Physics · Thermal Physics and Properties of Matter

Properties of Solids and Liquids formulas for JEE

Every Properties of Solids and Liquids formula you need for JEE, grouped by concept.

FormulasRevision notes
63 formulas6 concepts
01Elasticity of Solids02Fluid Statics03Fluid Dynamics04Surface Tension05Thermal Expansion and Calorimetry06Heat Transfer
01

Elasticity of Solids

21 formulas

Depression (Sagging) of a Beam

δ=Wl34bd3Y\delta = \frac{W l^3}{4 b d^3 Y}

Sag of a rectangular beam loaded at its center.

applies whenLoad at center, supported exactly at ends.
beamsagbending

Bulk Modulus

B=−pΔV/VB = \frac{-p}{\Delta V/V}

Ratio of hydraulic stress (pressure) to volume strain.

applies whenApplicable to solids, liquids, and gases.
modulusbulkpressure

Compressibility

k=1B=−1ΔpΔVVk = \frac{1}{B} = -\frac{1}{\Delta p} \frac{\Delta V}{V}

The reciprocal of the bulk modulus.

compressibilitybulkelasticity

Elastic Potential Energy

U=12FΔL=12YA(ΔL)2LU = \frac{1}{2} F \Delta L = \frac{1}{2} \frac{Y A (\Delta L)^2}{L}

Total work done stored as energy in a stretched wire.

applies whenWire under Hooke's Law valid region.
energyworkelasticity

Elongation Due to Own Weight

ΔL=ρgL22Y=MgL2AY\Delta L = \frac{\rho g L^2}{2Y} = \frac{M g L}{2 A Y}

Extension of a massive uniform vertical rod due to its own gravity.

applies whenUniform cross-section hanging vertically.
elongationgravityjee-advanced

Elastic Energy Density

u=12σϵ=σ22Y=12Yϵ2u = \frac{1}{2} \sigma \epsilon = \frac{\sigma^2}{2Y} = \frac{1}{2} Y \epsilon^2

Potential energy stored per unit volume of a stretched wire.

applies whenWithin elastic limit.
energydensityelasticity

Hooke's Law

Stress=k×Strain\text{Stress} = k \times \text{Strain}

For small deformations, stress is directly proportional to strain.

applies whenOnly valid in the linear proportional limit of the material.
hookestressstrain

Maximum Height of a Mountain

h=Elastic Limitρgh = \frac{\text{Elastic Limit}}{\rho g}

Calculated by equating base shear stress to the elastic limit of the rock.

applies whenAssumes mountain acts as uniform block exerting purely vertical normal load generating shear.
mountainshearlimit

Poisson's Ratio

ν=Δd/dΔL/L\nu = \frac{\Delta d/d}{\Delta L/L}

Ratio of lateral strain to longitudinal strain.

applies whenWithin elastic limit.
poissonstrainlateral

Shear Modulus (Modulus of Rigidity)

G=F/Aθ=F⋅LA⋅ΔxG = \frac{F/A}{\theta} = \frac{F \cdot L}{A \cdot \Delta x}

Ratio of shearing stress to shearing strain.

applies whenApplicable only to solids.
modulusshearrigidity

Elongation of Rotating Rod

ΔL=ρω2L33Y=Mω2L23AY\Delta L = \frac{\rho \omega^2 L^3}{3Y} = \frac{M \omega^2 L^2}{3 A Y}

Longitudinal extension of a uniform rod hinged at one end and rotating.

applies whenRotating on a smooth horizontal plane with constant angular velocity \omega.
rotationelongationjee-advanced

Shearing Strain

γ=ΔxL=tan⁡θ≈θ\gamma = \frac{\Delta x}{L} = \tan \theta \approx \theta

Ratio of relative displacement of faces to the length, equivalent to the angular displacement.

applies whenAngle \theta must be very small (expressed in radians).
strainshearelasticity

Longitudinal Strain

ϵ=ΔLL\epsilon = \frac{\Delta L}{L}

Ratio of the change in length to the original length.

applies whenDeformation must be small.
strainlongitudinalelasticity

Stress

σ=FA\sigma = \frac{F}{A}

Restoring force developed per unit area of a body.

