Mechanical Properties of Fluids — NEET notes

Pressure in Fluids

Pressure is the normal force per unit area ( P = F/A ). In a static fluid, pressure increases linearly with depth: P = P_0 + g h . Pascal's law says any change in pressure is transmitted equally throughout the fluid — the principle behind hydraulic machines. NEET loves gauge pressure vs absolute pressure and force-amplification in hydraulic lifts.

P = P_0 + g h | F_1 A_1 = F_2 A_2

Pressure

Pressure is defined as the component of force perpendicular to a surface divided by the area of that surface. It is a scalar quantity.

P = F A (SI unit: Pa = N/m^2)

Unlike force (which is a vector), pressure is a scalar. At any point inside a fluid, pressure acts equally in all directions — up, down, sideways. It has no preferred direction. A fluid at rest exerts pressure perpendicular to any surface in contact with it.

• SI unit: pascal (Pa) = N/m ^2 . Also: 1 atm = 1.013 10^5 Pa = 760 mmHg = 760 torr
• Dimensions: [ML^ -1 T^ -2 ]
• Pressure is a scalar — it has magnitude but no direction
• At a point in a fluid at rest, pressure is the same in all directions (this is the basis of Pascal's law)

Pressure Variation with Depth

In a fluid at rest, pressure increases linearly with depth due to the weight of the fluid column above. This gives us the fundamental hydrostatic pressure formula.

Consider a horizontal surface at depth h below the free surface of a liquid. The fluid above this surface has weight g h A , which pushes down on the surface. Adding the atmospheric pressure at the top gives the total (absolute) pressure at depth h .

P = P_0 + g h

• P_0 = atmospheric pressure at the free surface ( 1.013 10^5 Pa at sea level)
• g h = gauge pressure (the extra pressure due to the fluid column alone)
• P = P_0 + g h = absolute pressure (total pressure including atmosphere)
• Pressure depends ONLY on depth h , NOT on the shape or width of the container
• At the same depth in a connected fluid, pressure is the same — this is the hydrostatic paradox

Pascal's Law

A change in pressure applied to an enclosed fluid is transmitted equally to every point of the fluid and to the walls of the container. This is the physics behind hydraulic machines.

Pascal's Law: A change in pressure at any point in an enclosed incompressible fluid at rest is transmitted undiminished to every other point in the fluid. This means if you push harder on one side, every part of the fluid 'feels' the same increase in pressure.

• Applies to enclosed, incompressible fluids at rest
• The pressure CHANGE ( P ) is transmitted equally — not the total pressure (which still varies with depth)
• Basis of all hydraulic machines: lifts, brakes, jacks
• The law holds for both liquids and gases (though gases are compressible, it approximately works at moderate pressures)

Hydraulic Lift

A hydraulic lift uses Pascal's law to amplify force: a small force on a small piston creates the same pressure as a large force on a large piston. You trade force for distance.

In a hydraulic lift, two pistons of different cross-sectional areas ( A_1 and A_2 ) are connected by an incompressible fluid. By Pascal's law, the pressure is the same everywhere at the same height. So applying force F_1 on the smaller piston creates a larger force F_2 on the bigger piston.

F_1 A_1 = F_2 A_2 F_2 = F_1 A_2 A_1

• Force amplification factor = A_2 / A_1 (ratio of piston areas)
• Energy is conserved: F_1 d_1 = F_2 d_2 . You gain force but lose distance
• The smaller piston moves a larger distance; the larger piston moves a smaller distance
• This is NOT a violation of energy conservation — work input = work output (ideal case)

Quick recall

A hydraulic lift has piston areas 10 cm ^2 and 500 cm ^2 . What force on the small piston lifts a 2500 N car?

F_1 = F_2 A_1 A_2 = 2500 10 500 = 50 N. A mere 50 N (about 5 kg-force) lifts a car!

