Board exam focus — Mechanical Properties of Fluids (CBSE & HBSE)

Mechanical Properties of Fluids covers both fluids at rest and in motion. The hydrostatics part introduces pressure, Pascal's law and its applications (hydraulic lift and brakes), atmospheric pressure, and Archimedes' principle with buoyancy and flotation. The hydrodynamics part develops streamline and turbulent flow, the equation of continuity (conservation of mass) and Bernoulli's principle (conservation of energy), with applications such as the venturimeter, aerofoil lift and the Magnus effect. The chapter then treats viscosity, Stokes' law and terminal velocity, and finally surface tension, surface energy, angle of contact and capillary rise. CBSE and HBSE frequently test Bernoulli-based reasoning, terminal velocity numericals and capillarity.

Pressure, Pascal's Law and Archimedes' Principle

Pressure in Fluids

Pressure is the normal force per unit area exerted by a fluid:

$$P = \frac{F}{A}$$

SI unit: pascal (Pa); dimensions [ML⁻¹T⁻²]. Pressure is a scalar.

Pressure due to a fluid column of height h and density ρ:

$$P = \rho g h$$

Absolute pressure at depth h: P = P₀ + ρgh, where P₀ is atmospheric pressure. The term ρgh is the gauge pressure.

Pressure at a point in a fluid is the same in all directions and depends only on depth, not on the shape of the container (hydrostatic paradox).

Pascal's Law

A change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to every point of the fluid and the walls of the container.

Applications: Hydraulic lift, hydraulic brakes, hydraulic press. In a hydraulic lift:

$$\frac{F_1}{A_1} = \frac{F_2}{A_2}$$

so a small force on a small piston balances a large load on a large piston (force multiplication).

Atmospheric Pressure

The pressure exerted by the weight of the atmosphere; measured by a mercury barometer.

• Standard atmospheric pressure = 1.013 × 10⁵ Pa = 76 cm of mercury = 1 atm.

Archimedes' Principle and Buoyancy

When a body is immersed wholly or partly in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced:

$$F_b = V \rho_{fluid}\, g$$

Law of flotation: A floating body displaces a weight of fluid equal to its own weight. A body floats if its average density ≤ density of the fluid.

Streamline Flow, Continuity and Bernoulli's Principle

Types of Flow

• Streamline (laminar) flow: every fluid particle passing a point follows the same path with the same velocity; layers slide smoothly.
• Turbulent flow: irregular, with eddies; occurs at high speeds.
• Critical velocity marks the transition, governed by the Reynolds number:

$$R_e = \frac{\rho v D}{\eta}$$

Flow is generally laminar for Rₑ < 1000 and turbulent for Rₑ > 2000.

Equation of Continuity

For an incompressible fluid in steady flow, the mass flow rate is constant:

$$A_1 v_1 = A_2 v_2 = \text{constant}$$

This is a statement of conservation of mass — where a pipe narrows, the fluid speeds up. (This is why a stream of water narrows as it falls.)

Bernoulli's Principle

For an ideal (non-viscous, incompressible) fluid in streamline flow, the total energy per unit volume is constant along a streamline:

$$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$

• P = pressure energy per unit volume
• ½ρv² = kinetic energy per unit volume
• ρgh = potential energy per unit volume

It is a statement of conservation of energy for flowing fluids.

Where speed is high, pressure is low. This explains lift on an aeroplane wing, the spin of a cricket ball (Magnus effect), the action of an atomizer, and the venturimeter.

Viscosity, Stokes' Law, Surface Tension and Capillarity

Viscosity

Viscosity is the internal friction between adjacent layers of a fluid in relative motion. The viscous force is:

$$F = -\eta A \frac{dv}{dx}$$

where η is the coefficient of viscosity (SI unit: Pa·s or poiseuille; dimensions [ML⁻¹T⁻¹]) and dv/dx is the velocity gradient.

Viscosity of liquids decreases with temperature, while that of gases increases with temperature.

Stokes' Law and Terminal Velocity

For a small sphere of radius r moving with velocity v through a fluid of viscosity η, the viscous drag is:

$$F = 6\pi\eta r v$$

A body falling through a fluid attains a constant terminal velocity when weight = buoyancy + viscous drag:

$$v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$$

where ρ is the density of the sphere and σ the density of the fluid.

Surface Tension

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

$$T = \frac{F}{L}$$

SI unit: N/m; dimensions [MT⁻²]. It arises from cohesive forces and tends to minimise surface area (why drops are spherical).

Excess pressure: inside a liquid drop, P = 2T/r; inside a soap bubble (two surfaces), P = 4T/r.

Angle of Contact and Capillarity

• Angle of contact is the angle between the tangent to the liquid surface and the solid surface at the line of contact (acute for wetting liquids like water on glass, obtuse for mercury).
• Capillarity: the rise or fall of a liquid in a narrow tube. Capillary rise:

$$h = \frac{2T\cos\theta}{r\rho g}$$

Water rises in a capillary (θ < 90°) but mercury is depressed (θ > 90°). This explains the rise of sap in plants and oil in a wick.

Frequently asked questions

Are these Mechanical Properties of Fluids notes free?

Yes — the Mechanical Properties of Fluids notes for Physics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Mechanical Properties of Fluids notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Mechanical Properties of Fluids chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Mechanical Properties of Fluids.