Properties of Bulk Matter

Elasticity is the intrinsic property of a material body by virtue of which it regains its original shape and size upon removal of deforming external forces. When deforming forces displace atoms from their equilibrium lattice positions, internal restorative interatomic forces develop. Plasticity is the property whereby a body fails to regain its initial dimensions, retaining permanent deformation.

Stress ($\\sigma$): The internal restoring force developed per unit cross-sectional area of the deformed body:

SI Unit: $\text{N/m}^2$ or Pascal ($\text{Pa}$). Dimensional formula: $[\text{M}\text{L}^{-1}\text{T}^{-2}]$. Types of stress include:

• Longitudinal Stress: Normal force per unit area producing change in length (Tensile stress causes elongation; Compressive stress causes shortening).
• Shearing / Tangential Stress: Deforming force acting tangentially to the surface, altering shape at constant volume.
• Volumetric / Hydraulic Stress: Uniform perpendicular fluid pressure acting inward over the entire surface, changing volume without changing shape.

Strain ($\epsilon$): The fractional change in dimension produced in a body by deforming forces:

Strain is a dimensionless and unitless ratio. Longitudinal strain $\epsilon_l = \frac{\Delta L}{L}$; Shearing strain $\theta = \frac{\Delta x}{h} = \tan\theta$; Volume strain $\epsilon_v = \frac{\Delta V}{V}$.

Hooke's Law: Within the elastic limit of a material, stress is directly proportional to strain:

Poisson's Ratio ($\sigma$): The ratio of lateral strain to longitudinal strain:

Theoretical limits: $-1 \le \sigma \le 0.5$. Practical limits for most engineering materials: $0 \le \sigma \le 0.5$ (e.g., steel $\approx 0.28$, rubber $\approx 0.49$).

When a wire of length $L$ and area $A$ is stretched by length $\Delta L$, work is done against internal elastic forces. For an elongation $x$, restoring force is $F = \frac{Y A x}{L}$:

The Elastic Strain Energy Density ($u$, energy per unit volume):

A fluid is a substance that cannot sustain a static shear stress and deforms continuously under tangential forces. Normal force exerted by a fluid per unit surface area is hydrostatic pressure:

SI Unit: Pascal ($1\,\text{Pa} = 1\,\text{N/m}^2$). Standard atmospheric pressure: $1\,\text{atm} = 1.01325 \times 10^5\,\text{Pa} = 760\,\text{mm of Hg} = 1.013\,\text{bar}$.

Pressure Variation with Depth: Consider a cylindrical fluid element of cross-sectional area $A$ and height $h$ in a static fluid of uniform density $\rho$. Vertical force equilibrium requires:

If point 1 is at the open liquid surface exposed to atmospheric pressure $P_0$, the absolute pressure at depth $h$ is:

Gauge Pressure: The excess pressure over atmospheric pressure: $P_{\text{gauge}} = P - P_0 = \rho g h$.

Pascal's Principle: Any change in pressure applied to an enclosed, incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its containing vessel.

Archimedes' Principle: When a body is wholly or partially immersed in a fluid at rest, it experiences an upward buoyant force (upthrust) equal in magnitude to the weight of the fluid displaced by the body:

The buoyant force acts vertically upward through the Center of Buoyancy, which is the center of gravity of the displaced fluid.

Conditions for Flotation: For a body of mass $M$, volume $V$, and average density $\rho_b$ placed in a fluid of density $\rho_f$:

• Sinks: $\rho_b > \rho_f \implies W > F_{B,\max}$. The body accelerates downward to the bottom. Apparent weight: $W_{\text{app}} = W - F_B = V(\rho_b - \rho_f)g$.
• Neutral Equilibrium: $\rho_b = \rho_f \implies W = F_{B,\max}$. The body remains suspended at any depth where it is placed.
• Floats: $\rho_b < \rho_f$. The body floats partially submerged such that weight equals buoyant force: $V \rho_b g = V_{\text{sub}} \rho_f g \implies \frac{V_{\text{sub}}}{V} = \frac{\rho_b}{\rho_f}$.

Viscosity is the internal friction of a fluid that resists relative motion between adjacent parallel layers. In streamline laminar flow over a solid flat boundary, fluid layers move with velocities increasing with distance $x$ from the boundary.

Newton's Law of Viscosity: The tangential viscous force $F$ between two adjacent liquid layers of contact area $A$ is directly proportional to the area $A$ and the transverse velocity gradient $\frac{dv}{dx}$:

where $\eta$ is the coefficient of dynamic viscosity. The negative sign denotes that viscous drag opposes relative motion.

