Dimensional Formulae and Dimensional Equations

1.5 Dimensional Formulae and Dimensional Equations

Every physical quantity can be expressed in terms of the seven base quantities (mass, length, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity). The way a particular quantity depends on these base quantities is its dimension. The dimensional formula is the compact algebraic expression that shows this dependence.

For example, volume is a product of three lengths, so its dimension in length is 3, and it has no dependence on mass or time. Its dimensional formula is written as $[M^0 L^3 T^0]$. Speed is length divided by time, so its dimensional formula is $[M^0 L T^{-1}]$. Acceleration is length divided by time squared, giving $[M^0 L T^{-2}]$. Mass density is mass divided by volume, so it is $[M L^{-3} T^0]$.

A dimensional equation is simply the equation you get when you set a physical quantity equal to its dimensional formula. For instance, if $[V]$ denotes the dimension of volume, then the dimensional equation for volume is $[V] = [M^0 L^3 T^0]$. Similarly, for speed $v$, force $F$, and density $\rho$, we write:

The dimensional formula of any quantity is derived from the physical relation that defines it. For example, from Newton's second law, force equals mass times acceleration ($F = ma$). Since acceleration has dimensions $[L T^{-2}]$, force must have dimensions $[M L T^{-2}]$. This is how the dimensional formulae for a vast range of quantities are built — from the equations that connect them. …