ⟹ θ = arctan(2/9.81) ≈ 11.5°
Definition
Angle between tilted liquid surface and horizontal plane during horizontal acceleration.
Term
Effective Gravity
Example
Horizontal acceleration creates an apparent sideways gravity component.
Definition
In a non-inertial frame, the combination of true gravity and pseudo-force due to acceleration.
Diagrams To Know
- Tilted free surface in rectangular tank showing angle θ from horizontal
- Pressure distribution diagram showing iso-pressure lines perpendicular to tilted surface
- Tank end-view showing liquid height difference from front to back
Formulas
Formula
p = γ h (1 + a/g) [upward acceleration]
Meaning
p = gauge pressure; a = upward acceleration; h = vertical depth; + sign for upward
Watch Out
Sign: +a for upward. Pressure > γh. If a = g (free fall down), use p = γ h (1 − 1) = 0 gauge pressure.
When To Use
Container accelerating upward (elevator going up, rocket launch); pressure increases
Formula
p = γ h (1 − a/g) [downward acceleration]
Meaning
p = gauge pressure; a = downward acceleration; − sign for downward
Watch Out
Sign: −a for downward. When a/g = 1 (free fall), p = 0 everywhere (liquid feels weightless).
When To Use
Container accelerating downward; pressure decreases
Formula
Effective gravity: g_eff = g ± a
Meaning
g_eff = effective gravitational acceleration felt by liquid
Watch Out
Not usually written this way in exams, but helps intuition: pressure ∝ effective weight.
When To Use
Conceptual understanding; pressure scales as γ_eff × h = γ(g_eff/g) × h
Formula
Free surface remains horizontal
Meaning
No tilt—surface stays parallel to ground even during vertical acceleration
Watch Out
Easy to confuse with horizontal acceleration. Vertical ⟹ horizontal surface; horizontal ⟹ tilted surface.
When To Use
Sketch diagrams: surface is always horizontal, only pressure changes with depth
Common Values
Value
9.81 m/s²
Symbol
g
Quantity
Gravitational acceleration
Value
1–2 m/s²
Symbol
a_elev
Quantity
Typical elevator acceleration (upward)
Value
9.81 m/s² (downward)
Symbol
a_ff
Quantity
Free-fall acceleration
Section Title
Vertical Acceleration (Up or Down)
Important Facts
- Free surface stays HORIZONTAL during vertical acceleration—never tilts.
- Upward acceleration (a > 0): effective gravity increases ⟹ pressure increases faster with depth.
- Downward acceleration (a < 0): effective gravity decreases ⟹ pressure increases slower with depth.
- At a = g (free fall): gauge pressure = 0 everywhere (liquid and container are weightless relative to each other).
- Pressure always increases linearly with vertical depth in the accelerating frame.
- In an elevator: at rest, p_bottom = γ h; accelerating up, p_bottom = γ h (1 + a/g); accelerating down, p_bottom = γ h (1 − a/g).
- For closed tanks, the gas pressure above the liquid adjusts if the liquid depth is constant (fixed volume).
Key Definitions
Term
Free Fall (Special Case)
Example
Astronaut in orbit or a dropped bucket; water exerts no force on container walls (zero normal force).
Definition
Downward acceleration a = g; the liquid and container fall together; gauge pressure everywhere = 0.
Term
Effective Unit Weight
Example
Upward a = 5 m/s²: γ_eff = 9.81(1 + 5/9.81) ≈ 14.9 kN/m³
Definition
γ_eff = γ(1 ± a/g), the apparent weight per unit volume in the accelerating frame.
Term
Gauge Pressure vs Absolute Pressure
Example
Water 2 m deep with upward a = 3 m/s²: gauge p = 27.6 kPa; absolute ≈ 128.6 kPa.
Definition
Gauge = pressure above atmospheric (used in this chapter); Absolute = gauge + atmospheric ≈ 101 kPa.
Diagrams To Know
- Vertical cross-section showing horizontal surface with pressure distribution increasing downward faster (upward a) or slower (downward a)
- Effective gravity vector diagram showing g and a components combining
- Free-fall scenario (a = g): liquid floating in container, no pressure
Formulas
Formula
z(r) = (ω² r²) / (2g)
Meaning
z = height of free surface above the lowest point (vertex) at radius r; ω = angular velocity (rad/s); r = radial distance from axis; g = 9.81 m/s²
Watch Out
Must convert rpm to rad/s: ω = (2π N) / 60. Omitting the factor 2 in denominator is a classic error.
When To Use
Open cylinder or vessel rotating about vertical axis; find surface shape at any radius
Formula
z_max = (ω² R²) / (2g)
Meaning
Maximum rise of surface at the rim (radius R) above the vertex (center, lowest point)
Watch Out
R is the OUTER radius of the vessel. If diameter D is given, use R = D/2.
