Mechanical Properties of Fluids — Class 11 Physics
Class 11 Physics · Chapter 10

Mechanical Properties
of Fluids

A complete visual guide for NEET & JEE — concepts, numericals, MCQs, and mnemonics all in one place.

🔴 Pressure ⚙️ Pascal's Law 🛟 Buoyancy ✈️ Bernoulli 🧮 Numericals 📝 MCQs ⚡ Revision
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Start with curiosity

Why should you care about fluids?

Before formulas, before equations — let real-life questions spark your curiosity. Every concept you'll learn answers one of these!

🚢
Why do massive ships float but a tiny needle sinks?
A ship is hollow — it displaces a huge volume of water. The buoyant force equals the weight of displaced water, which exceeds the ship's weight. A needle displaces barely any water!
🚢
🚿
Why does water flow faster when you narrow a pipe with your thumb?
Same volume of water must pass through a smaller area — so it speeds up. Bernoulli adds: faster fluid → lower pressure. That's why perfume sprays work!
💨
🪲
Why can small insects walk on water without sinking?
Water's surface acts like a stretched elastic membrane (surface tension). The insect's weight is too small to break this "skin". Add soap → surface tension drops → insect sinks!
🕷️
🩸
Why does blood flow faster in thin arteries than in the aorta?
Continuity of flow: A₁v₁ = A₂v₂. Smaller cross-sectional area → higher velocity. Same reason highways slow down when lanes merge!
❤️
🧠
Master Mnemonic for the whole chapter:
"Please Apply Some Very Beautiful Concepts"
Pressure → Archimedes (Pascal) → Surface Tension → Viscosity → Bernoulli → Continuity
🔴
Pressure in Fluids
P = F/A · Hydrostatic pressure · Pascal's law
🛟
Archimedes' Principle
Buoyant force · Floating condition · Apparent weight
🫧
Surface Tension
T = F/L · Soap bubble · Capillarity
✈️
Bernoulli's Theorem
Energy conservation · Airplane lift · Venturimeter
Topic 1

Fluid Basics

A fluid is any substance that can flow — both liquids and gases qualify.

💧
Liquid
Definite volume, no definite shape
Has definite volume but takes the shape of its container. Weakly compressible. Examples: water, oil, mercury, blood.
🌬️
Gas
No definite volume or shape
No definite volume or shape — expands to fill its container. Highly compressible. Examples: air, steam, CO₂, natural gas.
Density
ρ = m / V
SI unit: kg/m³  |  Water: 1000 kg/m³  |  Mercury: 13,600 kg/m³  |  Air: 1.225 kg/m³
Relative Density (Specific Gravity) — dimensionless!
RD = ρ_substance / ρ_water
RD of iron ≈ 7.8  |  RD of ice ≈ 0.92 (why ice floats!)  |  RD of cork ≈ 0.2
⚠️
Critical exam point: Relative density has NO units. Writing kg/m³ after it costs you marks! Density has units; relative density does NOT.
Topic 2

Pressure in Fluids

Pressure is force per unit area. In fluids, it acts in all directions and increases with depth.

Fundamental definition
P = F / A
SI unit: Pascal (Pa) = N/m²  |  1 atm = 1.013 × 10⁵ Pa  |  1 bar = 10⁵ Pa
Hydrostatic pressure at depth h
P = P₀ + ρgh
P₀ = atmospheric pressure, ρ = fluid density, g = 9.8 m/s², h = depth below surface
👠
High heels vs flat shoes: Same weight (force), but high heels have tiny area → HUGE pressure → you sink into soft ground! Flat shoes spread the force → less pressure. Same logic explains why sharp knives cut easily.

Key rules to remember

Pressure increases linearly with depth (P ∝ h)
Double the depth → double the pressure. The P vs h graph is a straight line through the origin.
Same horizontal level → same pressure in a connected fluid
Regardless of the shape of the container. This is why a U-tube manometer works!
Pressure acts equally in all directions at a point
Unlike solid surfaces, fluid pressure pushes on every surface — up, down, sideways.
Gauge pressure = Absolute pressure − Atmospheric pressure
Your tyre pressure gauge shows gauge pressure, not absolute. P_gauge = P_abs − P₀
⚡ Interactive: Pressure Calculator
Depth h (m) 10 m
Fluid type
98,000
Hydrostatic P (Pa)
199,325
Absolute P (Pa)
2.0×
× Atmosphere
Topic 3

Pascal's Law & Hydraulic Machines

Pressure applied to an enclosed fluid is transmitted equally in all directions — the basis of all hydraulic machines.

