notes
Dimensional Formula and Dimensional Equation
Dimensional Formula:-
The dimensional formula of a physical quantity is an expression telling us how and which of the fundamental quantities enter into the unit of that quantity.
It is customary to express the fundamental quantities by a capital letter, e.g., length(L), mass (AT), time (T), electric current (I), temperature (K) and luminous intensity (C).
Physical quantity |
Unit |
Dimensional formula |
Acceleration or acceleration due to gravity |
ms^{–2} |
LT^{–2} |
Angle (arc/radius) |
rad |
M^{o}L^{o}T^{o} |
Angular displacement |
rad |
M^{o}l^{o}T^{o} |
Angular frequency (angular displacement/time) |
rads^{–1} |
T^{–1} |
Angular impulse (torque x time) |
Nms |
ML^{2}T^{–1} |
Angular momentum (Iω) |
kgm^{2}s^{–1} |
ML^{2}T^{–1} |
Angular velocity (angle/time) |
rads^{–1} |
T^{–1} |
Area (length x breadth) |
m^{2} |
L^{2} |
Boltzmann’s constant |
JK^{–1} |
ML^{2}T^{–2}θ^{–1} |
Bulk modulus (ΔP.VΔV.) |
Nm^{–2}, Pa |
M^{1}L^{–1}T^{–2} |
Calorific value |
Jkg^{–1} |
L^{2}T^{–2} |
Coefficient of linear or areal or volume expansion |
OC^{–1 }or K^{–1} |
θ^{–1} |
Coefficient of surface tension (force/length) |
Nm^{–1 }or Jm^{–2} |
MT^{–2} |
Coefficient of thermal conductivity |
Wm^{–1}K^{–1} |
MLT^{–3}θ^{–1} |
Coefficient of viscosity (F =ηAdvdx) |
poise |
ML^{–1}T^{–1} |
Compressibility (1/bulk modulus) |
Pa^{–1}, m^{2}N^{–2} |
M^{–1}LT^{2} |
Density (mass / volume) |
kgm^{–3} |
ML^{–3} |
Displacement, wavelength, focal length |
m |
L |
Electric capacitance (charge/potential) |
CV^{–1}, farad |
M^{–1}L^{–2}T^{4}I^{2} |
Electric conductance (1/resistance) |
Ohm^{–1 }or mho or siemen |
M^{–1}L^{–2}T^{3}I^{2} |
Electric conductivity (1/resistivity) |
siemen/metre or Sm^{–1} |
M^{–1}L^{–3}T^{3}I^{2} |
Electric charge or quantity of electric charge (current x time) |
coulomb |
IT |
Electric current |
ampere |
I |
Electric dipole moment (charge x distance) |
Cm |
LTI |
Electric field strength or Intensity of electric field (force/charge) |
NC^{–1}, Vm^{–1} |
MLT^{–3}I^{–1} |
Electric resistance (potential difference current) |
ohm |
ML^{2}T^{–3}I^{–2} |
Emf (or) electric potential (work/charge) |
volt |
ML^{2}T^{–3}I^{–1} |
Energy (capacity to do work) |
joule |
ML^{2}T^{–2} |
Energy density (energyvolume) |
Jm^{–3} |
ML^{–1}T^{–2} |
Entropy (ΔS=ΔQ/T) |
Jθ^{–1} |
ML^{2}T^{–2}θ^{–1} |
Force (mass x acceleration) |
newton (N) |
MLT^{–2} |
Force constant or spring constant (force/extension) |
Nm^{–1} |
MT^{–2} |
Frequency (1/period) |
Hz |
T^{–1} |
Gravitational potential (work/mass) |
Jkg^{–1} |
L^{2}T^{–2} |
Heat (energy) |
J or calorie |
ML^{2}T^{–2} |
Illumination (Illuminance) |
lux (lumen/metre2) |
MT^{–3} |
Impulse (force x time) |
Ns or kgms^{–1} |
MLT^{–1} |
Inductance (L) (energy =12LI2) or coefficient of self-induction |
henry (H) |
ML^{2}T^{–2}I^{–2} |
Intensity of gravitational field (F/m) |
Nkg^{–1} |
L^{1}T^{–2} |
Intensity of magnetization (I) |
Am^{–1} |
L^{–1}I |
Joule’s constant or mechanical equivalent of heat |
Jcal^{–1} |
M^{o}L^{o}T^{o} |
Latent heat (Q = mL) |
Jkg^{–1} |
M^{o}L^{2}T^{–2} |
Linear density (mass per unit length) |
kgm^{–1} |
ML^{–1} |
Luminous flux |
