Bulk matter — whether a steel girder, water in a pipe, or a soap film — responds to forces and to heat in ways governed by a handful of quantitative laws. This unit is a reliable source of 4–6 NEET questions every year, and most of them are direct applications of a formula plus a careful reading of units and geometry.
Elastic Behaviour of Solids
A solid resists a change in its shape or size because its atoms sit in a potential-energy minimum; displacing them sets up internal restoring forces. Elasticity is the tendency of a body to regain its original configuration once the deforming force is withdrawn; if it does not recover at all, it is plastic (e.g. putty).
Two quantities describe the deformed state:
- Stress = internal restoring force per unit area = $F/A$, unit N m⁻² (pascal). It is longitudinal (tensile or compressive) when the force is normal to the area, shearing when it is tangential, and hydraulic when the body is squeezed uniformly from all sides.
- Strain = fractional deformation, a dimensionless number: $\Delta L/L$ (longitudinal), $\Delta x/L = \theta$ (shear), $\Delta V/V$ (volume).
Hooke's law states that for small deformations stress is directly proportional to strain. The constant of proportionality is a modulus of elasticity:
- Young's modulus $Y = \dfrac{F L}{A,\Delta L}$ — resistance to stretching of a wire or rod. Steel has a larger $Y$ than copper, hence steel is more elastic in the physics sense.
- Shear (rigidity) modulus $G = \dfrac{F}{A\theta}$ — defined only for solids, since fluids cannot support a static shear.
- Bulk modulus $B = -\dfrac{\Delta P}{\Delta V/V}$ — the minus sign keeps $B$ positive because volume shrinks when pressure rises. Its reciprocal $1/B$ is the compressibility; gases are far more compressible than liquids, which in turn are far more compressible than solids.
Poisson's ratio $\sigma = \dfrac{-\Delta r/r}{\Delta L/L}$ (lateral contraction per unit longitudinal extension) lies between about 0.2 and 0.5 for real materials and has no unit.
On a tensile stress–strain graph for a metal you should be able to identify: the straight proportional region (Hooke's law valid), the elastic limit / yield point beyond which permanent set appears, a plastic plateau, the ultimate tensile strength (maximum stress), and the fracture point. Brittle materials break soon after the elastic limit; ductile materials show a long plastic region.
Work done in stretching is stored as elastic potential energy: $$U = \tfrac{1}{2},(\text{stress})(\text{strain})\times \text{volume} = \frac{1}{2}\frac{Y A (\Delta L)^2}{L}$$ so the energy density is $\frac{1}{2}\times$ stress $\times$ strain.
Fluids at Rest: Pressure and Buoyancy
A fluid at rest exerts a force normal to any surface in contact with it; the pressure $P = F_\perp/A$ is a scalar and at a point is the same in every direction. In a fluid of density $\rho$ the pressure increases with depth: $$P = P_0 + \rho g h$$ where $P_0$ is the pressure at the free surface. Consequences worth memorising:
- Pressure depends only on vertical depth, not on the shape or cross-section of the container (the "hydrostatic paradox").
- Gauge pressure is $P - P_{\text{atm}}$; a manometer or the "pressure" quoted for a tyre is gauge pressure.
- 1 atm ≈ $1.013\times10^5$ Pa ≈ 76 cm of mercury.
Pascal's law: a change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to every part of it. This is the basis of the hydraulic lift and hydraulic brakes, where a small force on a narrow piston ($F_1/A_1$) produces a large force on a wide piston ($F_2 = F_1 A_2/A_1$). The device multiplies force, not energy — the small piston moves through a proportionally larger distance.
Archimedes' principle: a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced, $F_B = \rho_{\text{fluid}} V_{\text{displaced}}, g$, acting at the centre of buoyancy. A floating body displaces fluid equal to its own weight, so the fraction submerged equals $\rho_{\text{body}}/\rho_{\text{fluid}}$.
Fluids in Motion: Continuity, Bernoulli and Viscosity
Flow is streamline (laminar) when every particle passing a given point follows the same path with the same velocity; it becomes turbulent above a critical speed, with eddies and mixing. The dimensionless Reynolds number $$R_e = \frac{\rho v D}{\eta}$$ predicts the regime: roughly $R_e < 1000$ laminar, $> 2000$ turbulent.
For steady flow of an incompressible fluid the equation of continuity expresses conservation of mass: $A_1v_1 = A_2v_2$, i.e. the fluid speeds up where the tube narrows.
Bernoulli's theorem applies energy conservation per unit volume to non-viscous, incompressible, steady, streamline flow: $$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$ Standard applications:
- Torricelli's law (speed of efflux from a small orifice at depth $h$): $v = \sqrt{2gh}$, the same as free fall.
- Venturimeter — measures flow rate from the pressure drop at a constriction.
- Dynamic lift on an aerofoil, the swing of a spinning cricket ball (Magnus effect), the "lifting" of a roof in a storm, and the working of an atomizer.
Real fluids have viscosity: adjacent layers moving with different velocities exert tangential drag on each other. For laminar flow, $$F = -\eta A \frac{dv}{dx}$$ defining the coefficient of viscosity $\eta$ (SI unit Pa s = decapoise; CGS unit poise). Liquid viscosity falls with rising temperature, whereas gas viscosity rises.