Pascal's Law — Force multiplication
F₁/A₁ = F₂/A₂  →  F₂ = F₁ × (A₂/A₁)
Small force on small piston → Large force on large piston. Area ratio is the force multiplier!
✨
Area Magic Trick: If A₂ = 100 × A₁, output force is 100× the input. But you're NOT creating energy — the small piston moves 100× more distance. Work in = Work out (energy is always conserved).
⚡ Hydraulic Lift Simulator
Input force F₁ (N) 100 N
Small piston A₁ 5 cm²
Large piston A₂ 100 cm²
2.00
Pressure (×10⁵ Pa)
2,000
Output F₂ (N)
20×
Force amplification

Real-world applications

🚗 Car lift
Small pump lifts entire car. Force × 100 easily achieved at a service station.
🛑 Hydraulic brakes
Foot pedal force transmitted equally to all four wheel cylinders simultaneously.
🔩 Hydraulic press
Industries crush and mold metal with enormous force from small input.
🦷 Dentist chair
Foot pump raises entire chair + patient smoothly and safely.
Topic 4

Archimedes' Principle & Buoyancy

When an object is immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced.

Buoyant force (Upthrust)
F_b = ρ_fluid × V_displaced × g
V_displaced = volume of fluid pushed aside — not necessarily the object's full volume (for partial immersion)
Apparent weight
W_apparent = mg − F_b = (ρ_obj − ρ_fluid) × V × g
If W_apparent < 0, the object floats upward. If = 0, it is neutrally buoyant.

🟢 Object FLOATS if...

  • ρ_object < ρ_fluid
  • F_b ≥ weight of object
  • Only partially submerged
  • Examples: wood, ice, ship

🔴 Object SINKS if...

  • ρ_object > ρ_fluid
  • F_b < weight of object
  • Apparent weight is positive
  • Examples: iron, stone, coin
🧠
Golden Mnemonic: "Less dense → Lives (floats)" — If density is less than the fluid, the object lives on the surface!
⚡ Buoyancy Calculator
Object density (kg/m³) 800
Volume (×10⁻³ m³) 10
Fluid
0.098
Buoyant force (N)
0.078
Weight (N)
FLOATS
Behaviour

Special cases for exams

Ice in water — 90% submerged
ρ_ice = 917, ρ_water = 1000. Fraction submerged = 917/1000 = 0.917. Only 8.3% is visible — origin of "tip of the iceberg"!
Iron ship floats because of trapped air
Average density (steel hull + air) is less than 1000 kg/m³. The large enclosed air volume reduces effective density.
Salt water gives more buoyancy than fresh water
ρ_seawater = 1025 > ρ_fresh = 1000. Ships float slightly higher at sea than in rivers (Plimsoll line accounts for this).
Topic 5

Surface Tension & Capillarity

The surface of a liquid behaves like a stretched elastic membrane — molecules at the surface have higher energy than those inside.

Surface tension — force per unit length
T = F / L
SI unit: N/m = J/m²  |  Water at 20°C: T ≈ 0.073 N/m  |  Mercury: 0.465 N/m  |  Soap solution: ~0.025 N/m
Excess pressure — soap bubble (2 surfaces)
ΔP = 4T / r
Two surfaces (inner + outer) → 4T/r
Excess pressure — water drop (1 surface)
ΔP = 2T / r
One surface only → 2T/r
Capillary rise height
h = 2T cosθ / (ρgr)
θ = contact angle, r = capillary radius. Smaller radius → greater rise. Jurin's Law!
🧠
SOAP = 4, DROP = 2: Soap bubble has 4T/r (4 letters in SOAP). Water drop has 2T/r (2 letters in DR). Count the surfaces — bubble has two, drop has one!
🌡️
Temperature effect: Surface tension DECREASES with temperature. Hot water has lower T — that's why hot water + detergent cleans better than cold water alone. At boiling point, T → 0.
💧 Spherical droplets
Sphere has minimum surface area for a given volume → minimum surface energy. Nature always minimizes energy.
📍 Floating needle
If placed gently without breaking the surface, needle floats due to surface tension acting along its length.
🌱 Plant water transport
Capillary action in thin xylem tubes draws water upward against gravity to heights of 100+ meters in tall trees.
🌡️ Mercury meniscus
Contact angle > 90° → convex meniscus → mercury depresses in a capillary tube (opposite of water).
Topic 6 & 7

Viscosity & Fluid Flow

Viscosity is the internal friction between fluid layers moving at different speeds — the resistance to flow.

