Properties of Solids and Liquids formulas for JEE Main & Advanced

24 formula entries for Properties of Solids and Liquids, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Stress and Strain

Stress = \frac{F}{A}

Strain = \frac{Δ L}{L}

(longitudinal),

Shear Strain = \frac{Δ x}{h}

Volumetric Strain = \frac{- Δ V}{V}

When to use. When analyzing deformations in solids under external forces.

Trap. Confusing stress with force; stress is force per unit area, not just force.

Elastic Moduli

Y = \frac{Tensile Stress}{Longitudinal Strain} = \frac{F / A}{Δ L / L}

B = - \frac{Δ P}{Δ V / V}

G = \frac{Shear Stress}{Shear Strain} = \frac{F / A}{Δ x / h}

When to use. When calculating deformation from applied forces or pressure changes in solids.

Trap. Forgetting the negative sign in Bulk modulus definition (pressure increase reduces volume).

Stress-Strain Curve and Elastic Limit

Elastic limit = maximum stress without permanent deformation

When to use. When interpreting material behavior under load, especially for safety factors.

Trap. Thinking the proportional limit and elastic limit are the same; they are close but not identical.

Poisson's Ratio

ν = - \frac{Lateral Strain}{Longitudinal Strain} = - \frac{Δ d / d}{Δ L / L}

When to use. When a tensile or compressive force causes both axial and lateral dimensional changes.

Trap. Forgetting the negative sign; lateral strain is opposite in sign to longitudinal strain.

Elastic Energy Stored per Unit Volume

U = \frac{1}{2} \times stress \times strain = \frac{1}{2} Y (strain)^{2} = \frac{(stress)^{2}}{2 Y}

When to use. When calculating energy stored in a stretched wire, compressed spring, or any elastic deformation.

Trap. Using the formula for plastic deformation; it only applies within elastic limit.

Fluid Pressure and Pascal's Law

P = \frac{F}{A}

Δ P = ρ g h

(for a column of fluid),

P = P_{0} + ρ g h

When to use. When analyzing forces on submerged surfaces, hydraulic lifts, or pressure at depth.

Trap. Thinking pressure is a vector; it's a scalar (magnitude only, acts perpendicular to surfaces).

Buoyancy and Archimedes' Principle

F_{b} = ρ_{fluid} \times V_{displaced} \times g

When to use. When an object is floating or submerged in a fluid, to find apparent weight or floating condition.

Trap. Using the density of the object instead of the fluid in buoyant force calculation.

Equation of Continuity

A_{1} v_{1} = A_{2} v_{2}

(for incompressible fluid)

When to use. When fluid flows through pipes of varying cross-section, to find velocity changes.

Trap. Applying to compressible fluids (gases) without considering density changes.

Bernoulli's Equation

P + \frac{1}{2} ρ v^{2} + ρ g h = constant

When to use. When analyzing fluid flow in pipes, nozzles, Venturi meters, or lift on airplane wings.

Trap. Applying Bernoulli's equation to viscous fluids or turbulent flow without modifications.

Torricelli's Theorem (Efflux Speed)

v = \sqrt{2 g h}

When to use. When calculating the speed of liquid coming out of a hole in a container.

Trap. Forgetting that the formula assumes the tank is open to atmosphere and the orifice is small compared to tank area.

Viscosity and Newton's Law of Viscosity

τ = η \frac{d v}{d y}

, where

η

is coefficient of viscosity.

When to use. When calculating drag force on a moving object in a fluid, or flow through pipes.

Trap. Confusing dynamic viscosity (η) with kinematic viscosity (ν = η/ρ).

Stokes' Law and Terminal Velocity

F_{d} = 6 π η r v

v_{t} = \frac{2}{9} \frac{(ρ_{s} - ρ_{f}) g r^{2}}{η}

When to use. When a sphere falls through a fluid (e.g., raindrop, ball bearing in oil).

Trap. Using Stokes' law for large velocities or turbulent flow; it's valid only for low Reynolds number (laminar flow).

Poiseuille's Law (Laminar Flow in a Pipe)

Q = \frac{π R^{4} Δ P}{8 η L}

When to use. When calculating flow rate or pressure drop in a pipe for viscous fluids (e.g., blood flow, oil pipelines).

Trap. Forgetting that the formula is for laminar flow only; turbulent flow has different behavior.

Surface Tension and Surface Energy

γ = \frac{F}{L}

γ = \frac{Work}{Δ A}

When to use. When analyzing liquid surfaces, bubbles, droplets, capillary rise, or wetting phenomena.

Trap. Thinking surface tension and surface energy per unit area are different; they are numerically and dimensionally the same ( $N/m = {J/m}^{2}$ ).

Excess Pressure in Drops and Bubbles

Δ P_{drop} = \frac{2 γ}{R}

Δ P_{bubble} = \frac{4 γ}{R}

When to use. When calculating the pressure inside a liquid droplet, soap bubble, or air bubble in liquid.

Trap. Using the bubble formula for a liquid drop; a drop has one surface, a bubble has two.

Capillary Rise/Depression

h = \frac{2 γ \cos θ}{ρ g r}

, where

θ

is contact angle.

When to use. When calculating the height of liquid in a capillary tube, or the radius from rise height.

