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Chapter 2 of 12

Fluid Statics

In the RSMSSB JE Civil (Degree) syllabus under Fluid Mechanics & Hydraulics · 2 parts

Fluid Statics & Pressure Measurement · Hydrostatic Forces, Buoyancy & Floatation

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📑 Contents (12 sections)

Part 1 of 2

Fluid Statics & Pressure Measurement

Last reviewed 16 Sept 2026 · 6 min read

Pressure

Pressure (intensity of pressure) is the normal force per unit area exerted by a fluid: $p = F / A$ (N/m² = pascal; 1 bar = 10⁵ Pa).

Pascal's law

At a point in a fluid at rest, the pressure is the same in all directions: $p_{x} = p_{y} = p_{z}$ . (Pressure acts normal to any surface; there is no shear in a static fluid.)

Applications: hydraulic press, hydraulic jack, hydraulic brakes — a small force on a small piston produces a large force on a large piston: $\frac{F _{1}}{A _{1}} = \frac{F _{2}}{A _{2}}$ .

Hydrostatic law

In a static fluid, pressure increases with depth:

\frac{d p}{d z} = - ρ g \Rightarrow p = ρ g h (gauge, below a free surface)

Pressure is the same at all points on a horizontal plane within the same continuous fluid.
The shape of the container does not matter (hydrostatic paradox) — only the vertical depth.
Pressure head $h = \frac{p}{ρ g}$ : the height of a column of the fluid that produces the pressure. Converting between liquids: $h_{1} S_{1} = h_{2} S_{2}$ .

Absolute, gauge and vacuum pressure

p_{ab s} = p_{a t m} + p_{g a ug e}, p_{v a c uu m} = p_{a t m} - p_{ab s}

Atmospheric pressure at sea level ≈ 101.325 kPa = 10.33 m of water = 760 mm of mercury.
Gauge pressure is measured relative to local atmosphere (positive or negative).
Vacuum (negative gauge) pressure — below atmospheric; absolute zero is a perfect vacuum.
Atmospheric pressure is measured by a barometer (mercury or aneroid).

Pressure measuring devices

Piezometer

A simple vertical tube connected to the pipe: $p = ρ g h$ . Limitations: cannot measure negative (vacuum) pressures (air enters), impractical for high pressures (very tall tube), unsuitable for gases.

Simple U-tube manometer

A U-tube containing a heavier manometric liquid (usually mercury, $S_{2}$ ) connected to the pipe carrying fluid of $S_{1}$ . Write pressures along the tube, equating pressures at the same level in the same liquid.

FormulaSimple U-tube manometer (pipe on the left, right limb open)

Positive pressure at A, fluid rising $h_{1}$ above the datum in the left limb, manometric liquid difference $h_{2}$ :

p_{A} + ρ_{1} g h_{1} = ρ_{2} g h_{2} \Rightarrow p_{A} = ρ_{2} g h_{2} - ρ_{1} g h_{1}

Negative (vacuum) pressure at A (manometric liquid higher in the left limb):

p_{A} = - ρ_{1} g h_{1} - ρ_{2} g h_{2}

Differential U-tube manometer

Measures the difference in pressure between two points A and B. For two pipes at the same level carrying the same fluid ( $S_{1}$ ) with mercury ( $S_{2}$ ) reading $x$ :

p_{A} - p_{B} = x g (ρ_{2} - ρ_{1}) \Rightarrow head difference = x (\frac{S _{2}}{S _{1}} - 1)

For water with mercury: $h = 12.6 x$ .

Inverted U-tube manometer

A lighter manometric fluid (air or oil) in an inverted U: used for small pressure differences in liquids: $p_{A} - p_{B} = x g (ρ_{1} - ρ_{l i g h t})$ (same-level points).

Micromanometers and inclined manometers

Inclined manometer — the limb is inclined at $θ$ , so a small vertical difference $h = L sin θ$ produces a longer reading $L$ — higher sensitivity.
Micromanometer — enlarged reservoir (well) on one limb so that only one limb is read; used for very small differences.