Measuring Pressure

Atmospheric pressure is measured with a mercury barometer (Torricelli's experiment). Gauge pressure of enclosed gases is measured with open-tube manometers.

Torricelli's barometer: Fill a long glass tube with mercury, invert it in a mercury trough. The mercury column drops until it's about 76 cm (760 mm) high. The space above mercury is nearly a vacuum. The atmospheric pressure supports this column.

P_ atm = _ Hg g h = 13600 9.8 0.76 1.013 10^5 Pa

Open-tube manometer: A U-tube partially filled with mercury, one end open to the atmosphere and the other connected to the gas whose pressure is to be measured. The height difference h between mercury levels in the two arms gives the gauge pressure.

P_ gas = P_ atm + _ Hg g h (if gas pushes mercury down on its side)

• Mercury is used because of its high density (13,600 kg/m ^3 ) — gives a manageable column height
• If water were used instead: h = P_ atm _w g 10^5 1000 10 = 10 m — impractically tall!
• 1 atm = 76 cm Hg = 10.3 m of water column
• Gauge pressure = manometer reading = P - P_ atm . Can be positive or negative

Archimedes' Principle & Buoyancy

When a body is immersed in a fluid, it experiences an upward force equal to the weight of the fluid displaced — this is the buoyant force. The body floats if buoyant force weight, sinks if buoyant force < weight. Apparent weight = true weight - buoyant force. NEET loves floating/sinking conditions and 'fraction submerged' problems.

F_B = _ fluid V_ displaced g

Buoyant Force

Any body partially or fully submerged in a fluid experiences an upward force called buoyancy, caused by the pressure difference between the bottom and top surfaces.

Why does a stone feel lighter in water? The fluid pressure at the bottom of the stone (deeper) is greater than at the top (shallower). This pressure difference creates a net upward force called the buoyant force or upthrust.

F_B = _ fluid V_ submerged g

• Buoyant force acts at the centre of buoyancy (centre of gravity of the displaced fluid)
• It depends on: fluid density _f , volume displaced V , and g
• It does NOT depend on: object density, object mass, or depth of submersion (for incompressible fluids)
• Buoyant force exists in ALL fluids — liquids AND gases (hot air balloons float due to buoyancy in air)

Archimedes' Principle

The buoyant force on a submerged body equals the weight of the fluid it displaces. This elegant result connects buoyancy to a simple measurement.

Archimedes' Principle: When a body is partially or fully immersed in a fluid, the buoyant force on it equals the weight of the fluid displaced by the body. If the body displaces a volume V of fluid with density _f , the buoyant force is _f V g .

F_B = W_ fluid displaced = _f V_ displaced g

• For a fully submerged body: V_ displaced = V_ body (entire volume displaces fluid)
• For a floating body: V_ displaced < V_ body (only the submerged part displaces fluid)
• Apparent weight = True weight - Buoyant force = W - F_B = ( _ body - _ fluid ) V g
• In free fall, g_ eff = 0 , so buoyant force = 0. A submerged ball in a freely falling container won't float!

Floating & Sinking

A body floats when the buoyant force can balance its weight. This reduces to a simple density comparison: float if object density < fluid density.

Three cases when a body is released inside a fluid:

Fraction submerged for a floating body: At equilibrium, weight = buoyant force:

_ body V_ body g = _ fluid V_ sub g V_ sub V_ body = _ body _ fluid

Quick recall

A wooden block ( = 600 kg/m ^3 ) floats in water. What fraction is above the surface?

Fraction submerged = 600/1000 = 0.6 . So fraction above surface = 1 - 0.6 = 0.4 = 40%.

Fluid Dynamics & Continuity

Moving fluids are described by streamlines (steady flow) or turbulent eddies. The equation of continuity ( A_1 v_1 = A_2 v_2 ) says fluid speeds up where the pipe narrows — a direct consequence of mass conservation. The Reynolds number predicts whether flow is laminar or turbulent. NEET tests continuity equation applications heavily.

