Unit 7: Properties of Solids and Liquids

This is a content-heavy unit covering both mechanical properties of solids (elasticity) and mechanics of fluids (hydrostatics, hydrodynamics, viscosity, surface tension). NEET sets 2–3 MCQs here, and questions are mostly formula-recall with light arithmetic. Surface tension and capillary rise yield assertion-reason questions almost every year.

Concept Map

• Elasticity
  • Stress, strain
  • Young's, bulk, shear moduli; Poisson's ratio
  • Stress-strain curve
  • Elastic PE
• Hydrostatics
  • Pressure variation with depth
  • Pascal's law
  • Archimedes' principle, buoyancy
• Hydrodynamics
  • Streamline vs turbulent flow
  • Equation of continuity
  • Bernoulli's equation and applications
• Viscosity
  • Newton's law of viscosity
  • Stokes' law, terminal velocity
  • Reynolds number
• Surface tension
  • Capillary rise
  • Excess pressure (drops, bubbles)

Topic 1: Elasticity

Sub-topic A: Stress and Strain

Stress $= F/A$ (force per unit area), SI unit N/m² $=$ Pa.

Three types:

• Longitudinal (tensile/compressive): along length.
• Volumetric: uniform pressure.
• Shear (tangential): force parallel to surface.

Strain $=$ change/original (dimensionless):

• Longitudinal: $\Delta L/L$.
• Volumetric: $\Delta V/V$.
• Shear: $\tan\phi \approx \phi$ (angular deformation).

Sub-topic B: Hooke's Law and Elastic Moduli

Within the elastic limit: stress $\propto$ strain. The proportionality constant is a modulus.

Compressibility $= 1/K$. Liquids have $K$ only (no $Y$, no $\eta$).

Sub-topic C: Poisson's Ratio

When stretched along length, lateral dimensions contract. Define

Typically $0 \le \sigma \le 0.5$. For most metals $\sigma \approx 0.3$.

Relations among moduli (theoretical limits):

Sub-topic D: Stress-Strain Curve

For a typical ductile material:

• Proportional limit: stress ∝ strain holds.
• Elastic limit / yield point: beyond here, permanent deformation.
• Plastic region: strain hardening.
• Ultimate stress: max stress before necking.
• Fracture point.

Brittle materials (cast iron, glass) fracture near the elastic limit. Ductile (steel, copper) show extensive plastic region.

Sub-topic E: Elastic PE

Energy stored per unit volume:

For a wire of natural length $L$, area $A$, stretched by $x$: $U = \tfrac{1}{2} (Y A/L) x^2$, i.e. behaves like a spring with $k = Y A/L$.

Topic 2: Hydrostatics

Sub-topic A: Pressure

Atmospheric pressure $\approx 1.013 \times 10^5\ \text{Pa} = 760\ \text{mm Hg} = 1\ \text{atm}$.

Sub-topic B: Pressure in a Fluid Column

The pressure depends only on depth, not shape of vessel. This explains the hydrostatic paradox.

Sub-topic C: Pascal's Law

Pressure applied to a confined fluid is transmitted equally in all directions. Used in hydraulic press, lift, brakes.

Hydraulic press: Effort $F_1$ on small piston of area $A_1$ produces load $F_2$ on big piston of area $A_2$:

Sub-topic D: Archimedes' Principle

A body wholly or partly submerged experiences an upward buoyant force equal to the weight of fluid displaced:

Floating body: total buoyancy = weight, so

An iceberg ($\rho_\text{ice} = 0.917 \rho_\text{water}$) floats with ~92% submerged.

Topic 3: Hydrodynamics

Sub-topic A: Streamline and Turbulent Flow

• Streamline: velocity at each point is constant in time (steady flow), particles follow smooth paths.
• Turbulent: chaotic, swirling.
• Transition controlled by Reynolds number:

Re < 2000 streamline; Re > 4000 turbulent; in between, unstable.

Sub-topic B: Equation of Continuity

For an incompressible fluid:

Narrower pipe ⟹ faster flow.

Sub-topic C: Bernoulli's Equation

For steady, incompressible, non-viscous flow along a streamline:

Three terms: pressure energy, kinetic energy per volume, potential energy per volume.

Sub-topic D: Applications of Bernoulli

• Venturi meter: pressure difference $\Delta P = \tfrac{1}{2} \rho (v_2^2 - v_1^2)$.
• Pitot tube: measures fluid speed.
• Aerodynamic lift: faster flow over curved wing top ⟹ lower pressure on top.
• Spinning ball curve (Magnus effect).
• Torricelli's theorem: speed of efflux from a small hole at depth $h$ below the free surface:

For a vessel of cross-section $A$, hole of area $a$, time to drain from height $h_0$ to $h_1$ is

Topic 4: Viscosity

Sub-topic A: Newton's Law of Viscous Force

For a fluid with velocity gradient $dv/dy$ across layers,

where $\eta$ is the coefficient of viscosity (dynamic viscosity). SI unit Pa·s = poise/10. Dimensions $[M L^{-1} T^{-1}]$.