Newton's law of viscosity
F = η × A × (dv/dy)
η (eta) = coefficient of viscosity (Pa·s = N·s/m²). dv/dy = velocity gradient between layers.
Stokes' law — drag on sphere
F = 6πηrv
r = radius, v = velocity of sphere through fluid
Terminal velocity
v_t = 2r²(ρ−σ)g / 9η
ρ = sphere density, σ = fluid density. v_t ∝ r²!
Reynolds number (predicts flow type)
Re = ρvD / η
Re < 2000 → Laminar (smooth)  |  2000–3000 → Transitional  |  Re > 3000 → Turbulent (chaotic)

🌊 Laminar (Streamline) Flow

  • Smooth, orderly motion
  • Fluid layers don't mix
  • Low velocity, small diameter
  • Re < 2000

🌀 Turbulent Flow

  • Chaotic, irregular motion
  • Layers mix vigorously
  • High velocity, large diameter
  • Re > 3000
⚠️
NEET Trap — Viscosity and Temperature:
For liquids: viscosity DECREASES with temperature (honey flows easier when warm).
For gases: viscosity INCREASES with temperature.
Completely opposite behaviour! Specify the phase in your answer.
Topic 8

Bernoulli's Theorem

Energy conservation applied to fluid flow: the sum of pressure, kinetic, and potential energy remains constant along a streamline.

Bernoulli's equation (per unit volume)
P + ½ρv² + ρgh = constant
P = pressure energy/volume  |  ½ρv² = kinetic energy/volume  |  ρgh = potential energy/volume
Equation of continuity (prerequisite)
A₁v₁ = A₂v₂   (Volume flow rate Q = Av = constant)
Narrow section → faster flow. Wide section → slower flow. Same volume per second must pass through.
✈️
Golden Rule — memorize this forever:
"Faster fluid → Lower pressure"
When v↑, P must ↓ (total energy = constant). This single rule explains airplane lift, venturimeter, atomizer, and storm roof damage!

Key applications

✈️ Airplane wing (Aerofoil lift)
Wing is curved on top → air travels farther/faster on top → lower pressure on top → net upward force = lift! This is why planes can fly.
🌀 Venturimeter
Narrow constriction speeds up fluid → pressure drops. Measure pressure difference with a manometer → calculate flow velocity using Bernoulli.
💨 Perfume atomizer / spray
Squeeze bulb → fast air jet over narrow tube → low pressure above tube → atmospheric pressure pushes liquid up → fine mist spray.
🏏 Magnus effect (spinning ball)
Spinning ball drags air with it → faster on one side → lower pressure → ball curves! Cricket swing, football curve, tennis topspin — all Magnus effect.
🏠 Storm blowing off roofs
Storm wind blows fast over roof → low pressure above → higher atmospheric pressure below → net upward force → roof lifts off!
⚡ Bernoulli Explorer (Horizontal pipe, ρ = 1000 kg/m³)
v₁ at wide section 5 m/s
Area ratio A₁/A₂ 4
P₁ (×10⁵ Pa) 3
20.0
v₂ at narrow (m/s)
1.22
P₂ (×10⁵ Pa)
1.78
ΔP drop (×10⁵ Pa)
Practice Problems

Step-by-Step Numericals

Follow the 5-step method: Identify concept → Write formula → Substitute → Check units → Final answer.

