Class 11 · Chapter 10

Mechanical Properties of Fluids

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Mechanical Properties of Fluids Formula Sheet

12 formulas across 5 topics in Mechanical Properties of Fluids.

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Updated 2026-07-03 · v1.0.1

Fluid Pressure & Pascal's Law

Pressure in a static fluid depends on depth alone — not on the shape or width of the container.

P = P₀ + ρgh

Pressure at depth h

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
P₀	pressure at the surface (usually atmospheric)1 atm = 1.013 × 10⁵ Pa = 760 mm of Hg	N/m²	[ML⁻¹T⁻²]
ρ	density of the fluid	kg/m³	[ML⁻³]
h	depth below the surface	m	[L]

Valid when

Static fluid, uniform density
Points at the same depth in a CONNECTED fluid have equal pressure

Common mistakes

In an accelerating container the free surface tilts with tanθ = a/g and pressure varies along the horizontal too

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F₂ = F₁ (A₂/A₁)

Hydraulic lift (Pascal's law)

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Equal pressure transmitted: F₁/A₁ = F₂/A₂

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
F₁, F₂	forces on the small and large pistons	N	[MLT⁻²]
A₁, A₂	piston areas	m²	[L²]

Valid when

Force multiplies, but work does not — the small piston moves proportionally farther

Read the concept →

Worth remembering

A barometer reads absolute pressure; a manometer reads gauge pressure (difference from atmospheric)

Buoyancy & Archimedes' Principle

A submerged body is pushed up by the weight of the fluid it displaces — nothing more, nothing less.

F(B) = ρ(fluid) V(sub) g

Buoyant force (Archimedes' principle)

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
F(B)	buoyant force (upward)	N	[MLT⁻²]
V(sub)	SUBMERGED volume of the body	m³	[L³]
ρ(fluid)	density of the fluid	kg/m³	[ML⁻³]

Valid when

Floating: weight = buoyant force, so fraction submerged = ρ(body)/ρ(fluid)
Apparent weight when fully submerged = W(1 − ρ(fluid)/ρ(body))

Common mistakes

Ice melting in a glass of water does NOT change the level — the melt exactly fills the displaced volume
Buoyant force acts at the centre of the DISPLACED fluid (centre of buoyancy), not the body's COM

Read the concept →

Worth remembering

In a freely falling lift (or orbit), buoyancy vanishes — no effective gravity, no pressure gradient

Fluid Dynamics — Continuity & Bernoulli

Continuity is mass conservation; Bernoulli is energy conservation per unit volume. Together they solve every ideal-flow problem.

A₁v₁ = A₂v₂

Equation of continuity

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Mass conservation for incompressible steady flow

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
A	cross-sectional area of the tube	m²	[L²]
v	flow speed at that section	m/s	[LT⁻¹]

Valid when

Incompressible fluid, steady flow
Narrow section → FAST flow

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P + ½ρv² + ρgh = constant

Bernoulli's equation

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Work-energy theorem per unit volume of flowing fluid

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
P	pressure	N/m²	[ML⁻¹T⁻²]
v	flow speed	m/s	[LT⁻¹]
h	height of the point	m	[L]

Valid when

Ideal fluid: non-viscous, incompressible, steady, irrotational — along a streamline

Common mistakes

Fast flow means LOW pressure — students invert this; it explains aerofoil lift, roofs blowing off, and the Venturi meter

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v = √(2gh)

Torricelli's law (speed of efflux)

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Bernoulli between the free surface and the hole, with the tank wide enough that the surface barely moves

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v	efflux speed from the hole	m/s	[LT⁻¹]
h	depth of the hole below the free surface	m	[L]

Valid when

Open tank, small hole
Same as free-fall speed from height h

Common mistakes

Range of the emerging jet from a hole at depth h in a tank of height H: x = 2√(h(H−h)), maximum when the hole is at mid-height

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Worth remembering

Bernoulli fails in viscous or turbulent flow — that is why real pipes need pumps

NEET asks Bernoulli applications conceptually (aerofoil, atomiser); JEE computes with the Venturi meter and moving-tank efflux setups.

Viscosity & Stokes' Law

Viscosity is internal friction between fluid layers. It caps the speed of anything falling through a fluid at the terminal velocity.

F = −ηA (dv/dx)

Newton's law of viscosity

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
η	coefficient of viscosity1 poise (CGS) = 0.1 Pa·s	Pa·s (poiseuille)	[ML⁻¹T⁻¹]
dv/dx	velocity gradient between layers	s⁻¹	[T⁻¹]
A	area of the layer	m²	[L²]

Valid when

Liquids: η DECREASES with temperature; gases: η INCREASES with temperature

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F = 6πηrv; v(t) = 2r²(ρ − σ)g / 9η

Stokes' law & terminal velocity

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Weight = buoyancy + viscous drag at terminal speed

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
r	radius of the sphere	m	[L]
ρ, σ	densities of the sphere and the fluid	kg/m³	[ML⁻³]
v(t)	terminal velocity	m/s	[LT⁻¹]

Valid when

Small sphere, laminar flow
v(t) ∝ r² — a drop of double radius falls four times faster

Common mistakes

When n identical drops merge, the big drop's terminal velocity = n²ᐟ³ × (small drop's) — radius grows as n¹ᐟ³, v(t) as r²

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Re = ρvd / η

Reynolds number

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
Re	Reynolds number≲ 1000 laminar, ≳ 2000 turbulent, in-between unsteady	dimensionless	[M⁰L⁰T⁰]
d	characteristic dimension (pipe diameter)	m	[L]

Read the concept →

Worth remembering

Raindrops land gently because of terminal velocity — without air viscosity they would arrive at hundreds of m/s

Surface Tension & Capillarity

Surface tension is energy per unit area of a liquid surface — it makes drops spherical, powers capillary rise, and sets the excess pressure inside bubbles.

T = F/L = (surface energy)/(area)

Surface tension

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
T	surface tension	N/m (= J/m²)	[MT⁻²]
L	length of the line on the surface	m	[L]

Valid when

Decreases with temperature; vanishes at the critical point
Soap LOWERS water's surface tension

Common mistakes

Work to blow a soap BUBBLE of radius r: W = 8πr²T (two surfaces) — a drop has one surface: W = 4πr²T

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ΔP = 2T/r (drop); ΔP = 4T/r (soap bubble)

Excess pressure inside a drop / bubble

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Balancing pressure force against surface tension around the equator; the bubble's factor 4 comes from its TWO surfaces

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
ΔP	pressure inside minus outside	N/m²	[ML⁻¹T⁻²]
r	radius	m	[L]

Valid when

SMALLER bubble → HIGHER inside pressure — connect two bubbles and the small one empties into the big one

Common mistakes

Air bubble INSIDE water is a single surface: ΔP = 2T/r, not 4T/r

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h = 2T cosθ / (ρgr)

Capillary rise

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Weight of the raised column balanced by the vertical component of surface tension around the meniscus

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
h	rise (or fall) of the liquid in the tube	m	[L]
θ	contact angleθ > 90° (mercury–glass) makes h negative — depression	rad (or °)	[M⁰L⁰T⁰]
r	tube radius	m	[L]

Valid when

Thin tube
In a tube of insufficient length the liquid does NOT overflow — the meniscus radius adjusts instead

Common mistakes

h ∝ 1/r: halving the tube radius DOUBLES the rise

Read the concept →

Worth remembering

In free fall (satellite), capillary liquid rises to fill the ENTIRE tube — g → 0 removes the counterweight

NEET rotates between capillary rise, excess pressure and merged-drops energy release; JEE combines surface energy changes with temperature rise of the coalesced drop.

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