A_1 v_1 = A_2 v_2 (equation of continuity)

Streamline vs Turbulent Flow

In streamline (laminar) flow, fluid particles follow smooth, non-crossing paths. Above a critical velocity, the flow becomes turbulent with chaotic, irregular motion.

In streamline (laminar) flow, every fluid particle passing a given point follows the exact same path. The streamlines are smooth curves that never cross each other. The velocity at any point remains constant in time (steady flow).

In turbulent flow, the velocity at any point fluctuates randomly. Fluid particles follow irregular, chaotic paths with eddies and vortices. This happens when the fluid moves too fast, or when it encounters sharp obstacles.

• Streamline: The path a fluid element follows in steady flow. Tangent to the streamline = velocity direction at that point
• Streamlines NEVER cross — if they did, a particle at the intersection would have two velocities (impossible in steady flow)
• Tube of flow: A bundle of streamlines forming a tube. Fluid cannot cross the walls of a tube of flow
• Steady flow uniform flow. Steady means velocity at each point doesn't change with time; it CAN vary from point to point

Equation of Continuity

For an incompressible fluid in steady flow, the product of cross-sectional area and velocity is constant along a tube of flow. Where the pipe narrows, the fluid speeds up.

Consider a pipe whose cross-section changes from A_1 to A_2 . In steady flow, the mass of fluid entering per second must equal the mass leaving per second (no fluid accumulates or disappears). For an incompressible fluid ( = constant), this gives:

A_1 v_1 = A_2 v_2 (equation of continuity)

The product Av is called the volume flow rate (or discharge), often denoted Q . Its SI unit is m ^3 /s.

Q = Av = constant (along a streamline, for incompressible fluid)

• Narrow pipe → higher velocity, wide pipe → lower velocity (inverse relationship)
• Volume flow rate Q = Av is constant for incompressible fluids
• For a circular pipe: A = r^2 , so v 1/r^2 . Halving the radius quadruples the speed
• Assumptions: steady flow, incompressible fluid, no sources or sinks in between

Quick recall

Water flows through a pipe of diameter 4 cm at 2 m/s. It enters a section of diameter 2 cm. What is the new speed?

A_1 v_1 = A_2 v_2 . Since A d^2 : v_2 = v_1 (d_1/d_2)^2 = 2 (4/2)^2 = 2 4 = 8 m/s.

Reynolds Number

The Reynolds number is a dimensionless quantity that predicts whether a flow will be laminar or turbulent. Below ~2000 it's laminar; above ~3000 it's turbulent.

R_e = v d

Here = fluid density, v = flow velocity, d = diameter of the pipe (characteristic length), and = coefficient of viscosity. R_e is dimensionless — all units cancel out.

• R_e < 1000 : flow is definitely laminar (streamline)
• 1000 < R_e < 2000 : transition zone — flow is unstable
• R_e > 2000 : flow becomes turbulent (NCERT uses 2000 as critical value)
• R_e increases with: higher velocity, larger pipe diameter, higher fluid density, lower viscosity
• Physically, R_e = ratio of inertial forces to viscous forces. High R_e → inertia dominates → turbulence

Bernoulli's Principle

Bernoulli's equation is the energy conservation law for flowing fluids: P + 1 2 v^2 + g h = constant along a streamline. Where velocity is high, pressure is low — explaining airplane lift, atomizers, and venturimeters. NEET heavily tests Bernoulli applications and the speed-pressure inverse relationship.

P + 1 2 v^2 + g h = constant

Bernoulli's Equation

For an ideal fluid in streamline flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline.