Easy — Hydrostatic Pressure
A column of water is 10 m deep. Find the pressure at the bottom. (ρ_water = 1000 kg/m³, g = 9.8 m/s²)
1
Concept: Hydrostatic pressure at depth
2
Formula: P = ρgh
3
Substitute: P = 1000 × 9.8 × 10 = 98,000 Pa
4
Unit check: kg/m³ × m/s² × m = Pa ✓
✅ Answer: P = 9.8 × 10⁴ Pa = 98 kPa
Easy — Pascal's Law
A hydraulic lift has pistons of area 10 cm² and 500 cm². Input force = 200 N. Find the output force.
1
Concept: Pascal's law — equal pressure transmission
2
Formula: F₂ = F₁ × (A₂/A₁)
3
Substitute: F₂ = 200 × (500/10) = 200 × 50 = 10,000 N
✅ Answer: F₂ = 10,000 N = 10 kN (enough to lift a small car!)
NEET Level — Archimedes
A metal block of volume 200 cm³ and density 8000 kg/m³ is fully submerged in water. Find: (a) buoyant force, (b) apparent weight.
1
Given: V = 200 cm³ = 200 × 10⁻⁶ m³, ρ_obj = 8000 kg/m³, ρ_water = 1000 kg/m³
2
Buoyant force: F_b = 1000 × 200×10⁻⁶ × 9.8 = 1.96 N
3
Actual weight: W = 8000 × 200×10⁻⁶ × 9.8 = 15.68 N
4
Apparent weight: W_app = 15.68 − 1.96 = 13.72 N
✅ F_b = 1.96 N  |  Apparent weight = 13.72 N
NEET Level — Terminal Velocity
A sphere of radius 2 mm falls through glycerine (η = 1.5 Pa·s, ρ_glycerine = 1260 kg/m³). Density of sphere = 7800 kg/m³. Find terminal velocity.
1
Formula: v_t = 2r²(ρ−σ)g / 9η
2
Values: r = 2×10⁻³ m, ρ = 7800, σ = 1260, g = 9.8, η = 1.5
3
v_t = 2 × (4×10⁻⁶) × 6540 × 9.8 / (9 × 1.5)
4
= 512,736 × 10⁻⁶ / 13.5 ≈ 0.038 m/s
✅ Terminal velocity ≈ 3.8 cm/s
JEE Level — Bernoulli
Water flows in a horizontal pipe: radius 4 cm (section 1), radius 2 cm (section 2). v₁ = 3 m/s, P₁ = 2 × 10⁵ Pa. Find P₂. (ρ = 1000 kg/m³)
1
Continuity: A₁v₁ = A₂v₂ → π(0.04)²×3 = π(0.02)²×v₂ → v₂ = 12 m/s
2
Bernoulli (horizontal, ρgh cancels): P₁ + ½ρv₁² = P₂ + ½ρv₂²
3
P₂ = 2×10⁵ + ½ × 1000 × (9 − 144)
4
P₂ = 2×10⁵ − 67,500 = 132,500 Pa
✅ P₂ = 1.325 × 10⁵ Pa — pressure dropped as velocity rose!
Exam Practice

Practice MCQs

Click an option to get instant feedback with full explanation. Build your exam instinct!

NEET Type
A body floats with 1/3 of its volume above the surface of water. The density of the body is:
If 1/3 is above surface → 2/3 is submerged. Fraction submerged = ρ_body/ρ_water. So ρ_body = 1000 × 2/3 = 667 kg/m³. Key formula: fraction submerged = ρ_object / ρ_fluid.
NEET Type
The excess pressure inside a soap bubble of radius R is P. If radius doubles to 2R, excess pressure becomes:
ΔP = 4T/r → ΔP ∝ 1/r. If r doubles, ΔP becomes P/2. Surface tension T doesn't change with radius — only r changes. Remember: SOAP = 4T/r, DROP = 2T/r.
JEE Type
Water flows in a horizontal pipe. At a constriction, velocity increases. The pressure at the constriction:
Direct Bernoulli application! P + ½ρv² = constant. When v↑, P must ↓. "Faster fluid → Lower pressure" — the single most important rule in Bernoulli's theorem.
NEET Type
Terminal velocity of a sphere of radius r is v. If radius is doubled, new terminal velocity will be:
v_t = 2r²(ρ−σ)g/9η → v_t ∝ r². If r doubles → r² becomes 4r² → v_t becomes 4v. This squared relationship is the most common trap in viscosity numericals!
Assertion-Reason
Assertion: Ships are made hollow.  |  Reason: To reduce effective density of the ship below that of water.
Making ships hollow traps air. Average density = total mass / total volume (including air). This reduces effective density below 1000 kg/m³ → ship floats. The reason correctly and completely explains the assertion. Answer: A.
Error Analysis

Common Mistakes & Corrections

These errors appear in student answer sheets every year. Identify them, understand why they're wrong, and never repeat them.

