2Dimensional analysis

IA Dynamics and Relativity

2 Dimensional analysis
When considering physical theories, it is important to be aware that physical
quantities are not pure numbers. Each physical quantity has a dimension.
Roughly speaking, dimensions are what units represent, such as length, mass
and time. In any equation relating physical quantities, the dimensions must be
consistent, i.e. the dimensions on both sides of the equation must be equal.
For many problems in dynamics, the three basic dimensions are sufficient:
length, L
mass, M
time, T
The dimensions of a physical quantity
X
, denoted by [
X
] are expressible
uniquely in terms of L, M and T . For example,
[area] = L
2
[density] = L
3
M
[velocity] = LT
1
[acceleration] = LT
2
[
force
] =
LMT
2
since the dimensions must satisfy the equation
F
=
ma
.
[energy] = L
2
MT
2
, e.g. consider E = mv
2
/2.
Physical constants also have dimensions, e.g. [
G
] =
L
3
M
1
T
2
by
F
=
GMm/r
2
.
The only allowed operations on quantities with dimensions are sums and
products (and subtraction and division), and if we sum two terms, they must
have the same dimension. For example, it does not make sense to add a length
with an area. More complicated functions of dimensional quantities are not
allowed, e.g. e
x
again makes no sense if x has a dimension, since
e
x
= 1 + x +
1
2
x
2
+ ···
and if
x
had a dimension, we would be summing up terms of different dimensions.