## 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 Quantity | Derivable Formula | Unit | Dimensional Formula |
---|---|---|---|

Area (A) | length x breadth | m^{2} | [M^{0} L^{2} T^{0}] |

Volume (V) | length x breadth x height | m^{3} | [M^{0} L^{3} T^{0}] |

Density (d or ρ) | mass/volume | kg/m^{3} | [M L^{-3} T^{0}] |

Specific Gravity or Relative Density | density of body/density of water at 4°C | no unit | [M^{0} L^{0} T^{0}] |

Speed or Velocity (ν) | displacement/time | m/s | [M^{0} L^{1} T^{-1}] |

Acceleration (a) | change in velocity/time taken | m/s^{2} | [M^{0} L^{1} T^{-2}] |

Linear Momentum (p) | mass x velocity | kg⋅m/s | [M^{1} L^{1} T^{-1}] |

Acceleration due to gravity (g) | change in velocity/time taken | m/s^{2} | [M^{0} L^{1} T^{-2}] |

Force (F) | mass x acceleration | kg⋅m/s^{2} or newton (N) | [M^{1} L^{1} T^{-2}] |

Impulse (I) | force x time | Ns | [M^{1} L^{1} T^{-1}] |

Pressure (p) | force/area | N/m^{2} | [M^{1} L^{-1} T^{-2}] |

Work (W) or Energy (E) | force x displacement | kg⋅m^{2}/s^{2} or Joule | [M^{1} L^{2} T^{-2}] |

Torque or Moment of Couple (𝜏) | force x perpendicular distance | Nm | [M^{1} L^{2} T^{-2}] |

Power (P) | work/time | watt (W) | [M^{1} L^{2} T^{-3}] |

Wavelength (λ) | length of one wave i.e. distance | m | [M^{0} L^{1} T^{0}] |

Frequency (n) | number of vibrations/sec | hertz (Hz) | [M^{0} L^{0} T^{-1}] |

Angle (θ) | arc/radius | radian | [M^{0} L^{0} T^{0}] |

Solid Angle (Ω) | surface area/(radius)^{2} | steradian | [M^{0} L^{0} T^{0}] |

Angular Velocity or Angular Frequency (ω) | angle/time | radian/s | [M^{0} L^{0} T^{-1}] |

Angular Acceleration (α) | change in angular velocity/time taken | radian/s^{2} | [M^{0} L^{0} T^{-2}] |

Radius of Gyration (K) | distance | m | [M^{0} L^{1} T^{0}] |

Moment of Inertia (I) | mass x (distance)^{2} | kg m^{2} | [M^{1} L^{2} T^{0}] |

Angular Momentum (J) | momentum x distance | kg.m^{2}/s | [M^{1} L^{2} T^{-1}] |

Stress | force/area | N/m^{2} | [M L^{-1} T^{-2}] |

Strain | change in length (for volume)/original length (for volume) | no unit | [M^{0} L^{0} T^{0}] |

Coefficient of Elasticity | stress/strain | N/m^{2} | [M L^{-1} T^{-2}] |

Poisson’s Ratio (σ) | lateral strain/longitudinal strain | no unit | [M^{0} L^{0} T^{0}] |

Velocity Gradient | velocity/distance | s^{-1} | [M^{0} L^{0} T^{-1}] |

Coefficient of Viscosity (η) | F/[A (dν/dt)] | Ns/m^{2} | [M^{1} L^{-1} T^{-1}] |

Surface Tension (T) | force/length | N/m | [M^{1} L^{0} T^{-2}] |

Force Constant (k) | force/displacement | N/m | [M^{1} L^{0} T^{-2}] |

Universal Gravitational Constant (G) | G = Fr^{2}/m_{1}m_{2} | Nm^{2}/kg^{2} | [M^{-1} L^{3} T^{-2}] |

Intensity of Gravitational Field (E_{g}) | GM/r^{2} | N/kg | [M^{0} L^{1} T^{-2}] |

Gravitational Potential (V_{g}) | GM/r | J/kg | [M^{0} L^{2} T^{-2}] |

Mechanical Equivalent of Heat (J) | work energy/heat energy | J/calorie | [M^{0} L^{0} T^{0}] |

