Hydrostatics evaluates fluid statics, pressure distribution in liquids at rest, buoyancy forces, and ideal gas compression law interactions.
Core Concepts:
1. Hydrostatic Pressure Formula: p(h) = p₀ + ρ · g · h
where p₀ is atmospheric surface pressure (approx. 101,325 Pa or 1 bar), ρ is fluid density (water = 1000 kg/m³), g is gravitational acceleration (10 m/s² for dMAT calculations), and h is fluid depth.
2. Rule of Thumb: Every 10 meters of water depth adds approximately 1 bar (100,000 Pa or 100 kPa) of hydrostatic pressure.
3. Gauge vs Absolute Pressure:
• Gauge Pressure (p_gauge) = ρ · g · h (pressure exerted by liquid column alone).
• Absolute Pressure (p_abs) = p_atm + p_gauge.
4. Archimedes Buoyancy Principle: F_buoyant = ρ_fluid · V_displaced · g. An object floats when F_buoyant = F_gravity.
5. Trapped Air & Gas Compression (Boyle's Law): p₁V₁ = p₂V₂ at constant temperature (isothermal).
2. Worked Solution Example: Trapped Air & Submerged Container
Worked Problem:
A cylindrical diving bell of height H = 3 m open at the bottom is lowered vertically into freshwater (ρ = 1000 kg/m³, g = 10 m/s²) until its top is at a depth of 7 m below the surface. Surface atmospheric pressure is p₀ = 100,000 Pa (1 bar).
Calculate the height h_water of water that enters the bell when air inside compresses isothermally.
Step 1: Initial Air State (at surface before submergence):
• Initial Pressure p₁ = p₀ = 100,000 Pa.
• Initial Air Volume V₁ = A · H = 3A (where A is cross-sectional area).
Step 2: Final Air State (when submerged):
• Let air column height inside bell be h_air.
• Volume V₂ = A · h_air.
• Depth of air-water interface inside bell = 7 m (top depth) + h_air.
• Pressure of trapped air p₂ = p₀ + ρ · g · (7 + h_air) = 100,000 + 1000 · 10 · (7 + h_air) = 100,000 + 10,000(7 + h_air).
Step 3: Apply Boyle's Law p₁V₁ = p₂V₂:
100,000 · (3A) = [100,000 + 10,000(7 + h_air)] · (A · h_air)
Cancel A and divide by 10,000:
30 = [10 + 7 + h_air] · h_air
30 = (17 + h_air) · h_air
h_air² + 17 h_air - 30 = 0
Step 4: Solve Quadratic Equation:
(h_air + 18.6)(h_air - 1.6) ≈ 0
h_air ≈ 1.6 meters.
Water height inside bell h_water = H - h_air = 3.0 m - 1.6 m = **1.4 meters**.
3. Essential Hydrostatics Cheatsheet
• Hydrostatic Force on Flat Hatch: F = p_centroid · Area = (p₀ + ρ g h_centroid) · A
• Floating Fraction of Submerged Object: V_submerged / V_total = ρ_object / ρ_fluid
• U-Tube Manometer Pressure Difference: Δp = (ρ₂ - ρ₁) g h
Key Takeaway Strategy Rules
Keep absolute and gauge pressure distinct
Gauge pressure measures the increase above a reference pressure. Absolute pressure includes the surface pressure. Using water density 1000 kg/m³ and g = 10 m/s², a 10 m water column adds 100000 Pa, or 1 bar.
Add layers and subtract opposing pressure
For several liquid layers, add ρgh for each layer. A hatch experiences net force equal to the pressure difference across it multiplied by its area. A vertical gate has varying pressure, so use its centroid depth to find the total force.
Balance trapped air with the water column
In an airtight flooded space at constant temperature, p₁V₁ = p₂V₂ uses absolute pressures. Water entering compresses the air, while the rising internal water level changes the depth at which pressure balances. Solve both relationships together; the leak height alone does not set the final water level.
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Study focus: Hydrostatics
Relate pressure to density, gravitational acceleration and depth. Distinguish gauge pressure from absolute pressure and keep units consistent. For buoyancy, identify displaced fluid volume before comparing upward and downward forces.
GAM modules are subject-specific tests tailored to your intended degree field.
Vector Calculations
Study 2D/3D vectors, scalar dot products, cross products, magnitudes, and triple products with reading articles and DataSnap’s independent 10-minute, 8-question practice rounds.
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Analyze step-by-step transformations across 4×4 grids. Focus on perimeter movement, wall reflection, step acceleration, and shape rotation rules to deduce missing stages 5 and 6.
What is the difference between dMAT Core Modules and General Academic Modules?
Core Modules evaluate fundamental cognitive aptitude (Figure Sequences, Latin Squares, Mathematical Equations), whereas General Academic Modules (GAM) test domain-specific knowledge like Vector Calculations and Hydrostatics tailored to specific degree fields.
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