Dimensions

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Section 2.1 Dimensions

During problem-solving in practical applications, it is a useful strategy to keep track of the units of the various quantities. When the desired solution is numerical, it is essential that we use a consistent system of units. In theoretical derivations, on the other hand, where the desired answer is an equation, we may not need to choose a particular system of units. Nevertheless, as a sense-making strategy, it can be valuable to keep track of some of the information contained in units, i.e. the dimensions. Thus, we may not care whether speed is measure in meters per second or miles per hour, but it may still be helpful to keep track that the relevant dimensions are length per unit time.

Notation.

This information is often written:

\begin{equation*} \hbox{speed}\sim\frac{L}{T} \end{equation*}

or

\begin{equation*} \hbox{speed}=\left[\frac{L}{T}\right] \end{equation*}

where the tilde \(\sim\) or the square brackets \(\left[\right]\) are both commonly used notations for dimensions.

Similarly, the dimensions of (volume) mass density are mass per unit volume, \(M/L^3\text{,}\) without regard to whether this density is measured in kilograms per cubic meter or slugs per cubic inch.

The basic dimensions that we will use in this text are: length \((L)\text{,}\) time \((T)\text{,}\) mass \((M)\text{,}\) charge \((C)\text{,}\) and temperature \((K)\text{.}\) (Notice that the same symbol \((C)\) may be used for charge if you are thinking about dimensions and Coulombs if you are thinking about units. As you work through this book, you should learn to write all other derived physical quantities in terms of these basic dimensions. It may also be helpful to be able to recognize certain commonly recurring combinations such as

\begin{equation*} \hbox{force}\sim\frac{ML}{T^2} \end{equation*}

and

\begin{equation*} \hbox{energy}\sim\frac{ML^2}{T^2} \end{equation*}