applies whenForce F must be perpendicular or parallel to area A depending on stress type.
stresselasticity

Maximum Tension in Vertical Circle

Tmax=mg+mv2L=mg+mω2LT_{max} = mg + \frac{m v^2}{L} = mg + m \omega^2 L

Maximum tension acting as deforming force for a wire whirled in a vertical circle.

applies whenOccurs exactly at the lowest point of the circular path.
tensioncircular-motionmechanics

Thermal Stress

σ=YαΔT\sigma = Y \alpha \Delta T

Stress developed in a rigidly clamped rod subjected to a temperature change.

applies whenRod must be clamped at both ends preventing natural expansion.
thermalstressjee-advanced

Volume Strain

ϵV=ΔVV\epsilon_V = \frac{\Delta V}{V}

Ratio of change in volume to the original volume.

applies whenValid for uniform hydraulic compression.
strainvolumeelasticity

Relation between Y, B, and G

Y=9BG3B+GY = \frac{9BG}{3B + G}

Direct relation connecting the three primary elastic moduli.

applies whenIsotropic elastic materials.
modulirelationsjee-advanced

Young's, Bulk and Poisson Relation

Y=3B(1−2ν)Y = 3B(1 - 2\nu)

Relation between Young's Modulus, Bulk Modulus, and Poisson's Ratio.

applies whenIsotropic elastic materials.
modulirelationsjee-advanced

Young's, Shear and Poisson Relation

Y=2G(1+ν)Y = 2G(1 + \nu)

Relation between Young's Modulus, Shear Modulus, and Poisson's Ratio.

applies whenIsotropic elastic materials.
modulirelationsjee-advanced

Young's Modulus

Y=F/AΔL/L=F⋅LA⋅ΔLY = \frac{F/A}{\Delta L/L} = \frac{F \cdot L}{A \cdot \Delta L}

Ratio of longitudinal stress to longitudinal strain.

applies whenApplicable only to solids within the elastic limit.
modulusyoungelasticity
02

Fluid Statics

15 formulas

Absolute Pressure

P=Pa+ρghP = P_a + \rho gh

Absolute pressure at depth h below the surface of a liquid open to atmosphere.

applies whenIncompressible fluid at rest, open to atmospheric pressure.
pressureabsolutestatics

Accelerating Fluid Free Surface

tan⁡θ=ag\tan\theta = \frac{a}{g}

Angle made by the free surface of a fluid undergoing constant horizontal acceleration.

applies whenConstant horizontal acceleration a.
staticsaccelerationjee-advanced

Atmospheric Pressure (Barometer)

Pa=ρghP_a = \rho gh

Atmospheric pressure measured by a mercury barometer.

applies whenPressure above mercury column is zero (vacuum).
barometerpressure

Archimedes Principle

FB=ρfluidVsubmergedgF_B = \rho_{\text{fluid}} V_{\text{submerged}} g

Buoyant force acting on an object completely or partially submerged in a fluid.

applies whenFluid in equilibrium under gravity.
buoyancystaticsjee-advanced

Mass Density

ρ=mV\rho = \frac{m}{V}

Density of a fluid of mass m occupying volume V.

applies whenUniform distribution of mass.
densityfluids

Fluid Element Equilibrium

(P2−P1)A=mg(P_2 - P_1)A = mg

Force balance on a fluid column of area A and mass m.

applies whenFluid at rest.
staticsequilibrium

Gauge Pressure

Pg=P−Pa=ρghP_g = P - P_a = \rho gh

Excess pressure over atmospheric pressure at depth h.

applies whenIncompressible fluid at rest.
pressuregaugestatics

Hydraulic Lift Force

F2=F1(A2A1)F_2 = F_1 \left(\frac{A_2}{A_1}\right)

Force transmitted to a larger piston by applying force F1 on a smaller piston.

applies whenIncompressible fluid at rest.
pascalhydraulic

Pascal's Law Equality

FaAa=FbAb=FcAc\frac{F_a}{A_a} = \frac{F_b}{A_b} = \frac{F_c}{A_c}

Pressure exerted is same in all directions in a fluid at rest.

