Homogeneity of Equations

Equations where the units match on both sides

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Table of Contents

Using any unit

We do not need to use the SI units for the physical quantities involved in an equation in order to show that the equation is homogeneous. Consider the case of speed: $$ \begin{equation}\begin{aligned} speed=\frac{distance}{time}\\ \end{aligned}\end{equation} $$

We can use $m/s$, $m$ and $s$ respectively for the units of speed, distance and time: $$ \begin{equation}\begin{aligned} m/s&=\frac{m}{s}\\ m/s&=m/s\\ \end{aligned}\end{equation} $$

We can also use $km/h$, $km$ and $h$ respectively for the units of speed, distance and time: $$ \begin{equation}\begin{aligned} km/h&=\frac{km}{h}\\ km/h&=km/h\\ \end{aligned}\end{equation} $$

Using dimensional analysis notation

We can use the following symbols as placeholders for any of the units, be it standard or not, in order to show that a formula is homogeneous:

  • $[L]$ - length
  • $[T]$ - time
  • $[M]$ - mass
  • $[\theta]$ - temperature

The first $3$ are the dimensions of the physical quantities associated with mechanics.

The dimension of a number (a numerical constant is exactly one ($[1]$)).

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