Trap. Forgetting the contact angle; for water in glass, θ≈0°, cosθ=1; for mercury, θ≈140°, cosθ negative, giving depression.

Heat Transfer: Conduction, Convection, Radiation

\frac{d Q}{d t} = - k A \frac{d T}{d x}

(Fourier's law),

\frac{d Q}{d t} = σ ϵ A T^{4}

(Stefan-Boltzmann for radiation)

When to use. When analyzing heat flow through walls, rods, or from hot objects.

Trap. Confusing thermal conductivity k with thermal diffusivity α = k/(ρc).

Thermal Expansion of Solids and Liquids

Δ L = L_{0} α Δ T

Δ A = A_{0} β Δ T

with

β = 2 α

Δ V = V_{0} γ Δ T

with

γ = 3 α

When to use. When calculating expansion of rods, sheets, or liquid volumes due to temperature change.

Trap. Forgetting that for liquids, only volumetric expansion is considered; linear expansion doesn't apply.

Calorimetry and Specific Heat

Q = m c Δ T

Q = m L_{f}

(fusion),

Q = m L_{v}

(vaporization)

When to use. When calculating final temperature or amount of substance after mixing, or heat required for phase change.

Trap. Forgetting to account for phase changes; using mcΔT when the substance changes phase.

Newton's Law of Cooling

\frac{d T}{d t} = - k (T - T_{s})

When to use. When a hot object cools in a constant-temperature environment (e.g., cup of coffee).

Trap. Applying the law for large temperature differences; it's an approximation for small differences.

Ideal Gas Equation and Kinetic Theory

P V = n R T

P = \frac{1}{3} ρ v_{rms}^{2}

\frac{1}{2} m v_{rms}^{2} = \frac{3}{2} k T

When to use. When relating pressure, volume, temperature of gases, or calculating molecular speeds.

Trap. Using R = 8.314 J/molK in problems with atm and liters; need consistent units (SI: Pa, m³).

Degrees of Freedom and Specific Heats of Gases

C_{v} = \frac{f}{2} R

C_{p} = \frac{f + 2}{2} R

γ = 1 + \frac{2}{f}

When to use. When calculating heat capacities or adiabatic exponents for gases.

Trap. Forgetting that for monatomic gas f=3, diatomic (rigid) f=5, polyatomic f=6 (or more).

Thermodynamic Processes: Isothermal, Adiabatic, Isobaric, Isochoric

Isothermal:

P V = const

W = n R T \ln (V_{2} / V_{1})

; Adiabatic:

P V^{γ} = const

W = \frac{P_{2} V_{2} - P_{1} V_{1}}{1 - γ}

; Isobaric:

W = P Δ V

; Isochoric:

W = 0

When to use. When analyzing engines, compressors, or any thermodynamic cycle.

Trap. Using the adiabatic relation for isothermal processes and vice versa.

First Law of Thermodynamics

Δ U = Q - W

When to use. In all thermodynamic processes to relate heat, work, and internal energy.

Trap. Sign convention: W is work done BY the system; if work is done ON the system, W is negative.

Values to remember

$g = 9.8 {m/s}^{2}$ Fluid Pressure and Pascal's Law
$ρ_{water} = 1000 {kg/m}^{3}$ Fluid Pressure and Pascal's Law
$η_{water} \approx 10^{- 3} Pa \cdot s$ Viscosity and Newton's Law of Viscosity
$η_{air} \approx 1.8 \times 10^{- 5} Pa \cdot s$ Viscosity and Newton's Law of Viscosity
$γ_{water} \approx 0.072 N/m$ Surface Tension and Surface Energy
$γ_{mercury} \approx 0.465 N/m$ Surface Tension and Surface Energy
$σ = 5.67 \times 10^{- 8} {W/m}^{2} K^{4}$ Heat Transfer: Conduction, Convection, Radiation
$k_{copper} \approx 400 W/mK$ Heat Transfer: Conduction, Convection, Radiation
$k_{wood} \approx 0.1 W/mK$ Heat Transfer: Conduction, Convection, Radiation
$α_{steel} \approx 1.2 \times 10^{- 5} K^{- 1}$ Thermal Expansion of Solids and Liquids
$α_{glass} \approx 8.5 \times 10^{- 6} K^{- 1}$ Thermal Expansion of Solids and Liquids
$c_{water} = 4186 J/kgK$ Calorimetry and Specific Heat
$L_{f} = 3.34 \times 10^{5} J/kg$ Calorimetry and Specific Heat
$L_{v} = 2.26 \times 10^{6} J/kg$ Calorimetry and Specific Heat
$R = 8.314 J/molK$ Ideal Gas Equation and Kinetic Theory
$k = 1.38 \times 10^{- 23} J/K$ Ideal Gas Equation and Kinetic Theory
$N_{A} = 6.022 \times 10^{23} {mol}^{- 1}$ Ideal Gas Equation and Kinetic Theory
For monatomic: $C_{v} = 12.47 J/molK$ , $γ = 5 / 3$ Degrees of Freedom and Specific Heats of Gases
For diatomic: $C_{v} = 20.79 J/molK$ , $γ = 7 / 5$ Degrees of Freedom and Specific Heats of Gases

Read the full chapter: Properties of Solids and Liquids for JEE, with what JEE asks and the common traps.