1.4 Dimensions of physical quantities

Dimension Of Physical Quantities 

Relation which express physical quantities in terms of appropriate powers of fundamental units.
\(
\begin{aligned}
& \text { e.g. } \text { Density }=\frac{\text { Mass }}{\text { Volume }}=\frac{\text { Mass }}{(\text { Length })^3} \\
& \text { or } \quad \text { Density }=(\text { Mass })(\text { Length })^{-3}
\end{aligned}
\)

\(
\begin{aligned}
\text { force } & =\text { mass × acceleration } \\
& =\text { mass } \text { × } \frac{\text { velocity }}{\text { time }} \\
& =\text { mass } \text { × } \frac{\text { length } / \text { time }}{\text { time }} \\
& =\text { mass } \text { × } \text { length } \text { × }(\text { time })^{-2} .
\end{aligned}
\)
Thus, the dimensions of force are 1 in mass, 1 in length and -2 in time. The dimensions in all other base quantities are zero. Note that in this type of calculation the magnitudes are not considered. It is equality of the type of quantity that enters. Thus, change in velocity, initial velocity, average velocity, final velocity all are equivalent in this discussion, each one is length/time.

Dimensional representation of physical quantities

For convenience, the fundamental quantities are represented by one letter symbols. The dependence of all other physical quantities on these base quantities can be expressed in terms of their dimensions.
Thus, the seven dimensions of physical quantities are represented as follows
\(
\begin{array}{ll}
{[\mathrm{M}]} & \text { for mass } \\
{[\mathrm{L}]} & \text { for length } \\
{[\mathrm{T}]} & \text { for time } \\
{[\mathrm{A}]} & \text { for electric current } \\
{[\mathrm{K}] \text { or }[\theta]} & \text { for thermodynamic } \\
& \text { temperature } \\
{[\mathrm{cd}]} & \text { for luminous intensity } \\
{[\mathrm{mol}]} & \text { for amount of substance }
\end{array}
\)
The physical quantity that is expressed in terms of the base quantities is enclosed in square brackets.
Thus, dimensions of density can be represented as \(\left[\mathrm{ML}^{-3}\right]\).