About Dimensional Formula
Let us first define dimension before studying the dimensional formula. In math, a dimension is a measurement of length, width, or height extended in one direction. It is a measure of a point / line extended in one direction, according to dimension definition. Every shape in our environment has proportions. In mathematics, there’s no specific dimensional formula for the idea of dimension. The power to which the fundamental units are elevated to obtain one unit of any physical quantity is called its dimension. Let's have a look at the dimensional formula and see some examples at the conclusion. Get the list of Maths Formulas.
What Is the Dimensional Formula?
The phrase expressing the powers to which the fundamental units must be increased in order to obtain 1 unit of a derived quantity is known as the dimensional formula of any quantity. If Q is any physical quantity, the dimensional formula is represented by,
List of Dimensional Formula table
Q = MaLbTc
where M, L, and T are the basic dimensions of mass, length, and time, and a, b, and c are the exponents of each.
| Physical quantity | Unit | Dimensional formula |
|---|---|---|
| Length | m | L |
| Mass | kg | M |
| Time | s | T |
| Acceleration or acceleration due to gravity | ms-2 | LT-2 |
| Angle(arc/radius) | rad | M0L0T0 |
| Angular displacement | rad | M0L0T0 |
| 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-r | M2L2T-2θ-1 |
| Bulk modulus(ΔP X (V/ΔV)) | Nm-2,Pa | M1L-1T-2 |
| Calorific Value | Jkg-1 | L2T-2 |
| Coefficient of linear or areal or volume expansion | 0C-1or K-1 | θ-1 |
| Coefficient of surface tension(force/length) | Nm-1or Jm-2 | MT-2 |
| Coefficient of thermal conductivity | Wm-1K-1 | MLT-3θ-1 |
| Coefficient of viscosity(F = η x A x (dv/dx)) | poise | ML-1T-1 |
| Compressibility(1/bulk modulus) | Pai,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 × time) | coulomb | IT |
| Electric current | ampere | I |
| Electric dipole moment (charge × distance) | Cm | LTI |
| Electric dipole moment (charge × distance) | 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 (energy/volume) | Jm-3 | ML-1T-2 |
| Entropy (ΔS = ΔQ/T) | Jθ–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 = LI2 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 | M0L0T0 |
| Latent heat (Q = mL) | Jkg–1 | M0L2T-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 | Am (ampere–meter) | LI |
| Modulus of elasticity (stress/strain) | Nm–2, Pa | ML-1T-2 |
| Moment of inertia (mass × radius2) | Kgm2 | ML2 |
| Momentum (mass × velocity) | Kgms-1 | MLT-1 |
| Permeability of free space (μo=4πFd2m1m2)(μo=4πFd2m1m2) | Hm–1 or NA-2 | MLT-2I-2 |
| Permittivity of free space (εo=Q1Q24πFd2)(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) | - | M0L0T0 |
| 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 | M0LT0 |
| Radioactivity | disintegrations per second | M0L0T-1 |
| Ratio of specific heats | - | M0L0T0 |
| Refractive index | - | M0L0T0 |
| Resistivity or specific resistance | Ω–m | ML-3T-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) | - | M0L0T0 |
| Specific heat (Q = mst) | Jkg–1θ–1 | M0L2T-2θ–1 |
| Specific volume (1/density) | m3kg–1 | M-1L3 |
| Speed (distance/time) | ms-1 | LT-1 |
| Stefan’s constant (heat energyarea×time×temperature) (heat energyarea×time×temperature4) |
Wm-2θ–4 | ML0T-3θ–4 |
| Strain (change in dimension/original dimension) | - | M0L0T0 |
| Stress (restoring force/area) | Nm-2Pa | ML-1T-2 |
| Surface energy density (energy/area) | Jm-2 | MT-2 |
| Temperature | 0C or θ | M0L0T0θ |
| Temperature gradient (change in temperaturedistance)(change in temperaturedistance) | 0Cm-1 or θm-1 | M0L0T0θ |
| Thermal capacity (mass × specific heat) | Jθ-1 | ML2T-2θ-1 |
| Time period | second | T |
| Torque or moment of force (force × distance) | Nm | ML2T-2 |
| Universal gas constant (work/temperature) | Jmol-1θ-1 | ML2T-2θ-1 |
| Universal gravitational constant (F=G.m1m2d2)(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 × breadth × height) | m3 | L3 |
| Water equivalent | kg | ML0T0 |
| Work (force × displacement) | J | ML2T-2 |
| Decay constant | s-1 | M0L0T-1 |
| Potential energy | J | M1L2T-2 |
| Kinetic energy | J | M1L2T-2 |
Dimensional Formula and Dimensional Equations
In terms of dimensions, a dimensional equation is an equation that connects fundamental and derived units. The three base dimensions in mechanics are length, mass, time, temperature, and electric current, with the fundamental units being metre, kilogramme, second, ampere, kelvin, mole, and candela. In any dimensional equation, the dimensional formula of separate values is employed to construct a link between them. The following is an example of a dimensional equation:
Dimensional formula (equation) for the area:
Area = length × breadth
= length × length
= [L] × [L]
= [L]2
⇒ Dimensional formula (equation) for area (A) = [L2 M0 T0]
Applications of Dimensional Formula
It is used to verify the correctness of an equation. It is used to derive the relationship between different physical quantities. It is used to convert from one system of units to another for any given quantity.
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