Mechanical Properties of Fluids Class 11 Physics Quick Recall (Short Notes 2026-27)

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

• Pascal's Law: Pressure applied to an enclosed fluid is transmitted undiminished.
• Bernoulli's Theorem: P + ½ρv² + ρgh = Constant. (Conservation of energy).
• Equation of Continuity: A1 v1 = A2 v2. (Conservation of mass).
• Terminal Velocity: v_t = 2r²(ρ - σ)g / 9η.
• Surface Tension: S = F/L. Excess pressure in drop = 2S/R; Bubble = 4S/R. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

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Introduction

Fluids—liquids and gases—are materials that have no definite shape and yield to even the slightest external force. The study of fluids is divided into Hydrostatics (fluids at rest) and Hydrodynamics (fluids in motion). This chapter explores the fundamental laws that explain how airplanes fly, how hydraulic brakes stop a car, and why water droplets form perfect spheres. In this "Comprehensive" guide, we provide exhaustive derivations for Bernoulli’s Theorem, the Equation of Continuity, and Terminal Velocity—providing the technical rigor required for top-tier competitive exams like JEE and NEET.

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1. Fluid Statics: Pressure and Pascal's Law

I. Variation of Pressure with Depth

Derivation:

1. Consider a fluid column of height h and area A.
2. Weight of fluid W = m g = (ρ V) g = (ρ A h) g.
3. Pressure P = Force / Area = (ρ A h g) / A. Result: P = P_atmosphere + ρgh. Conclusion: Pressure increases linearly with depth.

II. Pascal’s Law

Theorem: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. Application: Hydraulic Lift. F1 / A1 = F2 / A2.

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2. Fluid Dynamics: The Laws of Flow

I. Equation of Continuity

Statement: For an incompressible, non-viscous fluid in a pipe, the rate of flow remains constant. Derivation:

1. Mass of fluid entering per second = Mass leaving per second (Conservation of Mass).
2. ρ1 A1 v1 = ρ2 A2 v2.
3. For incompressible fluids (ρ1 = ρ2):
   • A1 v1 = A2 v2. (Proven) Result: Where the pipe narrows, the velocity increases.

II. Bernoulli’s Theorem (The Master Derivation)

Statement: For an ideal fluid in steady flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant. P + 1/2 ρv² + ρgh = Constant

Derivation (Based on Work-Energy Theorem):

1. Work done by Pressure forces (W_p): (P1 - P2) ΔV.
2. Change in Kinetic Energy (ΔK): 1/2 ρ ΔV (v2² - v1²).
3. Change in Potential Energy (ΔU): ρ ΔV g (h2 - h1).
4. By Work-Energy Theorem: W_p = ΔK + ΔU.
5. (P1 - P2) ΔV = 1/2 ρ ΔV (v2² - v1²) + ρ ΔV g (h2 - h1).
6. Dividing by ΔV and rearranging:
   • P1 + 1/2 ρv1² + ρgh1 = P2 + 1/2 ρv2² + ρgh2. (Proven)

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3. Viscosity and Stokes’ Law

Viscosity is the internal friction between fluid layers.

I. Stokes’ Law

Statement: The viscous force F acting on a sphere of radius r moving with velocity v through a fluid of viscosity η is: F = 6πηrv.

II. Derivation: Terminal Velocity (v_t)

When a body falls through a viscous medium, it eventually reaches a constant speed called terminal velocity.

1. Forces acting on a sphere: Weight (Down) = Buoyancy (Up) + Viscous Force (Up).
2. mg = B + 6πηrv_t.
3. Substituting m = 4/3 πr³ ρ and B = 4/3 πr³ σ (where σ is fluid density):
   • 4/3 πr³ ρ g = 4/3 πr³ σ g + 6πηrv_t.
4. Solving for v_t:
   • v_t = [2r² (ρ - σ) g] / 9η. (Proven)

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4. Surface Tension and Capillarity

Surface tension is the result of cohesive forces between molecules at the surface of a liquid.

I. Excess Pressure

• Liquid Drop: ΔP = 2S / R.
• Soap Bubble (2 surfaces): ΔP = 4S / R.

II. Capillary Rise (Ascent Formula)

Derivation:

1. Force of surface tension = Weight of liquid column.
2. (2πR S cosθ) = (πR² h ρ g).
3. h = (2S cosθ) / (R ρ g). (Proven)

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Comprehensive Exam Strategy (Q&A)

Q1: Why do airplanes fly? Answer: According to Bernoulli’s Theorem, the wings are curved such that air travels faster over the top surface than the bottom. Higher velocity creates lower pressure on top, while higher pressure on the bottom provides the Lift Force.

Q2: What happens to the viscosity of (a) Liquids and (b) Gases as temperature increases? Answer:

• Liquids: Viscosity decreases. Molecular bonds weaken as kinetic energy increases, allowing layers to slide more easily.
• Gases: Viscosity increases. Molecular collisions (which cause gas viscosity) become more frequent at higher temperatures.

Q3: Why are raindrops spherical? Answer: Surface Tension. A sphere is the shape with the minimum surface area for a given volume. Since surface tension seeks to minimize potential energy by reducing surface area, raindrops naturally adopt a spherical shape.

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Related Revision Notes

• Chapter 8: Mechanical Properties of Solids
• Chapter 10: Thermal Properties of Matter
• Mastering Hydrodynamics: Advanced Problem Set

Conclusion

The physics of fluids is the physics of flow—from the blood in our veins to the hurricanes in our atmosphere. By mastering the mathematical laws of Bernoulli and the dynamics of terminal velocity, we gain the power to harness fluid energy and design everything from water pumps to spacecraft. Master the derivation of Bernoulli’s Theorem and the nuances of surface tension—these are the fluid principles that keep our world moving. Stay in the flow, watch your Reynolds number, and never let your pressure drop!

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Reference: Physics Classroom: Fluid Mechanics