Bernoulli's equation is really the work-energy theorem applied to fluid flow. Consider a fluid element flowing along a streamline. The work done by pressure forces and gravity equals the change in kinetic energy. Rearranging gives:

P + 1 2 v^2 + g h = constant (along a streamline)

Each term has dimensions of pressure (energy per unit volume):

• P = static pressure (the actual pressure of the fluid at that point)
• 1 2 v^2 = dynamic pressure (kinetic energy per unit volume)
• g h = hydrostatic pressure (potential energy per unit volume, h = height from reference)
• The sum of all three is called the total mechanical energy per unit volume — and it's constant along a streamline

• Valid ONLY for: ideal fluid (incompressible, non-viscous) in steady, irrotational flow
• Applied along a single streamline — different streamlines may have different constants
• NOT valid for: viscous fluids (energy lost to friction), turbulent flow, or compressible gases at high speed

Speed-Pressure Relationship

Bernoulli's key insight: in a horizontal flow, where fluid speed increases, pressure decreases. This counter-intuitive result explains a huge range of phenomena.

For horizontal flow ( h = constant), Bernoulli's equation simplifies to:

P + 1 2 v^2 = constant P_1 + 1 2 v_1^2 = P_2 + 1 2 v_2^2

If v_2 > v_1 , then P_2 < P_1 . Where fluid moves faster, the pressure is lower. This seems counter-intuitive — you'd expect fast-moving fluid to push harder. But the fluid's energy is split between pressure and kinetic energy. More kinetic energy → less pressure energy.

• In a horizontal pipe: narrower section → higher velocity (continuity) → lower pressure (Bernoulli)
• This explains why roofs blow OFF in storms — fast wind over the roof creates low pressure above, higher pressure below pushes the roof up
• A spinning ball curves in flight (Magnus effect) — air moves faster on one side, creating a pressure difference
• Atomizer/spray gun: fast air over the tube opening reduces pressure, drawing liquid up

Venturimeter

A venturimeter uses the pressure drop in a constriction to measure the flow velocity of a fluid. It combines the continuity equation and Bernoulli's equation.

A venturimeter is a horizontal pipe with a constriction (narrow throat). As fluid enters the throat, velocity increases (continuity) and pressure drops (Bernoulli). A manometer measures the pressure difference P between the wide and narrow sections.

v_1 = A_2 2(P_1 - P_2) (A_1^2 - A_2^2) = A_2 2 _m g h (A_1^2 - A_2^2)

Here A_1 and A_2 are the cross-sectional areas at the wide and narrow sections, is the fluid density, _m is the manometer liquid density, and h is the manometer height difference.

• Combines both continuity ( A_1 v_1 = A_2 v_2 ) and Bernoulli ( P_1 + 1 2 v_1^2 = P_2 + 1 2 v_2^2 )
• Volume flow rate: Q = A_1 v_1 once v_1 is found
• The constriction creates a measurable pressure difference — wider difference → easier measurement
• Used in water supply systems, carburetors, and industrial flow measurement

Torricelli's Theorem

The speed of liquid flowing out of a hole near the bottom of an open tank equals the speed of free fall from the liquid surface: v = 2gh . It's Bernoulli applied to a draining tank.

Apply Bernoulli's equation between the free surface (point 1: large area, so v_1 0 , pressure = P_0 , height = h ) and the orifice (point 2: velocity = v , pressure = P_0 , height = 0).

P_0 + 1 2 (0)^2 + g h = P_0 + 1 2 v^2 + 0

v = 2gh (Torricelli's theorem — speed of efflux)

• Efflux speed depends ONLY on the height of the liquid surface above the hole
• If the tank is closed with pressure P above the liquid: v = 2gh + 2(P - P_0)
• The stream of liquid from the hole follows projectile motion after exiting
• Range on ground (for hole at height h from bottom of tank of height H ): R = 2 h(H-h) , maximum when h = H/2

Quick recall

A tank filled with water to 5 m has a small hole at the bottom. What is the efflux speed?

v = 2 10 5 = 100 = 10 m/s. Same as dropping a ball from 5 m!