Viscosity of liquids decreases with temperature; of gases increases.

Sub-topic B: Stokes' Law

For a small sphere of radius $r$ moving with speed $v$ through a fluid:

Sub-topic C: Terminal Velocity

When the sphere falls under gravity through a viscous fluid, it reaches terminal speed when weight $-$ buoyancy $=$ drag:

Hence $v_t \propto r^2$ — bigger particles fall faster.

Topic 5: Surface Tension

Sub-topic A: Definition

Surface tension $T$ is the force per unit length on any line drawn in the surface, acting tangentially:

Equivalently, $T$ is the surface energy per unit area (units N/m or J/m²). Dimensions $[M T^{-2}]$.

Surface tension decreases with temperature and falls to zero at the critical temperature.

Sub-topic B: Angle of Contact

The angle between the tangent to the liquid surface and the solid wall (measured inside the liquid). Examples:

• Water-glass: $\theta \approx 0°$ (wets, rises in capillary).
• Mercury-glass: $\theta \approx 138°$ (does not wet, depresses).

Sub-topic C: Capillary Rise

A capillary tube of radius $r$ dipped vertically in a liquid:

The smaller the tube, the higher the rise (Jurin's law). For water in glass ($T \approx 0.073\ \text{N/m}$, $\theta = 0$), in a 1 mm tube, $h \approx 1.5$ cm.

If the tube is too short for the predicted height, the liquid does not overflow — the meniscus radius adjusts.

Sub-topic D: Excess Pressure

The pressure inside a curved surface exceeds outside by:

When two bubbles of radii $r_1$ and $r_2$ ($r_1 < r_2$) merge through a tube, the smaller one collapses into the larger one (lower pressure inside larger).

Sub-topic E: Work Done to Form a Drop / Bubble

Work done by external agent to form a drop = increase in surface area $\times T$:

• Spherical drop of radius $R$: $W = 4\pi R^2 T$.
• Soap bubble: $W = 8\pi R^2 T$ (two surfaces).
• Splitting drop of radius $R$ into $n$ identical droplets of radius $r$ (volume conservation gives $R^3 = n r^3$): surface energy increases by $\Delta W = 4\pi T (n r^2 - R^2) = 4\pi T R^2 (n^{1/3} - 1)$.

NEET Pattern MCQ Tips

• Young's modulus: thin wire problem, $\Delta L = FL/(AY)$.
• Bulk modulus: pressure-induced volume change.
• Elastic PE: $U = \tfrac{1}{2} \cdot \text{stress} \cdot \text{strain} \cdot V$.
• Pascal/hydraulic press: force ratio = area ratio.
• Archimedes: floating fraction = $\rho_\text{body}/\rho_\text{liquid}$.
• Bernoulli / Torricelli: $v = \sqrt{2gh}$.
• Capillary: smaller tube ⟹ higher rise; bubbles inside a liquid.
• Excess pressure: drop $2T/R$, soap bubble $4T/R$.
• Stokes & terminal velocity: $v_t \propto r^2$.

Common Confusions and Traps

• $Y$, $K$, $\eta_\text{shear}$ apply to solids; for fluids only $K$ exists.
• The capillary rise formula assumes the meniscus is hemispherical. The smaller the tube radius, the higher the rise.
• A soap bubble in air has TWO surfaces (inner and outer), so excess pressure is $4T/R$. A liquid drop in air or air bubble in liquid has ONE surface, so $2T/R$.
• Two bubbles merged: smaller transfers air to larger (smaller has higher pressure).
• Stokes' law applies to small spheres in laminar flow; large/fast objects experience inertial drag.
• Reynolds number is dimensionless — students sometimes give it units.
• Surface tension decreases with temperature (and with surfactants like soap), eventually vanishing at $T_c$.

Quick Revision Card

• $Y =$ stress/strain (longitudinal); $K =$ stress / fractional volume change; $\eta_\text{shear} =$ tangential stress / shear angle.
• Elastic PE per unit volume: $\tfrac{1}{2}\,\text{stress}\,\times\,\text{strain}$.
• Hydrostatic: $P = P_0 + \rho g h$.
• Buoyancy: $F = V \rho_\text{fluid} g$; floating fraction = $\rho_\text{body}/\rho_\text{liquid}$.
• Continuity: $A v =$ const.
• Bernoulli: $P + \tfrac{1}{2}\rho v^2 + \rho g h =$ const.
• Torricelli: $v = \sqrt{2gh}$.
• Stokes: $F = 6\pi\eta r v$; $v_t = 2 r^2 (\rho_s - \rho_f) g/(9\eta)$.
• Capillary: $h = 2T\cos\theta/(r\rho g)$.
• Drop: $\Delta P = 2T/R$; bubble (air-liquid): $\Delta P = 2T/R$; soap bubble (in air): $\Delta P = 4T/R$.