When To Use
Find total rise from center to edge of a circular container
Formula
Paraboloid shape: Free surface is a paraboloid of revolution
Meaning
Surface equation in cylindrical coordinates: z = (ω² r²) / (2g)
Watch Out
Not a cone, not a hemisphere—strictly parabolic. Volume under paraboloid = 0.5 × (cylinder volume).
When To Use
Describe or sketch the rotating liquid surface; always parabolic profile
Formula
V_paraboloid = (1/2) × V_cylinder = (1/2) × π R² h_avg
Meaning
Volume of liquid under paraboloid = half the volume of the cylinder enclosing it
Watch Out
Assumes paraboloid doesn't touch the bottom or overflow. If it does, recalculate with spill/drain condition.
When To Use
Check if liquid spills or to find average depth during rotation
Formula
Pressure: p = γ × h_vertical
Meaning
Pressure at any point = unit weight × vertical depth below curved surface (measured vertically, not radially)
Watch Out
Depth is VERTICAL (downward) from the paraboloid surface, not perpendicular to surface or radial distance.
When To Use
Find pressure at any point in rotating liquid
Common Values
Value
9.81 m/s²
Symbol
g
Quantity
Gravitational acceleration
Value
2π/60 ≈ 0.1047
Symbol
conversion factor
Quantity
Conversion: 1 rpm to rad/s
Value
1000–3000 rpm
Symbol
N_centrifuge
Quantity
Typical centrifuge speed
Section Title
Rotation (Constant Angular Velocity)
Important Facts
- Rotating liquid forms a paraboloid (parabolic profile), not a cone or cone frustum.
- The vertex (center, r = 0) is the LOWEST point of the surface.
- Surface rises quadratically with radius: z ∝ r² (doubling r → 4× rise).
- Volume of rotating liquid = 0.5 × volume of the enclosing cylinder (conserved).
- Pressure is hydrostatic: depends only on vertical depth below the curved surface.
- Isobar (constant-pressure) surfaces are also paraboloids (rotated versions of the free surface).
- If the paraboloid reaches the bottom (z at center < 0), liquid at the center is exposed (exposed axis condition).
- If the paraboloid exceeds the rim, liquid spills (need to recompute with reduced liquid volume).
- In the rotating frame, an effective 'gravity' acts: g_eff combines true gravity g (downward) and centrifugal acceleration ω² r (outward).
- Particle trajectories in the rotating frame are straight lines perpendicular to effective gravity.
Key Definitions
Term
Paraboloid of Revolution
Example
Coffee in a spinning cup; swirled honey in a bowl.
Definition
3D surface generated by rotating a parabola about its axis; the free surface of a rotating open liquid.
Term
Angular Velocity (ω)
Example
1000 rpm = 1000 × 2π/60 ≈ 104.7 rad/s
Definition
Rate of rotation in radians per second (rad/s); related to rpm by ω = 2πN/60.
Term
Vertex (of Paraboloid)
Example
In a spinning cup, the vertex is the deepest point at the center.
Definition
The lowest point of the rotating free surface, located at the axis of rotation (r = 0).
Term
Rim (of Rotating Vessel)
Example
Edge of a spinning centrifuge tube.
Definition
The outer edge of the container at radius R; where the surface rises to maximum height z_max.
Diagrams To Know
- Vertical cross-section of rotating cylinder showing parabolic free surface profile
- Top-view (plan view) showing circles (isobars) at the free surface
- Cylindrical coordinates (r, θ, z) with paraboloid surface labeled
- Comparison: vertex (low), rim (high), and center axis
- Pressure distribution in the vertical plane (parabolic isobars)
Formulas
Formula
Pressure = γ × h_vertical (all scenarios)
Meaning
Fundamental: gauge pressure always equals unit weight times vertical depth below free surface
Watch Out
h is measured VERTICALLY (downward), not along the surface or any other direction. This is the #1 source of errors.
When To Use
Any relative equilibrium problem after identifying the free surface shape
Formula
Effective gravity vector: g_eff = √[(g ± a_vertical)² + a_horizontal²]
Meaning
Magnitude of net acceleration in the non-inertial (moving) frame; conceptual only
Watch Out
This is derived thinking—actual exam problems ask for tan θ or pressure directly, not g_eff.
When To Use
Understand why surfaces tilt or curve the way they do (not often used directly in calculations)
Section Title
Pressure Distribution & Special Cases
Important Facts
- In all relative equilibrium scenarios, the pressure at a point depends only on its VERTICAL distance below the free surface.
- The free surface is always perpendicular to the direction of effective gravity (g_eff).
