3. Interlude: Dimensional Analysis
The essence of dimensional analysis is very simple: if you are asked how hot it is outside,
the answer is never “2 o’clock”. You’ve got to make sure that the units agree. Quantities
which come with units are said to have dimensions. In contrast, pure numbers such as
2 or ⇡ are said to be dimensionless.
In all the examples that we met in the previous section, the units are hiding within
the variables. Nonetheless, it’s worth our e↵ort to dig them out. In most situations,
it is useful to identify three fundamental dimensions: length L, mass M and time T .
The dimensions of all other quantities should be expressible in terms of these. We will
denote the dimension of a quantity Y as [Y ]. Some basic examples include,
[Area] = L2
1
[Speed] = LT
2
[Acceleration] = LT
2
[Force] = M LT
[Energy] = M L2 T 2
The first three should be obvious. You can quickly derive the last two by thinking of
your favourite equation and insisting that the dimensions on both sides are consistent.
For example, F = ma immediately gives the dimensions [F ], while E = 12 mv 2 will give
you the dimensions [E]. This same technique can be used to determine the dimensions of
any constants that appear in equations. For example, Newton’s gravitational constant
appears in the formula F = GM m/r2 . Matching dimensions on both sides tells us
that
1
[G] = M L3 T 2
You shouldn’t be too dogmatic in insisting that there are exactly three dimensions of
length, mass and time. In some problems, it will be useful to introduce further dimen-
sions such as temperature or electric charge. For yet other problems, it could be useful
to distinguish between distances in the x-direction and distances in the z-direction. For
example, if you’re a sailor, you would be foolish to think of vertical distances in the
same way as horizontal distances. Your life is very di↵erent if you mistakenly travel 10
fathoms (i.e. vertically) instead of 10 nautical miles (i.e. horizontally) and it’s useful
to introduce di↵erent units to reflect this.
– 40 –
, Conversely, when dealing with matters in fundamental physics, we often reduce the
number of dimensional quantities. As we will see in Section 7, in situations where special
relativity is important, time and space sit on the same footing and can be measured in
the same unit, with the speed of light providing a conversion factor between the two.
(We’ll have more to say on this in Section 7.3.3). Similarly, in statistical mechanics,
Boltzmann’s constant provides a conversion factor between temperature and energy.
Scaling: Bridgman’s Theorem
Any equation that we derive must be dimensionally consistent. This simple observation
can be a surprisingly powerful tool. Firstly, it provides a way to quickly check whether
an answer has a hope of being correct. (And can be used to spot where a mistake
appeared in a calculation). Moreover, there are certain problems that can be answered
using dimensional analysis alone, allowing you to avoid calculations all together. Let’s
look at this in more detail.
We start by noting that dimensionful quantities such as length can only appear in
equations as powers, L↵ for some ↵. We can never have more complicated functions.
One simple way to see this is to Taylor expand. For example, the exponential function
has the Taylor expansion
x2
ex = 1 + x + + ...
2
The right-hand side contains all powers of x and only makes sense if x is a dimensionless
quantity: we can never have eL appearing in an exponent otherwise we’d be adding a
length to an area to a volume and so on. A similar statement holds for sin x and log x,
for your favourite and least favourite functions. In all cases, the argument must be
dimensionless unless the function is simply of the form x↵ . (If your favourite function
doesn’t have a Taylor expansion around x = 0, simply expand around a di↵erent point
to reach the same conclusion).
Suppose that we want to compute some quantity Y . This must have dimension
[Y ] = M ↵ L T
for some ↵, and . (There is, in general, no need for these to be integers although
they are typically rational). We usually want to determine Y in terms of various
other quantities in the game – call them Xi , with i = 1, . . . n. These too will have
certain dimensions. We’ll focus on just three of them, X1 , X2 and X3 . We’ll assume
that these three quantities are “dimensionally independent”, meaning that by taking
– 41 –
The essence of dimensional analysis is very simple: if you are asked how hot it is outside,
the answer is never “2 o’clock”. You’ve got to make sure that the units agree. Quantities
which come with units are said to have dimensions. In contrast, pure numbers such as
2 or ⇡ are said to be dimensionless.
In all the examples that we met in the previous section, the units are hiding within
the variables. Nonetheless, it’s worth our e↵ort to dig them out. In most situations,
it is useful to identify three fundamental dimensions: length L, mass M and time T .
The dimensions of all other quantities should be expressible in terms of these. We will
denote the dimension of a quantity Y as [Y ]. Some basic examples include,
[Area] = L2
1
[Speed] = LT
2
[Acceleration] = LT
2
[Force] = M LT
[Energy] = M L2 T 2
The first three should be obvious. You can quickly derive the last two by thinking of
your favourite equation and insisting that the dimensions on both sides are consistent.
For example, F = ma immediately gives the dimensions [F ], while E = 12 mv 2 will give
you the dimensions [E]. This same technique can be used to determine the dimensions of
any constants that appear in equations. For example, Newton’s gravitational constant
appears in the formula F = GM m/r2 . Matching dimensions on both sides tells us
that
1
[G] = M L3 T 2
You shouldn’t be too dogmatic in insisting that there are exactly three dimensions of
length, mass and time. In some problems, it will be useful to introduce further dimen-
sions such as temperature or electric charge. For yet other problems, it could be useful
to distinguish between distances in the x-direction and distances in the z-direction. For
example, if you’re a sailor, you would be foolish to think of vertical distances in the
same way as horizontal distances. Your life is very di↵erent if you mistakenly travel 10
fathoms (i.e. vertically) instead of 10 nautical miles (i.e. horizontally) and it’s useful
to introduce di↵erent units to reflect this.
– 40 –
, Conversely, when dealing with matters in fundamental physics, we often reduce the
number of dimensional quantities. As we will see in Section 7, in situations where special
relativity is important, time and space sit on the same footing and can be measured in
the same unit, with the speed of light providing a conversion factor between the two.
(We’ll have more to say on this in Section 7.3.3). Similarly, in statistical mechanics,
Boltzmann’s constant provides a conversion factor between temperature and energy.
Scaling: Bridgman’s Theorem
Any equation that we derive must be dimensionally consistent. This simple observation
can be a surprisingly powerful tool. Firstly, it provides a way to quickly check whether
an answer has a hope of being correct. (And can be used to spot where a mistake
appeared in a calculation). Moreover, there are certain problems that can be answered
using dimensional analysis alone, allowing you to avoid calculations all together. Let’s
look at this in more detail.
We start by noting that dimensionful quantities such as length can only appear in
equations as powers, L↵ for some ↵. We can never have more complicated functions.
One simple way to see this is to Taylor expand. For example, the exponential function
has the Taylor expansion
x2
ex = 1 + x + + ...
2
The right-hand side contains all powers of x and only makes sense if x is a dimensionless
quantity: we can never have eL appearing in an exponent otherwise we’d be adding a
length to an area to a volume and so on. A similar statement holds for sin x and log x,
for your favourite and least favourite functions. In all cases, the argument must be
dimensionless unless the function is simply of the form x↵ . (If your favourite function
doesn’t have a Taylor expansion around x = 0, simply expand around a di↵erent point
to reach the same conclusion).
Suppose that we want to compute some quantity Y . This must have dimension
[Y ] = M ↵ L T
for some ↵, and . (There is, in general, no need for these to be integers although
they are typically rational). We usually want to determine Y in terms of various
other quantities in the game – call them Xi , with i = 1, . . . n. These too will have
certain dimensions. We’ll focus on just three of them, X1 , X2 and X3 . We’ll assume
that these three quantities are “dimensionally independent”, meaning that by taking
– 41 –
