NEET Physics — Chapter 10

Mechanical Properties of Fluids

Pressure, Pascal's law, Archimedes' principle, Bernoulli's equation, viscosity, Stokes' law, surface tension, capillarity — complete NEET notes with diagrams and exam traps.

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1. Pressure in Fluids and Pascal's Law

A fluid is any substance that can flow — liquids and gases. Fluids exert pressure equally in all directions at any given point.

Pressure at depth $h$ in a fluid of density $ho$ :

P = P_0 + ho g h

where $P_0$ is atmospheric pressure. Pressure increases linearly with depth. Pressure is the same at all points at the same horizontal level in a connected fluid at rest.

Pascal's Law: A pressure change applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of the container.

rac{F_1}{A_1} = rac{F_2}{A_2} implies F_2 = F_1 cdot rac{A_2}{A_1}

Hydraulic press (car lift, hydraulic brake): small force on small piston creates large force on large piston.

Gauge pressure: Pressure above atmospheric: $P_{gauge} = P - P_0 = ho g h$

Atmospheric pressure: $P_0 = 101325$ Pa $approx 10^5$ Pa $= 760$ mmHg $= 1$ atm

NEET tip: Barometric pressure is measured by a mercury barometer. In a barometer,

P_0 = ho_{Hg} g h

, giving

h approx 760

mm of mercury. If water were used instead, the column height would be about 10.3 m (since

ho_{water}/ ho_{Hg} approx 1/13.6

2. Buoyancy and Archimedes' Principle

Archimedes' Principle: When a body is partially or fully immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced:

F_b = ho_{fluid} cdot V_{submerged} cdot g

The buoyant force acts at the centre of buoyancy — the geometric centre of the submerged volume.

Conditions for floating, sinking, and neutral buoyancy:

Condition	Comparison	Result
Floats (partially submerged)	$ho_{body} < ho_{fluid}$	$F_b = W$ , partial immersion
Neutral buoyancy	$ho_{body} = ho_{fluid}$	Suspended anywhere
Sinks	$ho_{body} > ho_{fluid}$	$F_b < W$ , sinks to bottom

Fraction submerged for a floating body:

rac{V_{sub}}{V_{body}} = rac{ ho_{body}}{ ho_{fluid}}

An iceberg: $ho_{ice}/ ho_{seawater} approx 0.917$ , so about 91.7% is submerged.

Apparent weight: Weight of body in fluid = True weight − Buoyant force

W_{apparent} = W - F_b = mg - ho_{fluid}Vg = V( ho_{body} - ho_{fluid})g

Pro tip: Relative density (specific gravity) = density of substance / density of water =

W_{air}/(W_{air} - W_{water})

. This is a direct application of Archimedes' principle and commonly tested in NEET.

3. Fluid Dynamics — Equation of Continuity

Ideal fluid: Incompressible (density constant), non-viscous (no internal friction), steady flow (velocity at a point doesn't change with time), irrotational (no eddies).

Equation of Continuity (conservation of mass for incompressible flow):

A_1 v_1 = A_2 v_2 = ext{constant} = Q ext{ (volume flow rate)}

Fluid speeds up when the pipe narrows and slows down when it widens. This is why a garden hose has higher speed when you cover part of the opening with your thumb.

NEET tip: Volume flow rate

Q = Av

has SI unit m³/s. Mass flow rate =

ho Q = ho Av

(kg/s). The continuity equation applies at any cross-section of the same streamline tube.

4. Bernoulli's Equation and Applications

Bernoulli's Equation is the energy conservation equation for an ideal fluid along a streamline:

P + rac{1}{2} ho v^2 + ho g h = ext{constant}

Each term has units of pressure (Pa). Dividing by $ho g$ gives the "head" form: $P/ ho g + v^2/2g + h = ext{const}$ (metres).

Key Applications:

Venturimeter: Measures flow rate using pressure difference at constriction. $v_1 = A_2sqrt{2gDelta h/(A_1^2 - A_2^2)}$
Torricelli's theorem (hole in tank): Speed of efflux from a hole at depth $h$ below the surface: $v = sqrt{2gh}$ — same as free fall speed from height $h$
Pitot tube: Measures aircraft speed using static vs stagnation pressure. $v = sqrt{2(P_{stag} - P_{static})/ ho}$
Aerofoil (dynamic lift): Faster flow over curved upper surface → lower pressure → net upward force (lift)
Magnus effect: Spinning ball curves in flight due to pressure difference caused by combined translational and rotational velocity

Caution: Bernoulli's equation applies ONLY to ideal (non-viscous, incompressible) fluids in steady, irrotational flow. It does NOT apply to turbulent flow, viscous flow, or unsteady flow.