Viscosity

Viscosity is the internal friction of fluids — it resists the relative motion between adjacent fluid layers. Newton's law of viscosity: F = A dv dx . The coefficient of viscosity has SI unit Pa s (or poiseuille). Poiseuille's formula gives flow rate through a pipe. NEET tests viscosity concepts, units, and the temperature dependence.

F = A dv dx | Q = P r^4 8 l

Viscosity

When fluid layers move at different speeds, a friction-like force acts between them resisting the relative motion. This property of the fluid is called viscosity.

Imagine a fluid flowing over a fixed surface. The layer directly touching the surface is stationary (no-slip condition). The next layer moves slowly. Layers farther from the surface move faster. This velocity gradient means adjacent layers slide past each other, and friction-like forces act between them. This is viscosity.

• Viscosity is the fluid's resistance to flow — 'thickness' of the fluid in everyday language
• Honey has high viscosity; water has low viscosity; ideal fluids have zero viscosity
• The layer in contact with a solid surface has zero velocity (no-slip condition)
• Velocity increases linearly with distance from the surface (for laminar flow between parallel plates)

Newton's Law of Viscosity

The viscous force between fluid layers is proportional to the area of contact and the velocity gradient. The constant of proportionality is the coefficient of viscosity.

F = - A dv dx

Here F is the viscous force, A is the area of the layer, dv dx is the velocity gradient (rate of change of velocity with distance perpendicular to flow), and (eta) is the coefficient of viscosity.

• The negative sign indicates the force opposes relative motion (like friction)
• SI unit of : Pa s = N s/m ^2 = kg/(m s). Also called poiseuille (Pl)
• CGS unit: poise (P). 1 Pa s = 10 poise
• Dimensions of : [ML^ -1 T^ -1 ]
• Fluids obeying this linear law are called Newtonian fluids (water, air, most simple fluids)

Poiseuille's Formula

For laminar flow of a viscous fluid through a cylindrical pipe, the volume flow rate depends on the fourth power of the radius — a tiny change in pipe radius dramatically affects flow.

Q = P r^4 8 l = P r^4 8 l

Here Q = volume flow rate, P = pressure difference across the pipe, r = pipe radius, = coefficient of viscosity, and l = length of the pipe.

• Flow rate r^4 — doubling the radius increases flow 16 times! This is the most important feature
• Flow rate P — double the pressure, double the flow
• Flow rate 1/ — more viscous fluid flows slower
• Flow rate 1/l — longer pipe means more resistance
• Valid only for laminar, steady flow of a Newtonian fluid in a horizontal cylindrical pipe

Quick recall

If the radius of a pipe is halved, how does the flow rate change (other factors constant)?

Q r^4 . So Q_ new = Q (1/2)^4 = Q/16 . Flow drops to 1/16th of the original!

Stokes' Law & Terminal Velocity

A sphere moving through a viscous fluid experiences a drag force F = 6 r v (Stokes' law). When this drag plus buoyancy equals the sphere's weight, it reaches terminal velocity and stops accelerating: v_t = 2r^2( - )g 9 . NEET frequently tests the terminal velocity formula and how it depends on radius, density, and viscosity.

F = 6 r v | v_t = 2r^2( - )g 9

Stokes' Law

A small sphere moving slowly through a viscous fluid experiences a retarding force proportional to its velocity, radius, and the fluid's viscosity.

F = 6 r v

Here F is the viscous drag force, is the coefficient of viscosity of the fluid, r is the radius of the sphere, and v is its velocity. The force opposes the motion — it always acts to slow the sphere down.

• Valid for: small, smooth, rigid spheres moving slowly through a viscous medium
• Drag v (first power, NOT v^2 ) — this is the low Reynolds number regime
• Drag r (bigger sphere → more drag) and (more viscous → more drag)
• Assumptions: laminar flow around the sphere, infinite extent of fluid, no-slip at sphere surface
• At higher speeds, drag becomes proportional to v^2 (turbulent drag) — Stokes' law no longer applies

Terminal Velocity

A sphere falling through a viscous fluid initially accelerates under gravity, but as its speed increases, the drag force increases until it balances weight minus buoyancy. The sphere then moves at constant terminal velocity.