• SI Unit: $\text{N}\cdot\text{s/m}^2 = \text{Pa}\cdot\text{s}$ or Decapoise / Poiseuille ($\text{Pl}$).
• CGS Unit: $\text{dyn}\cdot\text{s/cm}^2 = \text{Poise} \implies 1\,\text{Pa}\cdot\text{s} = 10\,\text{Poise}$.
• Dimensional Formula: $[\eta] = \frac{[F]}{[A][dv/dx]} = \frac{[\text{MLT}^{-2}]}{[\text{L}^2][\text{LT}^{-1}/\text{L}]} = [\text{M}\text{L}^{-1}\text{T}^{-1}]$.
• Temperature Dependence: For liquids, $\eta$ decreases exponentially with temperature (molecules gain thermal energy to overcome cohesive bonds). For gases, $\eta$ increases with temperature ($\eta \propto \sqrt{T}$, due to increased molecular momentum transport).

Sir George Gabriel Stokes showed that when a small smooth spherical body of radius $r$ moves with velocity $v$ through an infinite, homogeneous, stationary viscous medium of viscosity $\eta$, the retarding viscous force is:

Conditions for validity: The sphere must be rigid and smooth, the fluid medium must be boundless compared to the sphere, and the motion must be non-turbulent with no slipping at the solid-fluid boundary.

Consider a small sphere of radius $r$ and density $\rho$ falling vertically under gravity through a viscous fluid of density $\sigma$ and viscosity $\eta$. As the sphere accelerates, downward speed $v$ increases, increasing the upward viscous drag $F_v$.

The three forces acting on the sphere are:

1. Downward true gravitational weight: $W = m g = \left(\frac{4}{3}\pi r^3 \rho\right)g$
2. Upward hydrostatic buoyant upthrust: $F_B = \left(\frac{4}{3}\pi r^3 \sigma\right)g$
3. Upward retarding Stokes viscous drag: $F_v = 6\pi \eta r v$

Net downward force: $F_{\text{net}} = W - F_B - F_v = m\frac{dv}{dt}$.

When the retarding forces grow to balance net downward gravity, net acceleration becomes zero ($\\frac{dv}{dt} = 0$), and the sphere continues falling at a constant maximum speed called the terminal velocity ($v_t$):

The nature of flow through a pipe of diameter $D$ is characterized by the dimensionless Reynolds Number ($R_e$), which represents the ratio of inertial forces to viscous forces:

• $R_e < 1000$: Flow is purely laminar (streamline, orderly).
• $1000 < R_e < 2000$: Flow is unsteady and transitional.
• $R_e > 2000$: Flow becomes turbulent (chaotic, eddying, mixing).

In streamline flow, every particle passing a given point follows the exact same path and possesses the exact same velocity as preceding particles. A bundle of streamlines forms a tube of flow across which fluid cannot cross.

Equation of Continuity: For steady, streamline flow of an incompressible non-viscous fluid through a tube of non-uniform cross-section:

This expresses the Conservation of Mass: the mass flow rate $\frac{dm}{dt} = \rho A v$ remains invariant along the stream. Where the pipe narrows, flow velocity increases proportionally ($v \propto 1/A$).

Statement: For the steady, streamline flow of an ideal fluid (incompressible, non-viscous, and irrotational), the total mechanical energy per unit volume—comprising pressure energy, kinetic energy, and gravitational potential energy—remains strictly constant along any streamline:

Dividing throughout by $\rho g$ yields the Head Formulation (each term with dimensions of length $[\text{L}]$):

• $\frac{P}{\rho g} = \text{Pressure Head}$
• $\frac{v^2}{2g} = \text{Velocity Head}$
• $h = \text{Elevation / Potential Head}$

1. Venturimeter: A device to measure liquid flow rate in a horizontal pipe ($h_1 = h_2$). By continuity $A_1 v_1 = A_2 v_2$. Bernoulli yields $P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2) = \rho_m g h$. The volume rate of flow is:$$Q = A_1 v_1 = A_1 A_2 \sqrt{\frac{2gh}{\left(\frac{\rho_m}{\rho}\right)(A_1^2 - A_2^2)}}$$
2. Torricelli's Law of Efflux: For an open tank with a small orifice at depth $h$ below the free surface, the speed of efflux is identical to the speed acquired in free fall from height $h$:$$v = \sqrt{2gh}$$

   Horizontal range of efflux on ground at depth $H$: $R = v t = \sqrt{2gh}\sqrt{\frac{2(H-h)}{g}} = 2\sqrt{h(H-h)}$, with maximum range $R_{\max} = H$ achieved when the hole is drilled at mid-depth $h = H/2$.

3. Aerodynamic Dynamic Lift: Airfoil geometry forces air to travel faster over the curved top surface ($v_{\text{top}} > v_{\text{bottom}}$), lowering pressure on top ($P_{\text{top}} < P_{\text{bottom}}$) and generating net upward dynamic lift.
4. Magnus Effect: A spinning cricket or tennis ball drags air layers, creating velocity differential and asymmetric pressure that deflects the ball's trajectory in flight.

Liquid molecules exert mutual attractive intermolecular forces within a sphere of molecular activity (radius $\approx 10^{-9}\,\text{m}$).