Worked NEET Examples

Example 1: Strain Energy in Wire

A steel wire (Y = 2 × 10¹¹ N/m²), length 1 m, cross-section 1 mm². Stretched by 1 mm. Strain energy?

$U = (1/2)Y\varepsilon^2 V = (1/2)(2 \times 10^{11})(10^{-3})^2 (10^{-6} \times 1) = (1/2)(2 \times 10^{11})(10^{-6})(10^{-6}) = 10^{-1}$ J = 0.1 J.

Example 2: Pressure at Depth

Pressure at 100 m below water surface: $P = P_0 + \rho g h = 10^5 + 10^3 \times 10 \times 100 = 1.1 \times 10^6$ Pa $\approx 11$ atm.

Example 3: Capillary Tube in Water

Water (T = 0.073 N/m, contact angle ≈ 0°) in glass capillary of 0.5 mm radius. Rise:

Example 4: Terminal Velocity of Droplet

A water droplet of radius 0.1 mm falls through air (η = 1.8 × 10⁻⁵ Pa s).

Example 5: Bernoulli on Tank Hole

A water tank with surface 5 m above a small hole at the side. Speed of water exiting:

Derivations

Bernoulli's Equation

For steady, incompressible, non-viscous flow, work-energy theorem on a fluid element gives:

Interpretation: pressure energy + KE per volume + PE per volume is constant along a streamline.

Stokes' Law

For laminar flow past a sphere, dimensional analysis gives drag $\propto \eta r v$. Detailed solution (no derivation expected at NEET level) gives prefactor $6\pi$:

Terminal Velocity

A sphere of density $\rho_s$ falling through a fluid of density $\rho_f$ experiences three forces:

1. Weight: $(4/3)\pi r^3 \rho_s g$ (down).
2. Buoyancy: $(4/3)\pi r^3 \rho_f g$ (up).
3. Drag: $6\pi\eta r v$ (up).

At terminal velocity, net force = 0:

Solving: $v_t = (2/9) r^2 (\rho_s - \rho_f) g/\eta$.

Capillary Rise

Surface tension $T$ acts along the contact line $2\pi r$ at angle $\theta$ to the wall. Vertical component supports the column of weight:

giving Jurin's law:

Excess Pressure in a Drop

Consider half of a drop of radius $R$. Surface tension along the equator pulls inward with force $T \cdot 2\pi R$. Excess pressure inside pushes the half outward over area $\pi R^2$. At equilibrium:

Soap bubble has two surfaces (inner and outer), so excess pressure is $4T/R$.

Special Hydrodynamic Cases

Vena Contracta

Water exiting a sharp-edged orifice constricts to about 62% of the orifice area before re-expanding. The discharge coefficient is ~0.62 for sharp edges, ~0.97 for rounded.

Venturi Meter

A constriction in a horizontal pipe. From continuity and Bernoulli (h constant):

So $\Delta P = \tfrac{1}{2}\rho(v_2^2 - v_1^2) > 0$ (pressure drop at constriction). Measure $\Delta P$ to deduce flow rate.

Pitot Tube

A bent tube facing the flow. Stagnation pressure $P_0 = P + \tfrac{1}{2}\rho v^2$. By measuring $P_0 - P$, the flow speed is

Surface Tension Phenomena

Why Insects Walk on Water

Water surface acts like a stretched membrane with $T \approx 0.073$ N/m. A water strider's legs push down on the surface, deforming it; surface tension provides upward force without breaking the surface.

Soap Films vs Water Films

Surfactants in soap lower water's surface tension (from 0.073 to about 0.025 N/m), allowing stable thin films. The drop in $T$ also makes droplets larger (capillary rise is less).

Sphericality of Drops

Free drops are spherical because a sphere minimises surface area for a given volume — surface tension minimises the surface energy.

Viscosity in Daily Life

• Engine oils have viscosity ratings (SAE 10W-40).
• Honey: $\eta \sim 10\ \text{Pa s}$.
• Water at 20 °C: $\eta = 10^{-3}\ \text{Pa s}$.
• Air at 20 °C: $\eta \sim 1.8 \times 10^{-5}\ \text{Pa s}$.

Viscosity of liquids decreases with temperature (molecules slide past each other more easily). Viscosity of gases increases with temperature (more momentum exchange between layers).

Formula Sheet