Three forces act on a sphere falling through a fluid: weight (down), buoyancy (up), and viscous drag (up). Initially, weight dominates → the sphere accelerates. As speed increases, drag increases ( F v ). At some speed, the net force becomes zero → constant velocity = terminal velocity.

At terminal velocity: Weight = Buoyancy + Drag

4 3 r^3 g = 4 3 r^3 g + 6 r v_t

Solving for v_t :

v_t = 2r^2( - )g 9

Here = density of the sphere, = density of the fluid, r = radius of the sphere, = viscosity of the fluid.

• v_t r^2 — doubling the radius quadruples the terminal velocity. Larger raindrops fall faster!
• v_t ( - ) — denser sphere (relative to fluid) falls faster
• v_t 1/ — more viscous fluid → slower terminal velocity
• If < : v_t is negative → the sphere rises (like an air bubble in water)
• If = : v_t = 0 → neutral buoyancy

Quick recall

Two identical spheres, one of density 3 and another of density 5 , fall through a fluid of density . What is the ratio of their terminal velocities?

v_1 v_2 = _1 - _2 - = 3 - 5 - = 2 4 = 1 2 .

Surface Tension

Surface tension ( T = F/l ) is the force per unit length acting along the surface of a liquid, caused by unbalanced intermolecular forces at the surface. It makes the surface behave like a stretched membrane. Excess pressure inside a drop: 2T/R ; inside a bubble: 4T/R . Angle of contact determines wetting. NEET frequently tests excess pressure, surface energy, and the angle of contact.

T = F l | P_ drop = 2T R | P_ bubble = 4T R

Surface Tension

Molecules at the surface of a liquid have fewer neighbors than bulk molecules, creating a net inward pull. This makes the surface contract to minimum area and gives rise to surface tension.

A molecule deep inside a liquid is pulled equally in all directions by neighbors — net force = 0. But a molecule at the surface has neighbors only below and to the sides, not above. This creates a net inward pull, making the surface want to contract to the smallest possible area.

T = F l (SI unit: N/m)

Surface tension T is the force per unit length acting on an imaginary line drawn on the liquid surface. The force acts tangentially to the surface and perpendicular to the line.

• SI unit: N/m. Dimensions: [MT^ -2 ] (same as force/length)
• Surface tension is a scalar property of the liquid — not a vector (the direction of force depends on which line you draw)
• Free liquid drops are spherical because a sphere has the minimum surface area for a given volume
• Surface tension decreases with temperature (thermal energy overcomes intermolecular attraction)
• Soap/detergent drastically reduces surface tension of water (from 0.073 to ~0.025 N/m)

Surface Energy

Creating new liquid surface requires work against intermolecular forces. This work is stored as surface energy. Surface energy per unit area is numerically equal to surface tension.

To increase the surface area of a liquid, you must pull molecules from the interior to the surface against the net inward intermolecular force. The work done is stored as surface energy.

W = T A

For a thin film (like a soap film) which has TWO free surfaces:

W = T 2 A (soap film — two surfaces)

• Surface energy per unit area = T (in J/m ^2 ), numerically same as surface tension (in N/m)
• A soap film has TWO surfaces (inner and outer) — total surface area = 2 area of the film
• A liquid drop has only ONE surface — total surface area = 4 R^2
• Breaking a drop into n smaller drops increases total surface area → requires work input: W = T (new area - old area)

Quick recall

A water drop of radius R splits into 1000 identical droplets. If surface tension is T , find the work done.

Volume conserved: 4 3 R^3 = 1000 4 3 r^3 r = R/10 . New area = 1000 4 r^2 = 1000 4 R^2/100 = 40 R^2 . Old area = 4 R^2 . W = T(40 R^2 - 4 R^2) = 36 R^2 T .