applies whenFluid at rest, points at the same depth.
pascalstatics

Variation of Pressure with Depth

P2−P1=ρghP_2 - P_1 = \rho gh

Pressure difference between two points separated by vertical depth h.

applies whenIncompressible fluid at rest under gravity.
pressuredepthstatics

Pressure at a Point

P=lim⁡ΔA→0ΔFΔAP = \lim_{\Delta A \to 0} \frac{\Delta F}{\Delta A}

Definition of pressure at a specific point in a fluid.

applies whenLimit as area approaches zero.
pressurestatics

Relative Density

Relative Density=ρρwater at 4∘C\text{Relative Density} = \frac{\rho}{\rho_{\text{water at } 4^\circ\text{C}}}

Ratio of a substance's density to the density of water at 4 degrees Celsius.

applies whenDensity of water taken as 1.0 x 10^3 kg m^-3.
densityfluids

Rotating Fluid Surface

y=ω2x22gy = \frac{\omega^2 x^2}{2g}

Equation of the parabolic free surface of a fluid in a uniformly rotating cylinder.

applies whenConstant angular velocity.
staticsrotationjee-advanced

Force on Side Wall

F=12ρgH2wF = \frac{1}{2}\rho g H^2 w

Total hydrostatic force on a vertical rectangular wall of height H and width w.

applies whenLiquid open to atmosphere, starting from surface (depth 0 to H).
staticsforcejee-advanced

Average Pressure

Pav=FAP_{av} = \frac{F}{A}

Average pressure defined as the normal force acting per unit area.

applies whenForce must be the normal component.
pressurestatics
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03

Fluid Dynamics

15 formulas

Fractional Speed Change (Aerofoil)

Δvvav≈ΔPρvav2\frac{\Delta v}{v_{av}} \approx \frac{\Delta P}{\rho v_{av}^2}

Fractional increase in air speed over an aerofoil.

applies whenHorizontal flight, small height difference across wing.
aerofoildynamicsapproximation

Bernoulli's Work-Energy Form

(P1−P2)ΔV=12ρΔV(v22−v12)+ρgΔV(h2−h1)(P_1 - P_2)\Delta V = \frac{1}{2}\rho \Delta V (v_2^2 - v_1^2) + \rho g \Delta V (h_2 - h_1)

Work done by pressure difference equals change in kinetic and potential energy.

applies whenSteady, incompressible, non-viscous flow.
bernoulliworkenergy

Equation of Continuity (Mass)

ρ1A1v1=ρ2A2v2\rho_1 A_1 v_1 = \rho_2 A_2 v_2

Conservation of mass in fluid flow.

applies whenSteady flow.
dynamicscontinuitymass

Equation of Continuity

A1v1=A2v2A_1 v_1 = A_2 v_2

Conservation of volume flux in fluid flow.

applies whenSteady, incompressible fluid flow.
dynamicscontinuityvolume

Speed of Efflux (Pressurised Tank)

v1=2(P−Pa)ρ+2ghv_1 = \sqrt{\frac{2(P - P_a)}{\rho} + 2gh}

Speed of efflux from a sealed tank with pressure P.

applies whenHole area much smaller than tank cross-section.
effluxpressuredynamics

Tank Emptying Time

t=Aa2g(H1−H2)t = \frac{A}{a}\sqrt{\frac{2}{g}}(\sqrt{H_1} - \sqrt{H_2})

Time taken for liquid level to drop from H1 to H2.

applies whenOrifice area 'a' is much smaller than tank cross-section 'A'.
dynamicseffluxtimejee-advanced

Pitot Tube Velocity

v=2ΔPρv = \sqrt{\frac{2\Delta P}{\rho}}

Velocity of fluid measured using stagnation pressure.

applies whenIncompressible fluid.
dynamicspitotjee-advanced

Poiseuille's Equation

Q=πΔPr48ηlQ = \frac{\pi \Delta P r^4}{8\eta l}

Volume flow rate of a viscous fluid through a cylindrical tube.

applies whenLaminar flow, steady state.
dynamicsviscosityflowjee-advanced

Reynolds Number

Re=ρvDηR_e = \frac{\rho v D}{\eta}

Dimensionless quantity used to predict fluid flow patterns (laminar vs turbulent).

applies whenPipe flow with characteristic dimension D.
dynamicsturbulencejee-advanced

