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.
Horizontal Acceleration (Linear Motion)
Formulas
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
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
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)
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
Relative Equilibrium
Example
Water in an accelerating truck bed—no slosh, just tilted surface.
Liquid moving as a rigid body with zero relative motion between particles; no shear stress; free surface adjusts to new shape.
Free Surface Tilt Angle (θ)
Example
a = 2 m/s², g = 9.81 m/s² ⟹ θ = arctan(2/9.81) ≈ 11.5°
Angle between tilted liquid surface and horizontal plane during horizontal acceleration.
Effective Gravity
Example
Horizontal acceleration creates an apparent sideways gravity component.
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
Vertical Acceleration (Up or Down)
Formulas
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
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
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
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
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
Free Fall (Special Case)
Example
Astronaut in orbit or a dropped bucket; water exerts no force on container walls (zero normal force).
Downward acceleration a = g; the liquid and container fall together; gauge pressure everywhere = 0.
Effective Unit Weight
Example
Upward a = 5 m/s²: γ_eff = 9.81(1 + 5/9.81) ≈ 14.9 kN/m³
γ_eff = γ(1 ± a/g), the apparent weight per unit volume in the accelerating frame.
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.
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
Rotation (Constant Angular Velocity)
Formulas
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
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
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
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
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
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
Paraboloid of Revolution
Example
Coffee in a spinning cup; swirled honey in a bowl.
3D surface generated by rotating a parabola about its axis; the free surface of a rotating open liquid.
Angular Velocity (ω)
Example
1000 rpm = 1000 × 2π/60 ≈ 104.7 rad/s
Rate of rotation in radians per second (rad/s); related to rpm by ω = 2πN/60.
Vertex (of Paraboloid)
Example
In a spinning cup, the vertex is the deepest point at the center.
The lowest point of the rotating free surface, located at the axis of rotation (r = 0).
Rim (of Rotating Vessel)
Example
Edge of a spinning centrifuge tube.
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)
Pressure Distribution & Special Cases
Formulas
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
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)
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
Non-Inertial (Accelerating) Reference Frame
Example
Inside a braking car, everything 'falls' forward (pseudo-force backward).
A frame attached to the accelerating container; in this frame, pseudo-forces appear to act on the liquid.
Hydrostatic Assumption in Relative Equilibrium
Example
Pressure p = γ × (vertical distance) applies in the accelerating frame, not the ground frame.
Even though the container moves, the liquid inside behaves hydrostatically relative to the new orientation—no turbulence or sloshing.
Spillage Condition
Example
Spinning cup too fast ⟹ parabola reaches the axis at center, exposing the bottom; cup tilted too steeply ⟹ one corner runs dry.
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)
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
Three Scenarios of Relative Equilibrium: Summary
| Scenario | Motion Type | Free Surface Shape | Key Formula | Pressure Formula | Sign Convention |
|---|---|---|---|---|---|
| Horizontal Acceleration | 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 |
| Vertical Acceleration (Up) | 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 |
| Vertical Acceleration (Down) | 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 |
| Rotation (Angular) | 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 |
Pressure Comparison: Rest vs. Motion
| Condition | Surface Shape | Pressure at Bottom (depth h) | Notes |
|---|---|---|---|
| At Rest (a = 0, ω = 0) | Horizontal plane | p = γ h | Standard hydrostatic pressure |
| Horizontal Acceleration a | Tilted plane | p = γ h_v (h_v = vertical depth at point) | Pressure at bottom depends on vertical depth, not slant distance |
| Vertical Acceleration a (up) | Horizontal plane | p = γ h (1 + a/g) | Effective gravity increases; pressure amplified |
| Vertical Acceleration a (down) | Horizontal plane | p = γ h (1 − a/g) | Effective gravity decreases; pressure reduced |
| Free Fall (a = g down) | Horizontal plane | p = γ h (1 − 1) = 0 | Gauge pressure zero everywhere; liquid weightless |
| Rotation ω at rim (r = R) | Paraboloid | p = γ × (vertical depth from paraboloid to bottom) | Depth increases with r; pressure at center may be 0 if parabola touches axis |
Common Unit Conversions & Constants
| Quantity | SI Unit | Value / Conversion | Note |
|---|---|---|---|
| g (gravity) | m/s² | 9.81 | Use 9.81 (or 10 for estimates); standard for PRC exams |
| γ (water) | kN/m³ | 9.81 | Numerically equals g in SI; unit weight of fresh water |
| Angular velocity | rad/s | ω = 2πN/60 from rpm | N in revolutions per minute; multiply by 2π/60 ≈ 0.1047 |
| 1 rpm to rad/s | rad/s | 2π/60 ≈ 0.1047 rad/s | Common conversion for rotating machinery problems |
| Pressure | kPa or Pa | 1 kPa = 1000 Pa | Use kPa for typical depths; Pa for very small pressures |
| Atmospheric pressure | kPa or atm | 101.325 kPa ≈ 1 atm | Reference for gauge vs. absolute pressure; use 101 for quick calculations |
Diagrams
Relative Equilibrium of Liquids — Concept Map
Hierarchical overview of three relative equilibrium scenarios and their key formulas. Shows how each motion type (horizontal, vertical, rotational) creates a distinct free surface shape and pressure distribution.
Problem-Solving Flowchart for Relative Equilibrium
Step-by-step decision tree for solving relative equilibrium problems. Guides student from problem identification through scenario classification, formula selection, and pressure calculation. Includes spillage check.
Free Surface Shapes by Scenario
State diagram showing transitions between rest and three motion scenarios. Illustrates the free surface shape that results from each motion type and identifies spillage conditions (boundary states) for each scenario.