Dimensions of Physical Quantities, Dimensional Formulae and Equations

Units And Measurements • Physics

Concept Summary

The nature of a physical quantity is described by its dimensions. All quantities represented by derived units can be expressed through combinations of the seven base quantities, which we call the seven dimensions of the physical world and denote with square brackets: length [L], mass [M], time [T], electric current [A], thermodynamic temperature [K], luminous intensity [cd] and amount of substance [mol]. The dimensions of a physical quantity are the powers (exponents) to which the base quantities are raised to represent it. For example, volume is length × length × length, so its dimensions are [L³], and it has zero dimension in mass and time. Force is mass × acceleration, so its dimensions are [M L T⁻²]. Only the kind of quantity matters, not its magnitude, so change in velocity, initial velocity, final velocity, average velocity and speed all have dimensions [L T⁻¹]. The expression that shows which base quantities enter and with what powers is the dimensional formula, for example [M⁰ L³ T⁰] for volume and [M L⁻³ T⁰] for mass density. An equation obtained by equating a quantity with its dimensional formula is called its dimensional equation, such as [F] = [M L T⁻²]. Dimensional formulae can be derived from the equations that relate physical quantities.

Key Notes
  • The seven dimensions: [L], [M], [T], [A], [K], [cd], [mol].
  • Dimensions of a quantity are the powers of the base quantities in its expression.
  • Square brackets [ ] mean "the dimensions of" the quantity.
  • Magnitudes are not considered; only the type of quantity matters (speed, velocity, average velocity all have [L T⁻¹]).
  • Dimensional formula: shows which base quantities, and with what powers, represent a quantity, e.g. volume [M⁰ L³ T⁰].
  • Dimensional equation: a quantity equated to its dimensional formula, e.g. [F] = [M L T⁻²].
  • In mechanics, all quantities can be written using only [L], [M] and [T].
  • Dimensionless quantities (angle, refractive index, strain) have dimensional formula [M⁰ L⁰ T⁰].
  • Dimensional formulae are derived from the equations that define the quantities.
Important Equations
\[ [V] = [M^{0}L^{3}T^{0}] \qquad [v] = [M^{0}L^{1}T^{-1}] \] \[ [a] = [M^{0}L^{1}T^{-2}] \qquad [F] = [M^{1}L^{1}T^{-2}] \] \[ [\rho] = [M^{1}L^{-3}T^{0}] \qquad [\text{Energy}] = [M^{1}L^{2}T^{-2}] \]

Deriving a dimensional formula:

\[ \text{Force} = \text{mass}\times\text{acceleration} = [M]\,\frac{[L]}{[T]^{2}} = [M L T^{-2}] \] \[ \text{Volume} = [L]\times[L]\times[L] = [L^{3}] \]
Flow Chart
Pick the physical quantity
Mind Map
  • Dimensions
    • Seven base dimensions
      • [L], [M], [T]
      • [A], [K]
      • [cd], [mol]
    • Dimensions = powers of base quantities
    • Dimensional formula
      • Volume [M⁰L³T⁰]
      • Speed [M⁰LT⁻¹]
      • Acceleration [M⁰LT⁻²]
      • Density [ML⁻³T⁰]
    • Dimensional equation
      • [F] = [M L T⁻²]
    • Magnitude is not considered
Graphic Organiser
QuantityRelationDimensional FormulaSI Unit
Volumelength × breadth × height[M⁰ L³ T⁰]m3
Speed / velocitylength / time[M⁰ L T⁻¹]m s-1
Accelerationvelocity / time[M⁰ L T⁻²]m s-2
Forcemass × acceleration[M L T⁻²]N (kg m s-2)
Mass densitymass / volume[M L⁻³ T⁰]kg m-3
Energyforce × distance[M L² T⁻²]J (kg m2 s-2)
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