Torricelli's Law

v1=2ghv_1 = \sqrt{2gh}

Speed of efflux from a small hole at depth h.

applies whenOpen tank, hole area much smaller than tank cross-section.
effluxtorricellidynamics

Venturimeter Flow Rate

Q=A1A22(P1−P2)ρ(A12−A22)Q = A_1 A_2 \sqrt{\frac{2(P_1 - P_2)}{\rho(A_1^2 - A_2^2)}}

Volume flow rate through a venturi tube.

applies whenHorizontal pipe, steady incompressible flow.
dynamicsventurimeterjee-advanced

Coefficient of Viscosity

η=F/Av/l\eta = \frac{F/A}{v/l}

Ratio of shearing stress to strain rate in a viscous fluid.

applies whenLaminar flow.
viscositydynamics

Bernoulli's Equation

P+12ρv2+ρgh=constantP + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}

Conservation of energy for flowing fluid.

applies whenSteady, incompressible, non-viscous, irrotational flow.
bernoullidynamicsenergy

Stokes' Law

F=6πηavF = 6\pi\eta a v

Viscous drag force on a spherical object falling through a fluid.

applies whenLaminar flow, spherical object of radius a.
viscositydragstokes

Terminal Velocity

vt=2a2(ρ−σ)g9ηv_t = \frac{2a^2(\rho - \sigma)g}{9\eta}

Constant velocity attained when viscous drag and buoyancy balance gravity.

applies whenSpherical object falling through an infinite viscous fluid.
viscosityvelocityterminal
04

Surface Tension

8 formulas

Work to Blow Bubble

W=8πS(r22−r12)W = 8\pi S (r_2^2 - r_1^2)

Work done to expand a soap bubble from radius r1 to r2.

applies whenSoap bubble (2 surfaces).
surface_tensionworkjee-advanced

Angle of Contact

Slacos⁡θ+Ssl=SsaS_{la} \cos\theta + S_{sl} = S_{sa}

Relationship between interfacial tensions at the line of contact.

applies whenEquilibrium of surface forces.
contact_anglesurface_tension

Laplace Excess Pressure

ΔP=S(1R1+1R2)\Delta P = S\left(\frac{1}{R_1} + \frac{1}{R_2}\right)

Excess pressure across an arbitrarily curved liquid surface with principal radii R1 and R2.

applies whenAny curved fluid interface.
surface_tensionpressurejee-advanced

Surface Energy

ΔW=SΔA\Delta W = S \Delta A

Work done or extra energy required to increase the surface area of a fluid.

applies whenIsothermal expansion of surface area.
surface_energysurface_tension

Surface Tension (Force)

S=F2lS = \frac{F}{2l}

Surface tension of a liquid film with two surfaces.

applies whenLiquid film stretched by a bar (2 surfaces).
surface_tensionforce

Excess Pressure (Soap Bubble)

Pi−Po=4SrP_i - P_o = \frac{4S}{r}

Pressure difference across a soap bubble.

applies whenTwo liquid surfaces (inner and outer).
excess_pressurebubblesurface_tension

Capillary Rise

h=2Scos⁡θρgah = \frac{2S \cos\theta}{\rho g a}

Height of liquid column in a capillary tube.

applies whenNarrow tube of radius a.
capillarysurface_tension

Excess pressure in drop

ΔP=2Tr\Delta P=\frac{2T}{r}

Excess pressure for liquid drop.

applies whenSingle surface.
fluidssurface_tension
05

Thermal Expansion and Calorimetry

3 formulas

Heat equation

Q=mcΔTQ=mc\Delta T

Heat required for temperature change.

applies whenNo phase change.
thermalheat

Latent heat

Q=mLQ=mL

Heat during phase change.

applies whenAt constant temperature.
thermalheat

Linear expansion

ΔL=αLΔT\Delta L=\alpha L\Delta T

Change in length with temperature.

applies whenSmall temperature change.
thermalexpansion
06

Heat Transfer

1 formula

Conduction rate

Qt=kAΔTL\frac{Q}{t}=kA\frac{\Delta T}{L}

Heat conduction through a slab.

applies whenSteady state.
thermaltransfer
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