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Dimensional Formulae and Dimensional Equations

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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

MoLoTo

Angular displacement

rad

MoloTo

Angular frequency (angular displacement/time)

rads–1

T–1

Angular impulse (torque x time)

Nms

ML2T–1

Angular momentum (Iω)

kgm2s–1

ML2T–1

Angular velocity (angle/time)

rads–1

T–1

Area (length x breadth)

m2

L2

Boltzmann’s constant

JK–1

ML2T–2θ–1

Bulk modulus (ΔP.VΔV.)

Nm–2, Pa

M1L–1T–2

Calorific value

Jkg–1

L2T–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–1K–1

MLT–3θ–1

Coefficient of viscosity (F =ηAdvdx)

poise

ML–1T–1

Compressibility (1/bulk modulus)

Pa–1, m2N–2

M–1LT2

Density (mass / volume)

kgm–3

ML–3

Displacement, wavelength, focal length

m

L

Electric capacitance (charge/potential)

CV–1, farad

M–1L–2T4I2

Electric conductance (1/resistance)

Ohm–1 or mho or siemen

M–1L–2T3I2

Electric conductivity (1/resistivity)

siemen/metre or Sm–1

M–1L–3T3I2

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–3I–1

Electric resistance (potential difference current)

ohm

ML2T–3I–2

Emf (or) electric potential (work/charge)

volt

ML2T–3I–1

Energy (capacity to do work)

joule

ML2T–2

Energy density (energyvolume)

Jm–3

ML–1T–2

Entropy (ΔSQ/T)

–1

ML2T–2θ–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

L2T–2

Heat (energy)

J or calorie

ML2T–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)

ML2T–2I–2

Intensity of gravitational field (F/m)

Nkg–1

L1T–2

Intensity of magnetization (I)

Am–1

L–1I

Joule’s constant or mechanical equivalent of heat

Jcal–1

MoLoTo

Latent heat (Q = mL)

Jkg–1

MoL2T–2

Linear density (mass per unit length)

kgm–1

ML–1

Luminous flux

lumen or (Js–1)

ML2T–3

Magnetic dipole moment

Am2

L2I

Magnetic flux (magnetic induction x area)

weber (Wb)

ML2T–2I–1

Magnetic induction (F = Bil)

NI–1m–1 or T

MT–2I–1

Magnetic pole strength (unit: ampere–meter)

Am

LI

Modulus of elasticity (stress/strain)

Nm–2, Pa

ML–1T–2

Moment of inertia (mass x radius2)

kgm2

ML2

Momentum (mass x velocity)

kgms–1

MLT–1

Permeability of free space (μo=4πFd2m1m2)

Hm–1 or NA–2

MLT–2I–2

Permittivity of free space (εo=Q1Q24πFd2.)

Fm–1 or C2N–1m–2

M–1L–3T4I2

Planck’s constant (energy/frequency)

Js

ML2T–1

Poisson’s ratio (lateral strain/longitudinal strain)

––

MoLoTo

Power (work/time)

Js–1 or watt (W)

ML2T–3

Pressure (force/area)

Nm–2 or Pa

ML–1T–2

Pressure coefficient or volume coefficient

OC–1 or θ–1

θ–1

Pressure head

m

MoLTo

Radioactivity

disintegrations per second

MoLoT–1

Ratio of specific heats

––

MoLoTo

Refractive index

––

MoLoTo

Resistivity or specific resistance

Ω–m

ML3T–3I–2

Specific conductance or conductivity (1/specific resistance)

siemen/metre or Sm–1

M–1L–3T3I2

Specific entropy (1/entropy)

KJ–1

M–1L–2T2θ

Specific gravity (density of the substance/density of water)

––

MoLoTo

Specific heat (Q = mst)

Jkg–1θ–1

MoL2T–2θ–1

Specific volume (1/density)

m3kg–1

M–1L3

Speed (distance/time)

ms–1

LT–1

Stefan’s constant(heat energy /area x time x temperature4).

Wm–2θ–4

MLoT–3θ–4

Strain (change in dimension/original dimension)

––

MoLoTo

Stress (restoring force/area)

Nm–2 or Pa

ML–1T–2

Surface energy density (energy/area)

Jm–2

MT–2

Temperature

oC or θ

MoLoToθ

Temperature gradient (change in temperaturedistance)

OCm1 or θm–1

MoL–1Toθ

Thermal capacity (mass x specific heat)

–1

ML2T–2θ–1

Time period

second

T

Torque or moment of force (force x distance)

Nm

ML2T–2

Universal gas constant (work/temperature)

Jmol–1θ–1

ML2T–2θ–1

Universal gravitational constant (F = G. m1m2d2)

Nm2kg–2

M–1L3T–2

Velocity (displacement/time)

ms–1

LT–1

Velocity gradient (dv/dx)

s–1

T–1

Volume (length x breadth x height)

m3

L3

Water equivalent

kg

MLoTo

Work (force x displacement)

J

ML2T–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.



 

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