A sphere of radius $r$ moving slowly through a viscous medium feels Stokes' drag $F = 6\pi\eta r v$. When drag plus buoyancy balances weight, the sphere attains a constant terminal velocity $$v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$$ with $\rho$ the sphere's density and $\sigma$ the fluid's. Note $v_t \propto r^2$ — this is why fine mist droplets appear to hang in the air.
Surface Tension and Capillarity
Molecules in the interior of a liquid are pulled equally in all directions, but a molecule at the surface has no liquid above it, so the net inward attraction makes the surface behave like a stretched membrane. Surface tension $S$ is the force per unit length acting along the surface, in N m⁻¹, and is numerically equal to the surface energy per unit area (J m⁻²). Because surface energy costs work, a free liquid mass minimises area and becomes spherical.
Key results:
- Work to increase surface area by $\Delta A$ is $W = S,\Delta A$ (remember a soap film has two surfaces).
- Excess pressure inside a spherical liquid drop: $\Delta P = 2S/R$; inside a soap bubble: $\Delta P = 4S/R$; inside an air bubble in a liquid: $2S/R$. Smaller bubbles have higher internal pressure, so when two bubbles are connected the smaller one empties into the larger.
- Angle of contact $\theta$ is measured inside the liquid between the solid surface and the tangent to the liquid surface. $\theta < 90^\circ$ (water–glass) means the liquid wets the solid and the meniscus is concave; $\theta > 90^\circ$ (mercury–glass) means non-wetting, convex meniscus and capillary depression.
- Capillary rise in a tube of radius $r$: $$h = \frac{2S\cos\theta}{r\rho g}$$ so narrower tubes give greater rise ($h \propto 1/r$, Jurin's law).
Surface tension decreases with temperature (vanishing at the critical temperature) and is lowered by detergents (surfactants), which is precisely why soapy water penetrates cloth fibres better.
Thermal Properties: Expansion, Calorimetry and Heat Transfer
Temperature measures the degree of hotness; heat is energy in transit due to a temperature difference. On heating, solids expand: linear $\Delta L = \alpha L\Delta T$, areal $\Delta A = \beta A \Delta T$, volumetric $\Delta V = \gamma V\Delta T$, with $\beta = 2\alpha$ and $\gamma = 3\alpha$ for an isotropic solid. Water is anomalous — it contracts on warming from 0 °C to 4 °C, so it is densest at 4 °C, which allows aquatic life to survive under ice.
Specific heat capacity $s$ satisfies $Q = m s \Delta T$; the molar specific heat uses moles instead of mass. In a calorimeter (an isolated system) heat lost by the hot body equals heat gained by the cold body — the principle of calorimetry.
During a change of state temperature stays constant while heat supplied breaks intermolecular bonding: $Q = mL$, where $L$ is the latent heat of fusion or vaporisation. For water, $L_f \approx 3.34\times10^5$ J kg⁻¹ and $L_v \approx 22.6\times10^5$ J kg⁻¹. A useful graphical picture is the temperature–heat graph for a substance warmed from ice through to steam: sloped segments (temperature rising, specific heat capacity governs the slope) alternate with flat plateaus (temperature constant while latent heat is absorbed at a phase change) — the plateau lengths are proportional to the respective latent heats.
Heat travels from a hotter region to a colder one by three mechanisms:
- Conduction — energy passed molecule to molecule without bulk motion, dominant in solids. For steady-state conduction through a slab of area $A$, thickness $d$, and thermal conductivity $K$, the rate of heat flow is $\dfrac{dQ}{dt} = \dfrac{KA,\Delta T}{d}$. Metals conduct well because of free electrons; for slabs in series the same heat current flows through each, while for slabs in parallel the temperature difference is common and the heat currents add.
- Convection — heat transfer by actual bulk movement of a fluid (natural convection driven by density differences, or forced convection as in a fan or pump). This is why land and sea breezes form, and why a room is heated from a floor-level radiator rather than a ceiling one.
- Radiation — energy transfer by electromagnetic waves, needing no medium at all; it is how the Sun's energy reaches the Earth. A perfectly black surface is both the best absorber and the best emitter, described by Stefan's law, $\dfrac{dQ}{dt} = \sigma A T^4$ for an ideal black body, while Newton's law of cooling — valid only for a small temperature excess over the surroundings — states that the rate of cooling is proportional to that excess temperature.
Common Mistakes and Exam Traps
- Using the linear, areal and volumetric expansion coefficients interchangeably. Remember $\beta = 2\alpha$ and $\gamma = 3\alpha$ hold only for an isotropic solid; treating area or volume expansion as numerically equal to linear expansion is a frequent slip.
- Forgetting water's anomalous expansion between 0 °C and 4 °C. Most substances contract steadily on cooling, but water contracts only down to 4 °C and then expands as it approaches 0 °C — a question about ice forming on a pond's surface or the survival of aquatic life underneath is testing exactly this anomaly.
- Confusing the excess-pressure formulas for a drop, a soap bubble, and a submerged bubble. A soap bubble in air has two liquid surfaces (excess pressure $4S/R$), while a liquid drop or an air bubble inside a liquid has only one surface (excess pressure $2S/R$) — swapping these is one of the most common errors in this topic.
- Assuming Bernoulli's theorem applies to viscous or unsteady flow. It strictly requires non-viscous, incompressible, steady, streamline flow; a rough pipe or a fluid with significant viscosity means the theorem must be corrected or abandoned, not applied blindly.