• Interior Molecule: Completely surrounded by neighbors, experiencing zero net resultant intermolecular force.
• Surface Film Molecule: Lacks upward neighbors, experiencing a net inward downward cohesive pull toward the bulk liquid.

Consequently, the liquid surface contracts to minimize its surface area, behaving like a stretched elastic membrane. Surface Tension ($T$ or $S$) is defined as the force acting per unit length of an imaginary line drawn on the liquid surface perpendicular to the line:

SI Unit: $\text{N/m}$. Dimensions: $[\text{M}\text{T}^{-2}]$.

Bringing molecules from the interior to the surface against the net inward pull requires work, stored as potential energy in the surface film. Surface energy per unit area is numerically equal to surface tension:

Units: $\text{J/m}^2 = \text{N/m}$.

Because of surface tension, the pressure on the concave side of a curved liquid surface is always strictly greater than on the convex side.

Angle of Contact ($\theta$): The angle between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid.

• $\theta < 90^\circ$ (Acute): Adhesive force exceeds cohesive force. Liquid wets the solid, forming a concave meniscus (e.g., pure water on clean glass $\theta \approx 0^\circ$).
• $\theta > 90^\circ$ (Obtuse): Cohesive force dominates. Liquid does not wet solid, forming a convex meniscus (e.g., mercury on glass $\theta \approx 135^\circ\text{--}140^\circ$).

Capillary Ascent Formula: When a clean narrow capillary tube of radius $r$ is dipped vertically into a wetting liquid of density $\rho$, the upward vertical component of surface tension supports the weight of the elevated liquid column:

Jurin's Law: $h \cdot r = \frac{2T\cos\theta}{\rho g} = \text{constant}$. The narrower the capillary bore, the higher the liquid rises.

Thermal expansion occurs because increasing temperature enhances atomic vibrational amplitudes about an asymmetric potential energy well.

• Linear Expansion: $\Delta L = L_0 \alpha \Delta T \implies L = L_0(1 + \alpha \Delta T)$
• Superficial (Area) Expansion: $\Delta A = A_0 \beta \Delta T \implies A = A_0(1 + \beta \Delta T)$
• Cubical (Volume) Expansion: $\Delta V = V_0 \gamma \Delta T \implies V = V_0(1 + \gamma \Delta T)$

For an isotropic solid body, differentiating dimensions reveals:

Anomalous Expansion of Water: Between $0^\circ\text{C}$ and $4^\circ\text{C}$, water contracts on heating (negative volume expansion coefficient $\gamma < 0$). Water reaches its maximum density at $4^\circ\text{C}$ ($1000\,\text{kg/m}^3$). This anomaly causes frozen ice ($0^\circ\text{C}$) to float on top of lakes, insulating $4^\circ\text{C}$ water below and preserving aquatic ecosystems through harsh winters.

Principle of Calorimetry: In an isolated, thermally insulated system, heat lost by warmer bodies equals heat gained by colder bodies until thermal equilibrium is established:

Sensible heat: $Q = m c \Delta T$. Latent heat of phase transformation: $Q = m L$. For water: Latent heat of fusion $L_f = 80\,\text{cal/g} = 3.36 \times 10^5\,\text{J/kg}$; Latent heat of vaporization $L_v = 540\,\text{cal/g} = 2.26 \times 10^6\,\text{J/kg}$.

1. Thermal Conduction (Fourier's Law): Heat flows down a temperature gradient via lattice vibrations (phonons) and free electron collisions without macroscopic material transport:

where $K$ is thermal conductivity (SI unit: $\text{W}/(\text{m}\cdot\text{K})$). Thermal resistance: $R_{\text{th}} = \frac{L}{KA}$ (analogous to Ohm's Law $\Delta T = I_{\text{heat}} R_{\text{th}}$).

2. Thermal Convection: Heat transfer accompanied by actual bulk movement of fluid matter driven by buoyant density gradients (natural convection) or external blowers/pumps (forced convection).

3. Thermal Radiation: Emission of electromagnetic energy requiring no material medium.

• Stefan-Boltzmann Law: Total radiant energy emitted per second per unit area by a black body at absolute temperature $T$ is $E = \sigma T^4$, where $\sigma = 5.67 \times 10^{-8}\,\text{W}/(\text{m}^2\cdot\text{K}^4)$. For an imperfect emitter of emissivity $\epsilon$ ($0 < \epsilon < 1$):$$\frac{dQ}{dt} = \epsilon \sigma A (T^4 - T_0^4)$$
• Wien's Displacement Law: The wavelength $\lambda_{\max}$ corresponding to peak spectral emittance is inversely proportional to absolute temperature:$$\lambda_{\max} T = b \approx 2.898 \times 10^{-3}\,\text{m}\cdot\text{K}$$
• Newton's Law of Cooling: For small temperature difference between body and surroundings ($\Delta T = T - T_0 \ll T_0$):$$\frac{dT}{dt} = -k(T - T_0) \implies \ln\left(\frac{T_1 - T_0}{T_2 - T_0}\right) = k t$$