Properties of Matter Cheat Sheet - NDA Physics Notes

Welcome to the comprehensive LibreTexts-style chapter guide for Properties of Matter, tailored specifically for UPSC NDA (National Defence Academy) aspirants. This study guide compiles key formulas, theoretical foundations, and high-yield concepts required to master General Science Physics section questions.

1. Elasticity & Mechanical Properties of Solids

Elasticity is the fundamental property of a body by virtue of which it tends to regain its original size and shape when deforming forces are removed.

• Stress (\(\sigma\)): Restoring force developed per unit area of cross-section. \[\sigma = \frac{F}{A} \quad [\text{SI Unit: N/m}^2 \text{ or Pa}]\]
• Strain (\(\epsilon\)): Fractional change in dimensions produced in the body. \[\text{Longitudinal Strain } \epsilon = \frac{\Delta L}{L}, \quad \text{Volumetric Strain } \epsilon_v = \frac{\Delta V}{V}\]
• Hooke's Law: Within the elastic limit, stress is directly proportional to strain. \[\sigma = E \cdot \epsilon\]

Elastic Potential Energy

Work done in stretching a wire is stored as elastic potential energy per unit volume (\(u\)):

\[u = \frac{1}{2} \times \text{Stress} \times \text{Strain} = \frac{1}{2} Y \epsilon^2 = \frac{\sigma^2}{2Y}\]

2. Surface Tension & Capillarity

Surface tension arises due to cohesive intermolecular forces acting on molecules at the liquid surface.

• Definition: Force per unit length acting perpendicular to an imaginary line drawn on the liquid surface. \[T = \frac{F}{l} \quad [\text{Unit: N/m or J/m}^2]\]
• Work Done (\(W\)) in Forming Surfaces:
  • Liquid Drop (1 free surface): \(W = T \cdot \Delta A = T (4\pi r^2)\)
  • Soap Bubble (2 free surfaces): \(W = T \cdot 2(\Delta A) = 8\pi r^2 T\)
• Excess Pressure (\(\Delta P\)):
  • Spherical liquid drop: \(\Delta P = \frac{2T}{r}\)
  • Soap bubble (2 surfaces): \(\Delta P = \frac{4T}{r}\)
  • Cylindrical drop: \(\Delta P = \frac{T}{r}\)
• Capillary Ascent/Descent Formula: \[h = \frac{2T \cos\theta}{r \varrho g}\] where \(\theta\) is angle of contact, \(\varrho\) is fluid density, and \(r\) is tube radius.

3. Hydrostatics & Fluid Pressure

• Fluid Pressure: \[P = \frac{F}{A}, \quad P = P_0 + \varrho gh\]
• Pascal's Law: Pressure applied to an enclosed, incompressible fluid is transmitted undiminished throughout the fluid in all directions.
• Archimedes' Principle: \[F_b = V_{\text{submerged}} \varrho_{\text{fluid}} g = \text{Weight of displaced fluid}\]

4. Hydrodynamics (Fluid Dynamics)

• Equation of Continuity: For incompressible fluid flow through non-uniform cross-section: \[A_1 v_1 = A_2 v_2 = \text{constant}\]
• Bernoulli's Theorem: Statement of conservation of energy for ideal liquid flow: \[P + \frac{1}{2}\varrho v^2 + \varrho gh = \text{constant}\]
• Torricelli's Law (Speed of Efflux): \[v = \sqrt{2gh}\]
• Venturi Meter Flow Speed: \[v = \sqrt{\frac{2gh(\varrho_m - \varrho)}{\varrho}}\]

5. Viscosity & Viscous Drag

Viscosity measures internal resistance to flow caused by momentum transfer between fluid layers.

• Newton's Law of Viscosity: \[F = -\eta A \frac{dv}{dx}\] where \(\eta\) is coefficient of viscosity and \(\frac{dv}{dx}\) is velocity gradient.

Capillary Rise Mechanism Diagram

```tikz \begin{tikzpicture}[scale=1.0] % Outer Container \draw[thick] (-2.5,0) -- (-2.5,3.5) -- (2.5,3.5) -- (2.5,0); % Liquid level in container \fill[blue!15] (-2.5,0) rectangle (2.5,2.0); \draw[thick, blue!80!black] (-2.5,2.0) -- (-0.6,2.0); \draw[thick, blue!80!black] (0.6,2.0) -- (2.5,2.0); % Capillary Tube \draw[thick, fill=white] (-0.6,0.5) rectangle (-0.4,4.2); \draw[thick, fill=white] (0.4,0.5) rectangle (0.6,4.2); % Liquid inside tube \fill[blue!25] (-0.4,0.5) rectangle (0.4,3.2); \draw[thick, blue!80!black] (-0.4,3.2) arc[start angle=180, end angle=360, radius=0.4cm]; % Dimension lines \draw[dashed] (-2.0,2.0) -- (1.5,2.0); \draw[dashed] (-0.4,3.2) -- (1.5,3.2); \draw[<->, thick] (1.2,2.0) -- (1.2,3.2) node[midway, right] {$h$}; % Annotations ode[right] at (1.8, 1.0) {Liquid (Density $\varrho$)}; ode[above] at (0, 4.3) {Capillary Tube (Radius $r$)}; \end{tikzpicture} ```