lumen or (Js^{–1}) |
ML^{2}T^{–3} |
Magnetic dipole moment |
Am^{2} |
L^{2}I |
Magnetic flux (magnetic induction x area) |
weber (Wb) |
ML^{2}T^{–2}I^{–1} |
Magnetic induction (F = Bil) |
NI^{–1}m^{–1 }or T |
MT^{–2}I^{–1} |
Magnetic pole strength (unit: ampere–meter) |
Am |
LI |
Modulus of ^{elasticity} (stress/strain) |
Nm^{–2}, Pa |
ML^{–1}T^{–2} |
Moment of inertia (mass x radius2) |
kgm^{2} |
ML^{2} |
Momentum (mass x velocity) |
kgms^{–1} |
MLT^{–1} |
Permeability of free space (μo=4πFd2m1m2) |
Hm^{–1 }or NA^{–2} |
MLT^{–2}I^{–2} |
Permittivity of free space (εo=Q1Q24πFd2.) |
Fm^{–1 }or C^{2}N^{–1}m^{–2} |
M^{–1}L^{–3}T^{4}I^{2} |
Planck’s constant (energy/frequency) |
Js |
ML^{2}T^{–1} |
Poisson’s ratio (lateral strain/longitudinal strain) |
–– |
M^{o}L^{o}T^{o} |
Power (work/time) |
Js^{–1 }or watt (W) |
ML^{2}T^{–3} |
Pressure (force/area) |
Nm^{–2 }or Pa |
ML^{–1}T^{–2} |
Pressure coefficient or volume coefficient |
OC^{–1 }or θ^{–1} |
θ^{–1} |
Pressure head |
m |
M^{o}LT^{o} |
Radioactivity |
disintegrations per second |
M^{o}L^{o}T^{–1} |
Ratio of specific heats |
–– |
M^{o}L^{o}T^{o} |
Refractive index |
–– |
M^{o}L^{o}T^{o} |
Resistivity or specific resistance |
Ω–m |
ML^{3}T^{–3}I^{–2} |
Specific conductance or conductivity (1/specific resistance) |
siemen/metre or Sm^{–1} |
M^{–1}L^{–3}T^{3}I^{2} |
Specific entropy (1/entropy) |
KJ^{–1} |
M^{–1}L^{–2}T^{2}θ |
Specific gravity (density of the substance/density of water) |
–– |
M^{o}L^{o}T^{o} |
Specific heat (Q = mst) |
Jkg^{–1}θ^{–1} |
M^{o}L^{2}T^{–2}θ^{–1} |
Specific volume (1/density) |
m^{3}kg^{–1} |
M^{–1}L^{3} |
Speed (distance/time) |
ms^{–1} |
LT^{–1} |
Stefan’s constant(heat energy /area x time x temperature4). |
Wm^{–2}θ^{–4} |
ML^{o}T^{–3}θ^{–4} |
Strain (change in dimension/original dimension) |
–– |
M^{o}L^{o}T^{o} |
Stress (restoring force/area) |
Nm^{–2 }or Pa |
ML^{–1}T^{–2} |
Surface energy density (energy/area) |
Jm^{–2} |
MT^{–2} |
Temperature |
oC or θ |
M^{o}L^{o}T^{o}θ |
Temperature gradient (change in temperaturedistance) |
OCm^{–}^{1} or θm^{–1} |
M^{o}L^{–1}T^{o}θ |
Thermal capacity (mass x specific heat) |
Jθ^{–1} |
ML^{2}T^{–2}θ–1 |
Time period |
second |
T |
Torque or moment of force (force x distance) |
Nm |
ML^{2}T^{–2} |
Universal gas constant (work/temperature) |
Jmol^{–1}θ^{–1} |
ML^{2}T^{–2}θ^{–1} |
Universal gravitational constant (F = G. m1m2d2) |
Nm^{2}kg^{–2} |
M^{–1}L^{3}T^{–2} |
Velocity (displacement/time) |
ms^{–1} |
LT^{–1} |
Velocity gradient (dv/dx) |
s^{–1} |
T^{–1} |
Volume (length x breadth x height) |
m^{3} |
L^{3} |
Water equivalent |
kg |
ML^{o}T^{o} |
Work (force x displacement) |
J |
ML^{2}T^{–2} |
Dimensional Equation :-
An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity. Thus, the dimensional equations are the equations, which represent the dimensions of a physical quantity in terms of the base quantities. For example, the dimensional equations of volume [V], speed [v], force [F] and mass density [ ρ ] may be expressed as [V] = [M0 L3 T0] [v] = [M0 L T–1] [F] = [M L T–2] [ ρ ] = [M L–3 T0]. The dimensional equation can be obtained from the equation representing the relations between the physical quantities.