Dimensional Analysis

Dimensional Analysis:

To find the dimension of a physical quantity, we first write the quantity in terms of basic quantities and proceed as illustrated in the examples given below.

(1) Density = mass/volume = mass/length x breadth x height

⇒ [ρ] = M/L x L x L = ML-3

where the symbol [ ] stands for “dimension of” the quantity written in the parenthesis.

(2) Force acting on a particle of mass m is related to the acceleration “a” is given by

F = ma

⇒ [F] = [m] [a]

Since the dimensional formula for acceleration is LT-2 i.e.

⇒ [F] = (M) (LT-2) = MLT-2

Physical QuantityDerivable FormulaUnitDimensional Formula
Area (A)length x breadthm2[M0 L2 T0]
Volume (V)length x breadth x heightm3[M0 L3 T0]
Density (d or ρ)mass/volumekg/m3[M L-3 T0]
Specific Gravity or Relative Densitydensity of body/density of water at 4°C no unit[M0 L0 T0]
Speed or Velocity (ν)displacement/timem/s[M0 L1 T-1]
Acceleration (a)change in velocity/time takenm/s2[M0 L1 T-2]
Linear Momentum (p)mass x velocitykg⋅m/s[M1 L1 T-1]
Acceleration due to gravity (g)change in velocity/time takenm/s2[M0 L1 T-2]
Force (F)mass x accelerationkg⋅m/s2 or newton (N)[M1 L1 T-2]
Impulse (I)force x timeNs[M1 L1 T-1]
Pressure (p)force/areaN/m2[M1 L-1 T-2]
Work (W) or Energy (E)force x displacementkg⋅m2/s2 or Joule[M1 L2 T-2]
Torque or Moment of Couple (𝜏)force x perpendicular distanceNm[M1 L2 T-2]
Power (P)work/timewatt (W)[M1 L2 T-3]
Wavelength (λ)length of one wave i.e. distancem[M0 L1 T0]
Frequency (n)number of vibrations/sechertz (Hz)[M0 L0 T-1]
Angle (θ)arc/radiusradian[M0 L0 T0]
Solid Angle (Ω)surface area/(radius)2steradian[M0 L0 T0]
Angular Velocity or Angular Frequency (ω)angle/timeradian/s[M0 L0 T-1]
Angular Acceleration (α)change in angular velocity/time takenradian/s2[M0 L0 T-2]
Radius of Gyration (K)distancem[M0 L1 T0]
Moment of Inertia (I)mass x (distance)2kg m2[M1 L2 T0]
Angular Momentum (J)momentum x distancekg.m2/s[M1 L2 T-1]
Stressforce/areaN/m2[M L-1 T-2]
Strainchange in length (for volume)/original length (for volume)no unit[M0 L0 T0]
Coefficient of Elasticitystress/strainN/m2[M L-1 T-2]
Poisson’s Ratio (σ)lateral strain/longitudinal strainno unit[M0 L0 T0]
Velocity Gradientvelocity/distances-1[M0 L0 T-1]
Coefficient of Viscosity (η)F/[A (dν/dt)]Ns/m2[M1 L-1 T-1]
Surface Tension (T)force/lengthN/m[M1 L0 T-2]
Force Constant (k)force/displacementN/m[M1 L0 T-2]
Universal Gravitational Constant (G) G = Fr2/m1m2Nm2/kg2[M-1 L3 T-2]
Intensity of Gravitational Field (Eg)GM/r2N/kg[M0 L1 T-2]
Gravitational Potential (Vg)GM/rJ/kg[M0 L2 T-2]
Mechanical Equivalent of Heat (J)work energy/heat energyJ/calorie[M0 L0 T0]
Specific Heat Capacity (s)heat energy/(mass x temperature difference)J/kg.K[M0 L2 T-2 θ-1]
Thermal Capacity heat energy/temperature differenceJ/K[M1 L2 T-2 θ-1]
Latent Heat (L)heat energy/massJ/kg[M0 L2 T-2]
Universal Gas Constant (R)PV/TJ/K[M1 L2 T-2 θ-1]
Thermal Conductivity (K)(Q/t)/[A(dθ/dt)W/m.K[M1 L1 T-3 θ-1]
Coefficient of Linear Expansion (α)change in length/(original length x temperature difference)K-1[M0 L0 T0 θ-1]
Coefficient of Volume Expansion (γ)volume/(original volume x temperature difference)K-1[M0 L0 T0 θ-1]
Refractive Index (µ or n)velocity of light in vacuum/velocity of light in mediumno unit[M0 L0 T0]
Electric Charge (Q)current x timeCoulomb[M0 L0 T1 A1]
Electric Field Intensity (E)electric force/chargeN/Coulomb[M1 L1 T-3 A-1]
Electric Potential (V)work/chargeJ/Coulomb or volt[M1 L2 T-3 A-1]
Permittivity (ε)Q1Q2/4πr2FCoulomb2/N.m2[M-1 L-3 T4 A2]
Dielectric Constant or Relative Permittivity (K)medium/permittivity of free spaceno unit[M0 L0 T0]
Capacitance (C)charge/potential differenceCoulomb/volt or farad[M-1 L-2 T4 A2]
Electrical Resistance (R)potential difference/currentohm[M L2 T-3 A-2]
Magnetic Field Induction (B)F/Qν.sin θweber/m2 or tesla[M1 L0 T-2 A-1]
Magnetic Flux (Φ)field x areavolt-second or weber[M1 L2 T-2 A-1]
Magnetic Permeability (µ)B/H henries/metre[M1 L1 T-2 A-2]
Magnetic Moment (M)torque/B.sin θAmpere.m2[M0 L2 T0 A1]
Intensity of Magnetisation (M)magnetic moment/volumeampere/m[M0 L-1 T0 A1]
Inductance (L)potential difference/(di/dt)henry[M1 L2 T-2 A-2]
Rate of Flowvolume/timem3/s[M0 L3 T-1]
Planck’s Constant (h)energy/frequencyJ.s[M1 L2 T-1]
Mass of Unit Length (m)mass/lengthkg/m[M1 L-1 T0]
Distance Travelled in nth Seconddistance/timem/s[M0 L1 T-1]
Avogadro’s Number (N)number of atoms/molecules in one gram atom/molemole-1[M0 L0 T0]
Reynold Number (NR)ρDν/ηno unit[M0 L0 T0]
Rydberg Constant (R)2mK2e4/ch3m-1[M0 L-1 T0]
Stefan’s Constant (σ)(energy emitted)/[(area x time) (temp)4]watt.m-2K-4[M1 L0 T-3 θ-4]
Boltzmann Constant (k)energy/temperatureJ.K-1[M1 L2 T-2 θ-1]

Important Concepts in Thermodynamics
Zeroth Law of Thermodynamics
First Law of Thermodynamics
Second and Third Law of Thermodynamics
Spontaneous and Non-Spontaneous Process
Enthalpy and Entropy
Gibbs Free Energy and Gibbs Helmholtz equation
Dimensional Analysis– Wikipedia

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