Mechanical gauges

Bourdon tube gauge — a curved elliptical tube tends to straighten under pressure; pointer motion; used for high pressures and vacuum.
Diaphragm gauge, bellows gauge, dead-weight gauge (used for calibration), and electronic pressure transducers.

Part 2 of 2

Hydrostatic Forces, Buoyancy & Floatation

Last reviewed 16 Sept 2026 · 6 min read

Total pressure on plane surfaces

Total pressure $F$ is the resultant force of fluid pressure on a surface; the centre of pressure is the point where it acts. Because pressure increases with depth, the centre of pressure lies below the centroid of a submerged vertical or inclined surface.

FormulaPlane surface (any inclination)

F = ρ g A \overset{ˉ}{h}

$\overset{ˉ}{h}$ = vertical depth of the centroid of the area below the free surface.

Depth of centre of pressure:

h^{*} = \overset{ˉ}{h} + \frac{I _{G} sin ^{2} θ}{A h ˉ}

$I_{G}$ = second moment of area about the centroidal axis parallel to the free surface; $θ$ = inclination of the surface to the horizontal ( $θ$ = 90° for vertical: $h^{*} = \overset{ˉ}{h} + I_{G} / (A \overset{ˉ}{h})$ ).

Horizontal surface: pressure uniform; $F = ρ g A h$ acting at the centroid.
The centre of pressure approaches the centroid as depth increases.
For a vertical rectangle with top edge at the surface: $h^{*} = \frac{2}{3} d$ . For a vertical triangle with base at the surface (apex down): $h^{*} = d /2$ ; apex at the surface (base down): $h^{*} = \frac{3}{4} d$ .

Pressure diagram method

For rectangular surfaces, the total force equals the volume of the pressure prism (area of the pressure diagram × width) acting through its centroid — convenient for gates and dam faces, and when water acts on both sides.

Curved surfaces

The force on a curved surface is found from components:

FormulaCurved surfaces

Horizontal component $F_{H}$ = force on the vertical projection of the curved surface: $F_{H} = ρ g A_{v} \overset{ˉ}{h}_{v}$ , acting at the centre of pressure of that projection.
Vertical component $F_{V}$ = weight of the liquid vertically above the curved surface up to the free surface (real or imaginary), acting through the centroid of that volume.
Resultant $F = F_{H}^{2} + F_{V}^{2}$ , inclined at $tan^{- 1} (F_{V} / F_{H})$ to the horizontal; for a circular surface it passes through the centre.

Applications: radial (Tainter) gates, curved dam faces, pipe bends, domes and spherical tanks.

Buoyancy

Archimedes' principle: a body wholly or partly immersed in a fluid experiences an upward buoyant force equal to the weight of fluid displaced, acting through the centre of buoyancy (centroid of the displaced volume).

A body floats when its weight equals the buoyant force of the immersed part.
Fraction submerged of a floating body = (specific gravity of body)/(specific gravity of liquid).

Metacentre and stability

When a floating body tilts slightly, the centre of buoyancy shifts. The metacentre M is the point where the vertical through the new centre of buoyancy meets the original vertical axis.

FormulaMetacentric height

B M = \frac{I}{V}, GM = B M - B G = \frac{I}{V} - B G

$I$ = second moment of the waterline plane area about the axis of tilting (the longitudinal axis for rolling — use the smaller I); $V$ = volume displaced; $B$ = centre of buoyancy; $G$ = centre of gravity ( $B G$ positive when G is above B).

Condition	Floating body	Submerged body
Stable	M above G ( $GM > 0$ )	B above G
Neutral	M coincides with G	B coincides with G
Unstable	M below G ( $GM < 0$ )	B below G

Typical metacentric heights: merchant ships about 0.3–1.2 m; warships higher. A large GM means a stiff ship with quick, uncomfortable rolling; small GM gives slow comfortable rolling but less reserve against capsizing.

Experimental metacentric height

Move a known weight $w$ across the deck by distance $x$ and measure the angle of heel $θ$ :

GM = \frac{w x}{W tan θ}

( $W$ = total weight including $w$ .)

Period of rolling

T = 2 π \frac{k ^{2}}{g GM}

$k$ = radius of gyration about the rolling axis.

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