Properties of Solids and Liquids

1. Solid Elasticity

Elasticity is the property of a material to regain its original shape and size after the removal of deforming force.

1.1 Stress and Strain

Stress: Restoring force developed per unit area.

Strain: Change in dimension per original dimension.

\[\text{Modulus of Elasticity} = \frac{\text{Stress}}{\text{Strain}}\]

Greater modulus of elasticity means greater stress developed for the same strain.

1.2 Young's Modulus

Young's modulus measures the resistance of a solid to change in its length when a force is applied perpendicular to a face.

\[Y = \frac{\sigma}{\epsilon} = \frac{F_n / A}{\Delta L / L_0} = \frac{F_n L_0}{A \Delta L}\]

1.3 Shear Modulus

The shear stress is defined as tangential force per unit area.

\[\tau = \frac{F_t}{A}\]

1.4 Bulk Modulus

Bulk modulus measures the resistance to volume change under pressure.

\[K = -V \frac{dP}{dV}\] \[K = \frac{\text{Normal stress}}{\text{Volumetric strain}} = -V \frac{\Delta P}{\Delta V}\]

Compressibility = \(\frac{1}{K}\)

2. Fluid Properties

2.1 Density and Pressure

The density (ρ) of a substance is defined as mass per unit volume.

\[\rho = \frac{M}{V}\]

SI units: kg/m³

Specific Gravity: Ratio of density to water density at 4°C (1000 kg/m³).

Pressure: Force per unit area at a point within the fluid.

\[p = \frac{F}{A}\]

SI unit: Pascal (Pa) = N/m²

2.2 Variation of Pressure with Depth

Pressure increases with depth in a fluid.

\[\frac{dp}{dh} = \rho g\] \[p = p_0 + \rho g h\]

2.3 Pascal's Law

Pascal's Law

A pressure applied to a confined fluid at rest is transmitted equally undiminished to every part of the fluid and the walls of the container.

2.4 Buoyancy and Archimedes' Principle

Archimedes' Principle

A body immersed in a fluid experiences an upward buoyant force equivalent to the weight of the fluid displaced by it.

2.5 Equation of Continuity

Statement of conservation of mass for fluid flow.

\[\rho_1 A_1 v_1 = \rho_2 A_2 v_2\]

For incompressible fluids (ρ₁ = ρ₂):

\[A_1 v_1 = A_2 v_2\]

2.6 Bernoulli's Equation

\[p + \rho g y + \frac{1}{2} \rho v^2 = \text{constant}\]

3. Surface Tension

Surface tension is the property of liquid surfaces to behave like stretched elastic membranes.

3.1 Definition and Formula

\[T = \frac{F}{l}\]

Unit: Newton/meter (N/m), Dimensions: [MT⁻²]

3.2 Surface Energy

Relation between surface tension and work done:

\[W = T \times \Delta A\] \[T = \frac{W}{\Delta A}\]

If ΔA = 1, then T = W. The work done in increasing surface area by unity equals surface tension.

3.3 Molecular Forces

Cohesive Force

Force of attraction between molecules of the same substance.

Adhesive Force

Force of attraction between molecules of different substances.

3.4 Angle of Contact

The angle inside the liquid between the tangent to the solid surface and the tangent to the liquid surface at the point of contact.

3.5 Capillarity

Rise or fall of liquid in a tube of fine diameter.

\[h = \frac{2T}{R\rho g}\]

Where R is the radius of the capillary tube, ρ is liquid density, and g is acceleration due to gravity.

Key Point:

As radius decreases, height increases - narrower tubes cause greater liquid rise.

4. Viscosity

Viscosity is the internal friction between layers of fluid in motion, opposing relative motion.

4.1 Newton's Law of Viscosity

According to Newton, viscous force F depends on:

Contact area A (F ∝ A)
Velocity gradient Δvₓ/Δz (F ∝ Δvₓ/Δz)

\[F \propto A \frac{\Delta v_x}{\Delta z}\] \[F = \pm \eta A \frac{\Delta v_x}{\Delta z}\]

Where η is the coefficient of viscosity.

4.2 Coefficient of Viscosity

\[\eta = \frac{F}{A(\Delta v_x / \Delta z)}\]

Dimensions: [ML⁻¹T⁻¹], Unit: kg/(meter-second)

4.3 Temperature Effect

Liquids

Viscosity decreases with temperature rise

Gases

Viscosity increases with temperature rise

4.4 Poiseuille's Formula

Volume of liquid flowing per second through a capillary tube:

\[q = \frac{\pi p a^4}{8\eta l}\]

4.5 Stoke's Law and Terminal Velocity

Stoke's Law: Viscous force on a sphere moving through fluid:

\[F = 6\pi \eta r v\]

Terminal Velocity: Constant velocity when viscous force equals net downward force:

\[v = \frac{2}{9} r^2 \frac{(\rho - \sigma)g}{\eta}\]

Key Point:

Terminal velocity is directly proportional to the square of the radius of the falling object.

5. Thermal Properties

5.1 Thermal Expansion

When temperature increases, the size of a body increases.

Linear Expansion

\[\alpha = \frac{\Delta L}{L\Delta T}\] \[L_t = L_0 (1 + \alpha\Delta T)\]

Superficial Expansion

\[\beta = \frac{\Delta A}{A\Delta T}\] \[A_t = A_0 (1 + \beta\Delta T)\]

Cubical Expansion

\[\gamma = \frac{\Delta V}{V\Delta T}\] \[V_t = V_0 (1 + \gamma\Delta T)\]

For isotropic materials: γ = 3α, β = 2α

5.2 Specific Heat and Heat Capacity

\[\text{Heat Capacity } C = \frac{Q}{\Delta T}\] \[\text{Specific Heat } c = \frac{Q}{m\Delta T}\]

SI unit of heat capacity: J/K

5.3 Heat Conduction

Transfer of energy due to temperature difference between adjacent parts.

\[\frac{dQ}{dt} = -kA \frac{dT}{dx}\]

Where k is thermal conductivity, dT/dx is temperature gradient.

5.4 Heat Radiation

Heat transfer without intervening medium.

Key Definitions:

Perfectly Black Body: Absorbs all incident radiation
Absorptive Power (a): Ratio of absorbed to incident radiation

5.5 Kirchhoff's Law

\[\frac{e}{a} = \text{constant}\] \[\frac{e_\lambda}{a_\lambda} = \text{constant}\]

A good absorber is a good emitter at a particular wavelength.

5.6 Stefan's Law

\[E = \sigma T^4 \quad \text{(for black body)}\] \[E = e\sigma T^4 \quad \text{(for other surfaces)}\]

σ = 5.67 × 10⁻⁸ Wm⁻²K⁻⁴ (Stefan's constant)

5.7 Newton's Law of Cooling

Rate of cooling ∝ Temperature difference between body and surroundings (for small differences).

5.8 Wien's Displacement Law

\[\lambda_m T = b = \text{constant}\] \[\lambda_m \propto \frac{1}{T}\]

b = 2.89 × 10⁻³ m·K (Wien's constant)