❌ Writing units (kg/m³) after relative density
✅ Relative density is DIMENSIONLESS — it's a ratio. Never add units. If you do, marks are deducted automatically.
❌ "Buoyant force depends on the weight of the object"
✅ F_b = ρ_fluid × V_displaced × g. It depends ONLY on the fluid's density and the volume displaced — NOT on the object's weight or density!
❌ "A floating object is always fully submerged"
✅ Only partially submerged. Fraction submerged = ρ_object / ρ_fluid. Ice at 0.92 → 92% under water. Cork at 0.2 → only 20% under water.
❌ "Soap bubble excess pressure = 2T/r (same as water drop)"
✅ Soap bubble has TWO surfaces: ΔP = 4T/r. Water drop has ONE surface: ΔP = 2T/r. Always identify — is it a bubble or a drop?
❌ "Viscosity of all fluids increases with temperature"
✅ For LIQUIDS: viscosity DECREASES with T. For GASES: viscosity INCREASES with T. Completely opposite! Always specify the phase.
❌ "Higher velocity means higher pressure" (applying Bernoulli incorrectly)
✅ Exactly OPPOSITE! Higher velocity → LOWER pressure. P + ½ρv² = constant. v↑ means P↓. This is the most misunderstood concept in this chapter.
❌ "Terminal velocity is proportional to radius r" (v_t ∝ r)
✅ v_t ∝ r² (radius SQUARED). v_t = 2r²(ρ−σ)g/9η. Double the radius → quadruple the terminal velocity. This is the most tested numerical relationship in Stokes' law.
❌ Forgetting to convert cm³ to m³ in numerical problems
✅ 1 cm³ = 10⁻⁶ m³. Always convert before substituting in SI formulas. A volume of 200 cm³ = 200 × 10⁻⁶ m³ = 2 × 10⁻⁴ m³.
5-Minute Power Revision

Quick Revision Sheet

Your complete exam-ready cheat sheet. All formulas, all mnemonics — in one place.

All formulas at a glance

Pressure
P = F/A = P₀ + ρgh
Pascal's Law
F₁/A₁ = F₂/A₂
Buoyant Force
F_b = ρ_f × V_d × g
Float condition
ρ_obj < ρ_fluid
Surface Tension
T = F/L = W/ΔA
Soap bubble ΔP
4T/r
Water drop ΔP
2T/r
Capillary rise
h = 2T cosθ / ρgr
Stokes' Law
F = 6πηrv
Terminal Velocity
v_t = 2r²(ρ−σ)g / 9η
Reynolds Number
Re = ρvD / η
Continuity Eq.
A₁v₁ = A₂v₂
Bernoulli Eq.
P + ½ρv² + ρgh = C
Fraction Submerged
ρ_obj / ρ_fluid
Relative Density
ρ_sub / ρ_water (no unit!)
Apparent Weight
W − F_b

Master mnemonics

🧠
"Please Apply Some Very Beautiful Concepts"
Pressure → Archimedes → Surface Tension → Viscosity → Bernoulli → Continuity
✈️
"Fast → Low" (Bernoulli)
Higher velocity = Lower pressure. Airplane lift, venturimeter, atomizer, storm damage — ALL use this one rule.
🛟
"Less dense → Lives (floats)" (Buoyancy)
Object density < fluid density → it floats. Object density > fluid density → it sinks.
🫧
"SOAP = 4, DROP = 2" (Surface Tension)
Soap bubble excess pressure = 4T/r. Water drop = 2T/r. Count the surfaces — bubble has two, drop has one.
🌊
"Re < 2000 = Calm, Re > 3000 = Chaos" (Reynolds)
Low Reynolds number = laminar, orderly flow. High = turbulent, chaotic flow. 2000–3000 = unpredictable transition.
Fluids is easy now! 🎉

You've covered every concept, solved every type of numerical, and practised exam-level MCQs. Go ace that paper!

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