Dimensions

No matter where you go, there you are.

—Buckaroo Banzai

Dimensions and derived units

Most of the physical quantities we actually deal with in science and also in our daily lives, have units of their own: volume, pressure, energy and electrical resistance are only a few of hundreds of possible examples.

It is important to understand, however, that all of these units can be expressed in terms of building blocks that we refer to here as fundamental dimensions.

These fundamental dimensions are

  • mass, \(M\)

  • length, \(L\)

  • time, \(t\)

  • temperature, \(T\)

(Electrical charge is also a fundamental dimension, but we won’t use it in this course.)

Consider, for example, the unit of volume, which we denote as \(V\). To measure the volume of a rectangular box, we need to multiply the lengths as measured along the three coordinates:

\[V = x \cdot y \cdot z\]

We say, therefore, that volume has the dimensions of length-cubed: \(dim (V) = L^{3}\).

Thus, any formula that calculates a volume must contain within it the \(\si{L^{3}}\) dimension.

In general, a unit will be represented by a product of the fundamental dimensions taken to some power, i.e.

\[dim(\text{unit}) = M^{C1} \, L^{C2} \, t^{C3} \, T^{C4}\]

For instance, for volume, \(C1=0\), \(C2=3\), \(C3=0\), and \(C4=0\), leading to

\[dim(V) = M^{0} L^{3} t^{0} T^{0} = L^{3}\]

Here is a table of values of \(C1\), \(C2\), and \(C3\) for some common units:

quantity

C1

C2

C3

SI unit, other typical units

mass

1

0

0

kilogram, gram, metric ton, pound

length

0

1

0

meter, foot, mile

time

0

0

1

second, day, year

volume

0

3

0

liter, \(\si{cm^{3}}\), quart, fluid ounce

density

1

-3

0

\(\si{kg/m^{3}}\) , \(\si{g/cm^{3}}\)

force

1

1

-2

newton, dyne

pressure

1

-1

-2

pascal, atmosphere, torr

energy

1

2

-2

joule, erg, calorie, electron-volt

power

1

2

-3

watt

Why are unit dimensions useful?

Why consider the dimensions of a unit:

  1. to help understand the relations between various units of measure and thereby get a better understanding of their physical meaning

  2. to understand how to calculate a desired quantity, using whatever specific units you wish (Note here the distinction between dimensions and units.)

  3. to assure consistency and provide an important check on the form of an equation and terms in that equation. Just as you cannot add apples to oranges, an expression such as \(a = b + c \, x^{2}\) is meaningless unless the dimensions of each term are identical. Also, many quantities must be dimensionless; for example, the variable \(x\) in expressions such as \(log(x)\), \(exp(x)\), and \(sin(x)\). Checking through the dimensions of such a quantity can help avoid errors.

The formal, detailed study of dimensions is known as dimensional analysis and is a topic in many engineering and physics courses.

Exercise: Dimensions

Part 1:

Write down the dimensions of the units for energy and velocity.

Using these expressions, find the fundamental dimensions of energy divided by the square of velocity.

Does your answer make sense?

Part 2:

You see the following equation in a book: \(y = \log (x / \alpha) + \beta\).

If the dimensions for \(x\) are \(M L / t^{2}\), what are the appropriate dimensions for \(y\) , \(\alpha\) , and \(\beta\)?