Properties of Matter

Stress = restoring force per unit area: $\sigma = F/A$ (N/m² or Pa). Strain = fractional change in dimension (dimensionless).

Hooke's Law: Within elastic limit, stress $\propto$ strain. Compressibility $K = 1/B$. Poisson's ratio $\sigma = -\dfrac{\text{lateral strain}}{\text{longitudinal strain}}$; $-1 < \sigma < 0.5$.

Elastic Potential Energy stored per unit volume: $u = \dfrac{1}{2} \times \text{stress} \times \text{strain} = \dfrac{1}{2}Y(\text{strain})^2$.

For a wire: $U = \dfrac{1}{2} \times \dfrac{YA(\Delta L)^2}{L}$. Force constant of wire: $k = YA/L$.

Stress-Strain Curve: Proportional limit → Elastic limit → Yield point → Ultimate strength → Fracture point. Region beyond elastic limit shows permanent (plastic) deformation.

Pressure in a fluid at depth $h$: $P = P_0 + \rho gh$, where $P_0$ = atmospheric pressure. Pascal's law: Pressure applied to enclosed fluid transmits equally in all directions (basis of hydraulic press).

Archimedes' Principle: Buoyant force = weight of displaced fluid. $$F_B = V_{\text{submerged}} \cdot \rho_{\text{fluid}} \cdot g$$

For floating body: $\rho_{\text{body}} V_{\text{total}} = \rho_{\text{fluid}} V_{\text{submerged}}$.

Equation of Continuity (incompressible flow): $A_1 v_1 = A_2 v_2$ (mass conservation).

Bernoulli's Equation (steady, incompressible, non-viscous, streamline flow): $$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$

Sum of pressure energy, kinetic energy, and potential energy per unit volume is constant along a streamline.

Torricelli's Law: Speed of efflux through small hole at depth $h$ below free surface: $$v = \sqrt{2gh}$$ (same as free-fall from height $h$).

Venturimeter: Used to measure flow rate. From Bernoulli + continuity: $$v_1 = a_2\sqrt{\frac{2gh}{a_1^2 - a_2^2}}$$ where $h$ = manometer height difference; $a_1, a_2$ = pipe and throat areas.

Viscosity: Internal friction in a fluid. Newton's law of viscous flow: $F = \eta A \dfrac{dv}{dx}$, where $\eta$ = coefficient of viscosity (Pa·s = $10$ poise). Velocity gradient $dv/dx$ exists between adjacent fluid layers.

Stokes' Law: Viscous drag on a sphere of radius $r$ moving with velocity $v$ in fluid of viscosity $\eta$: $$F = 6\pi\eta r v$$

Terminal Velocity of a sphere falling in a viscous fluid (when net force = 0): $$v_T = \frac{2r^2(\rho - \sigma)g}{9\eta}$$ where $\rho$ = sphere density, $\sigma$ = fluid density.

Reynolds Number $Re = \dfrac{\rho v d}{\eta}$. Flow is laminar if $Re < 2000$, turbulent if $Re > 3000$.

Surface Tension ($T$ or $S$): Force per unit length acting on liquid surface, tangentially. Units: N/m. Equivalently, surface energy per unit area (J/m²).

Excess Pressure:

Capillary Rise (Jurin's law): $h = \dfrac{2T\cos\theta}{\rho gr}$, where $r$ = capillary radius, $\theta$ = contact angle. Mercury falls in glass capillary ($\theta > 90°$).

Thermal Expansion:

Bimetallic strip bends due to different $\alpha$ values; used in thermostats.

Anomalous expansion of water: Maximum density at $4°$C; expansion below $4°$C.

Calorimetry:

• Heat absorbed/released: $Q = mc\Delta T$, where $c$ = specific heat capacity (J/kg·K).
• Latent heat: $Q = mL$ during phase change (constant temperature).
• Principle of calorimetry: heat lost = heat gained (in mixing).
• Specific heat of water = $4186$ J/kg·K; latent heat of fusion of ice = $3.36 \times 10^5$ J/kg; vaporisation of water = $2.26 \times 10^6$ J/kg.

Heat Transfer Mechanisms:

Thermal resistance: $R_{\text{th}} = L/(KA)$, similar to electrical resistance. For series: $R_{\text{eq}} = R_1 + R_2$; parallel: $1/R_{\text{eq}} = 1/R_1 + 1/R_2$.

Wien's Displacement Law: $\lambda_{\max} T = b$, where $b = 2.898 \times 10^{-3}$ m·K.

Newton's Law of Cooling: $\dfrac{dT}{dt} = -k(T - T_s)$, valid for small temperature differences.