5. Viscosity, Stokes' Law, and Reynolds Number

Viscosity is the property of a fluid that resists relative motion between its layers — the internal friction of fluids. Honey has higher viscosity than water.

Newton's law of viscosity: The tangential (viscous) force between fluid layers:

F = eta A rac{dv}{dx}

$eta$ = coefficient of dynamic viscosity, $dv/dx$ = velocity gradient. SI unit of $eta$ : Pa·s = N·s/m² (also Poise in CGS: 1 Pa·s = 10 Poise).

Stokes' Law: Viscous drag force on a sphere of radius $r$ moving with velocity $v$ in a fluid:

F_{viscous} = 6pieta r v

Terminal velocity: When a sphere falls through a viscous fluid, it reaches constant terminal velocity when drag + buoyancy = weight:

v_t = rac{2r^2( ho - sigma)g}{9eta}

$ho$ = density of sphere, $sigma$ = density of fluid. $v_t propto r^2$ — a larger sphere falls faster.

Reynolds Number (Re): Dimensionless number predicting laminar vs turbulent flow:

Re = rac{ ho v L}{eta}

$Re < 1000$ : laminar (streamlined). $Re > 2000$ : turbulent. $1000 < Re < 2000$ : transition zone.

6. Surface Tension and Capillarity

Surface tension (T or S) is the force per unit length along the liquid surface, or equivalently, the surface energy per unit area:

T = rac{F}{L} = rac{W}{A}

SI unit: N/m. Dimensional formula: $[MT^{-2}]$ . Surface tension decreases with increasing temperature.

Excess pressure inside curved surfaces:

Surface	Excess pressure
Liquid drop (1 surface)	$Delta P = 2T/R$
Soap bubble (2 surfaces)	$Delta P = 4T/R$

Capillarity: Rise or fall of liquid in a narrow tube due to surface tension:

h = rac{2Tcos heta}{ ho g r}

$heta$ = contact angle. $heta < 90°$ (wetting liquid, e.g., water in glass): rises ( $h > 0$ ). $heta > 90°$ (non-wetting, e.g., mercury in glass): depressed ( $h < 0$ ). Finer the tube, higher the rise.

NEET tip: Capillary rise is independent of the shape of the tube (same rise in cylindrical and conical tube of same radius at base). The work done in capillary rise is supplied by surface tension — not by any external agency.

7. NEET Traps & Formula Summary

Trap 1 — Pressure depends on depth, not shape: A tall thin container and a wide shallow container both with 1 m of water have the same pressure at the bottom. Pressure =

ho g h

, independent of container shape or amount of water.

Trap 2 — Soap bubble has TWO surfaces: Excess pressure inside a soap bubble =

4T/R

, not

2T/R

(which is for a single curved surface like a liquid drop). The factor of 2 is because a soap bubble has an inner and outer surface.

Trap 3 — Terminal velocity ∝ r² (not r): Stokes' drag ∝

r

, but buoyancy and weight ∝

r^3

. At terminal velocity,

v_t propto r^2

— doubling the radius quadruples the terminal speed.

Trap 4 — Bernoulli NOT applicable to viscous flow: Bernoulli assumes no energy loss. Real fluids lose energy to viscosity. For viscous flow, add a pressure drop term (Poiseuille's law for pipe flow).

Formula Sheet:

Fluid pressure	$P = P_0 + ho gh$
Buoyant force	$F_b = ho_{fluid} V_{sub} g$
Continuity	$A_1v_1 = A_2v_2$
Bernoulli	$P + rac{1}{2} ho v^2 + ho gh = C$
Torricelli (efflux)	$v = sqrt{2gh}$
Stokes' law	$F = 6pieta rv$
Terminal velocity	$v_t = 2r^2( ho-sigma)g/9eta$
Liquid drop excess P	$Delta P = 2T/R$
Soap bubble excess P	$Delta P = 4T/R$
Capillary rise	$h = 2Tcos heta/ ho gr$

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