- Horizontal acceleration ⟹ tilted plane surface (perpendicular to g_eff at angle θ).
- Vertical acceleration only ⟹ horizontal plane surface (no tilt).
- Rotation ⟹ paraboloid surface (perpendicular to combined g and centrifugal effect).
- Closed tanks: air pressure above liquid adjusts; gauge pressure at liquid bottom = γ h (1 ± a/g) for vertical acceleration.
- Open tanks: atmospheric pressure acts on free surface; gauge pressure starts at 0 on the surface.
- At maximum spillage (tilted surface just reaches a corner or rotating surface just reaches rim/axis), volume of liquid is conserved.
Key Definitions
Term
Non-Inertial (Accelerating) Reference Frame
Example
Inside a braking car, everything 'falls' forward (pseudo-force backward).
Definition
A frame attached to the accelerating container; in this frame, pseudo-forces appear to act on the liquid.
Term
Hydrostatic Assumption in Relative Equilibrium
Example
Pressure p = γ × (vertical distance) applies in the accelerating frame, not the ground frame.
Definition
Even though the container moves, the liquid inside behaves hydrostatically relative to the new orientation—no turbulence or sloshing.
Term
Spillage Condition
Example
Spinning cup too fast ⟹ parabola reaches the axis at center, exposing the bottom; cup tilted too steeply ⟹ one corner runs dry.
Definition
When the free surface (tilted or curved) reaches the rim or bottom of the container, liquid is exposed to overpressure or underpressure.
Diagrams To Know
- Schematic showing pressure increasing downward (at 90° to free surface normal in all three scenarios)
- Diagram of spillage: tilted surface hitting the corner; rotating surface exposing the axis
- Isopressure (isobar) contours in horizontal acceleration (parallel lines perpendicular to θ)
- Isopressure contours in rotation (concentric circles above the paraboloid)
Section Title
Board Exam Strategy & Problem-Solving Flow
Important Facts
- Step 1: Identify the motion type — horizontal, vertical, or rotational?
- Step 2: Sketch the free surface (plane tilted at angle θ, horizontal, or paraboloid).
- Step 3: Find the shape parameter (θ, ω, or a) using the given acceleration.
- Step 4: Locate the point of interest (depth, radius, height).
- Step 5: Calculate vertical distance from point to free surface.
- Step 6: Apply p = γ × h_vertical.
- For spillage problems: set surface at rim (max z_max) or axis (min z at center = 0); solve for ω or a.
- For closed tanks: consider gas pressure above liquid in addition to liquid pressure.
- For combined motions (e.g., horizontal + vertical): apply superposition carefully—use vector addition of accelerations.
Must Remember
- 1. HORIZONTAL ACCELERATION: Surface tilts at angle tan θ = a/g DOWN in direction of acceleration. Pressure = γ × vertical depth below tilted surface. Never use sine or cosine—only tangent.
- 2. VERTICAL ACCELERATION UP: Pressure increases as p = γ h (1 + a/g). Free surface stays HORIZONTAL—no tilt. Effective gravity increases.
- 3. VERTICAL ACCELERATION DOWN: Pressure decreases as p = γ h (1 − a/g). Free fall (a = g) → gauge pressure = 0 everywhere. Effective gravity decreases.
- 4. ROTATION: Free surface is a PARABOLOID z = ω² r² / (2g). Rise at rim = ω² R² / (2g). Convert rpm to rad/s using 2π/60. Pressure depends on vertical depth below the curved surface.
- 5. PRESSURE RULE (ALL CASES): p = γ × h_vertical, where h_vertical is measured DOWNWARD from the free surface. This is the fundamental rule—never measure along the surface or at an angle.
- 6. PARABOLOID VOLUME: Volume of rotating liquid = 0.5 × cylinder volume (conserved). If paraboloid reaches axis or rim, liquid is exposed (spillage condition).
- 7. CLOSED vs. OPEN TANKS: Open tanks: gauge pressure starts at 0 on surface (atmospheric = reference). Closed tanks: air pressure above liquid must be considered; gauge pressure at bottom = γ h (1 ± a/g) for vertical.
- 8. SPILLAGE: Tilted surface reaches corner or rim → solve by setting surface height at that location. Rotating surface reaches axis (z_center = 0) or rim (z_rim = tank height) → recalculate with boundary condition.
- 9. EFFECTIVE GRAVITY CONCEPT: Horizontal a tilts surface perpendicular to combined g and a_horizontal. Vertical a changes magnitude of pressure. Rotation: surface perpendicular to combined g (down) and ω² r (outward).