Excess Pressure

The curved surface of a drop or bubble creates a pressure difference across the surface. The pressure inside is always greater than outside — smaller radius means greater excess pressure.

A curved liquid surface generates a pressure difference across it. The concave side (inside) always has higher pressure. This excess pressure keeps the drop/bubble from collapsing.

Liquid drop: P = 2T R

Air bubble in liquid: P = 2T R

Soap bubble: P = 4T R (two surfaces!)

• Liquid drop (one surface): P = 2T/R
• Soap bubble (two surfaces): P = 4T/R — exactly double
• Smaller radius → greater excess pressure. Tiny bubbles have enormous internal pressure
• If two bubbles of different sizes are connected, the smaller one shrinks and the larger one grows (air flows from high to low pressure)

Angle of Contact

The angle of contact is the angle between the solid surface and the tangent to the liquid surface at the point of contact. It determines whether the liquid wets the surface or forms a bead.

At the boundary where a liquid meets a solid surface, three forces compete: cohesive forces (liquid-liquid attraction), adhesive forces (liquid-solid attraction), and gravity. The resulting shape of the meniscus (curved liquid surface) is described by the angle of contact .

• < 90° → adhesive > cohesive → liquid wets the surface → concave meniscus (e.g., water-glass: 0° )
• > 90° → cohesive > adhesive → liquid does NOT wet the surface → convex meniscus (e.g., mercury-glass: 140° )
• = 90° → liquid surface is flat at the contact point
• = 0° → liquid completely wets (spreads out) — water on a clean glass surface

Capillarity

Capillary action is the rise (or fall) of a liquid in a narrow tube due to the interplay of surface tension, adhesion, and cohesion. The capillary rise formula h = 2T r g shows that narrower tubes give greater rise. NEET tests this formula, the effect of tube radius, and the role of the angle of contact.

h = 2T r g

Capillary Rise

When a narrow tube is dipped in a wetting liquid, the liquid rises inside the tube against gravity. The height of rise depends on surface tension, tube radius, liquid density, and angle of contact.

Place a thin glass tube (capillary) vertically in water. The water rises inside the tube above the outside level. Why? The adhesive force between water and glass pulls the meniscus up. Surface tension along the contact ring provides the upward force. This continues until the weight of the risen liquid column balances the surface tension force.

Balancing the upward surface tension force with the weight of the liquid column:

2 r T = r^2 h g

h = 2T r g

• h 1/r — narrower tube → greater rise. This is why capillarity is significant only in thin tubes
• h T — higher surface tension → greater rise
• h — for wetting liquids ( < 90° ), > 0 → liquid rises. For non-wetting ( > 90° ), < 0 → liquid falls (depression)
• h 1/ — denser liquids rise less
• For water in glass: 0° , so 1 . Formula simplifies to h = 2T r g

Capillarity in Nature & Technology

Capillary action is essential in nature (plants, soil) and technology (wicks, paper, ink). Understanding when and why liquids rise or fall in narrow spaces.

• Plants: Water rises from roots to leaves through xylem vessels (narrow tubes) — capillarity assists (along with transpiration pull and root pressure)
• Oil lamps: Kerosene rises through the wick to the flame — classic capillary action
• Paper/cloth absorption: Paper towels soak up water via capillary action through tiny pores between fibers
• Soil: Underground water rises to the surface through capillary-sized gaps between soil particles
• Ink pens: Ink flows from the cartridge to the nib via capillary channels

Quick recall

Water ( T = 0.073 N/m, = 1000 kg/m ^3 , 0° ) rises in a glass tube of radius 0.5 mm. Find the height of rise. ( g = 10 m/s ^2 )

h = 2 0.073 1 0.5 10^ -3 1000 10 = 0.146 5 = 0.0292 m 2.9 cm.