Specific Heat Capacity (s) | heat energy/(mass x temperature difference) | J/kg.K | [M^{0} L^{2} T^{-2} θ^{-1}] |

Thermal Capacity | heat energy/temperature difference | J/K | [M^{1} L^{2} T^{-2} θ^{-1}] |

Latent Heat (L) | heat energy/mass | J/kg | [M^{0} L^{2} T^{-2}] |

Universal Gas Constant (R) | PV/T | J/K | [M^{1} L^{2} T^{-2} θ^{-1}] |

Thermal Conductivity (K) | (Q/t)/[A(dθ/dt) | W/m.K | [M^{1} L^{1} T^{-3} θ^{-1}] |

Coefficient of Linear Expansion (α) | change in length/(original length x temperature difference) | K^{-1} | [M^{0} L^{0} T^{0} θ^{-1}] |

Coefficient of Volume Expansion (γ) | volume/(original volume x temperature difference) | K^{-1} | [M^{0} L^{0} T^{0} θ^{-1}] |

Refractive Index (µ or n) | velocity of light in vacuum/velocity of light in medium | no unit | [M^{0} L^{0} T^{0}] |

Electric Charge (Q) | current x time | Coulomb | [M^{0} L^{0} T^{1} A^{1}] |

Electric Field Intensity (E) | electric force/charge | N/Coulomb | [M^{1} L^{1} T^{-3} A^{-1}] |

Electric Potential (V) | work/charge | J/Coulomb or volt | [M^{1} L^{2} T^{-3} A^{-1}] |

Permittivity (ε) | Q_{1}Q_{2}/4πr^{2}F | Coulomb^{2}/N.m^{2} | [M^{-1} L^{-3} T^{4} A^{2}] |

Dielectric Constant or Relative Permittivity (K) | medium/permittivity of free space | no unit | [M^{0} L^{0} T^{0}] |

Capacitance (C) | charge/potential difference | Coulomb/volt or farad | [M^{-1} L^{-2} T^{4} A^{2}] |

Electrical Resistance (R) | potential difference/current | ohm | [M L^{2} T^{-3} A^{-2}] |

Magnetic Field Induction (B) | F/Qν.sin θ | weber/m^{2} or tesla | [M^{1} L^{0} T^{-2} A^{-1}] |

Magnetic Flux (Φ) | field x area | volt-second or weber | [M^{1} L^{2} T^{-2} A^{-1}] |

Magnetic Permeability (µ) | B/H | henries/metre | [M^{1} L^{1} T^{-2} A^{-2}] |

Magnetic Moment (M) | torque/B.sin θ | Ampere.m^{2} | [M^{0} L^{2} T^{0} A^{1}] |

Intensity of Magnetisation (M) | magnetic moment/volume | ampere/m | [M^{0} L^{-1} T^{0} A^{1}] |

Inductance (L) | potential difference/(di/dt) | henry | [M^{1} L^{2} T^{-2} A^{-2}] |

Rate of Flow | volume/time | m^{3}/s | [M^{0} L^{3 }T^{-1}] |

Planck’s Constant (h) | energy/frequency | J.s | [M^{1} L^{2 }T^{-1}] |

Mass of Unit Length (m) | mass/length | kg/m | [M^{1} L^{-1 }T^{0}] |

Distance Travelled in nth Second | distance/time | m/s | [M^{0} L^{1 }T^{-1}] |

Avogadro’s Number (N) | number of atoms/molecules in one gram atom/mole | mole^{-1} | [M^{0} L^{0} T^{0}] |

Reynold Number (N_{R}) | ρDν/η | no unit | [M^{0} L^{0} T^{0}] |

Rydberg Constant (R) | 2π^{2}mK^{2}e^{4}/ch^{3} | m^{-1} | [M^{0} L^{-1} T^{0}] |

Stefan’s Constant (σ) | (energy emitted)/[(area x time) (temp)^{4}] | watt.m^{-2}K^{-4} | [M^{1} L^{0} T^{-3} θ^{-4}] |

Boltzmann Constant (k) | energy/temperature | J.K^{-1} | [M^{1} L^{2} T^{-2} θ^{-1}] |