- 10. EXAM PITFALLS: Sign errors in vertical acceleration (+a up, −a down); forgetting to convert rpm to rad/s (multiply by 2π/60); measuring depth along surface instead of vertically; confusing which scenario applies. Read carefully: 'tank accelerates horizontally' vs. 'vertically' vs. 'spins.'
Last Minute Tips
- TIP 1 — Identify the scenario first: Look for keywords 'accelerates horizontally' (tilt), 'accelerates vertically' (no tilt, pressure changes), 'rotates/spins' (paraboloid). This determines which formula family to use—don't guess.
- TIP 2 — Always draw a rough sketch of the free surface (plane tilted, plane horizontal, or paraboloid) and mark the point where you need pressure. Sketching prevents sign and direction errors and saves time on complex problems.
- TIP 3 — Double-check units for ω: If given in rpm, convert immediately to rad/s using ω = 2πN/60. Forgetting this factor causes the entire paraboloid calculation to be wrong by a factor ~4 (because it's squared in the formula).
- TIP 4 — Remember the hierarchy: First find the free surface shape and parameters (θ, a, ω). Then identify the vertical depth of your point of interest below that surface. Only then apply p = γ h. Skipping steps leads to errors.
- TIP 5 — For spill/boundary problems: If the problem states 'liquid just reaches the rim' or 'just exposes the axis,' that's your boundary condition—set z_max = tank height or z_center = 0, then solve for the unknown (ω or a). Don't over-complicate; use the constraint given.
Comparison Tables
Rows
Values
- Linear, perpendicular to gravity
- Tilted plane at angle θ
- tan θ = a/g
- p = γ h_vertical
- a always positive; θ in direction of acceleration
Property
Horizontal Acceleration
Values
- Linear, parallel to gravity (upward)
- Horizontal plane (no tilt)
- p = γ h (1 + a/g)
- p = γ h (1 + a/g)
- + sign for upward; pressure increases
Property
Vertical Acceleration (Up)
Values
- Linear, parallel to gravity (downward)
- Horizontal plane (no tilt)
- p = γ h (1 − a/g)
- p = γ h (1 − a/g)
- − sign for downward; pressure decreases
Property
Vertical Acceleration (Down)
Values
- Circular about vertical axis at ω
- Paraboloid of revolution
- z(r) = ω² r² / (2g)
- p = γ × h_vertical (depth below paraboloid)
- ω in rad/s; always positive; z increases outward
Property
Rotation (Angular)
Columns
- Scenario
- Motion Type
- Free Surface Shape
- Key Formula
- Pressure Formula
- Sign Convention
Table Title
Three Scenarios of Relative Equilibrium: Summary
Rows
Values
- Horizontal plane
- p = γ h
- Standard hydrostatic pressure
Property
At Rest (a = 0, ω = 0)
Values
- Tilted plane
- p = γ h_v (h_v = vertical depth at point)
- Pressure at bottom depends on vertical depth, not slant distance
Property
Horizontal Acceleration a
Values
- Horizontal plane
- p = γ h (1 + a/g)
- Effective gravity increases; pressure amplified
Property
Vertical Acceleration a (up)
Values
- Horizontal plane
- p = γ h (1 − a/g)
- Effective gravity decreases; pressure reduced
Property
Vertical Acceleration a (down)
Values
- Horizontal plane
- p = γ h (1 − 1) = 0
- Gauge pressure zero everywhere; liquid weightless
Property
Free Fall (a = g down)
Values
- Paraboloid
- p = γ × (vertical depth from paraboloid to bottom)
- Depth increases with r; pressure at center may be 0 if parabola touches axis
Property
Rotation ω at rim (r = R)
Columns
- Condition
- Surface Shape
- Pressure at Bottom (depth h)
- Notes
Table Title
Pressure Comparison: Rest vs. Motion
Rows
Values
- m/s²
- 9.81
- Use 9.81 (or 10 for estimates); standard for PRC exams
Property
g (gravity)
Values
- kN/m³
- 9.81
- Numerically equals g in SI; unit weight of fresh water
Property
γ (water)
Values
- rad/s
- ω = 2πN/60 from rpm
- N in revolutions per minute; multiply by 2π/60 ≈ 0.1047
Property
Angular velocity
Values
- rad/s
- 2π/60 ≈ 0.1047 rad/s
- Common conversion for rotating machinery problems
Property
1 rpm to rad/s
Values
- kPa or Pa
- 1 kPa = 1000 Pa
- Use kPa for typical depths; Pa for very small pressures
Property
Pressure
Values
- kPa or atm
- 101.325 kPa ≈ 1 atm
- Reference for gauge vs. absolute pressure; use 101 for quick calculations
Property
Atmospheric pressure
Columns
- Quantity
- SI Unit
- Value / Conversion
- Note
Table Title
Common Unit Conversions & Constants
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