Dimensional analysis in engineering
Dimensional analysis is a way to check whether an equation makes physical sense before doing detailed
numerical work. The main idea is simple: both sides of a valid physical formula must have the same
dimensions.
\[
[\text{left side}]=[\text{right side}]
\]
For example, speed has dimensions of length divided by time. Therefore any formula for speed must reduce
to the dimensions
\[
\mathrm{L}\mathrm{T}^{-1}.
\]
Base dimensions used in this calculator
The calculator tracks five common engineering base dimensions:
| Symbol |
Meaning |
Example unit |
| \(\mathrm{M}\) |
Mass |
\(\mathrm{kg}\) |
| \(\mathrm{L}\) |
Length |
\(\mathrm{m}\) |
| \(\mathrm{T}\) |
Time |
\(\mathrm{s}\) |
| \(\Theta\) |
Temperature |
\(\mathrm{K}\) |
| \(\mathrm{I}\) |
Electric current |
\(\mathrm{A}\) |
Example: checking \(v=\sqrt{2gh}\)
Suppose
\[
g:\mathrm{m\,s^{-2}},
\qquad
h:\mathrm{m}.
\]
The constant \(2\) is dimensionless, so it does not affect the dimensions.
\[
[2gh]
=
[g][h]
=
\left(\mathrm{L}\mathrm{T}^{-2}\right)\left(\mathrm{L}\right)
=
\mathrm{L}^{2}\mathrm{T}^{-2}
\]
Taking the square root gives
\[
\sqrt{\mathrm{L}^{2}\mathrm{T}^{-2}}
=
\mathrm{L}\mathrm{T}^{-1}.
\]
This is the dimension of velocity, so the formula is dimensionally consistent.
How the calculator treats operations
Multiplication adds dimension exponents, division subtracts dimension exponents, and powers multiply
dimension exponents.
\[
[ab]=[a][b],
\qquad
\left[\frac{a}{b}\right]=[a][b]^{-1},
\qquad
[a^n]=[a]^n
\]
Addition and subtraction are stricter. The terms being added or subtracted must already have the same
dimensions.
\[
[a+b]\ \text{is valid only if}\ [a]=[b].
\]
Functions and dimensions
Some functions require dimensionless inputs. For example, the argument of a sine, cosine, exponential,
or logarithm must be dimensionless.
\[
\sin(x),\ \cos(x),\ e^x,\ \ln(x)
\quad\text{require}\quad
[x]=1.
\]
The square-root function is different. It can act on dimensional quantities by halving each dimension
exponent.
\[
\left[\sqrt{x}\right]=[x]^{1/2}.
\]
Unit conversion inside formulas
A formula may be dimensionally correct but still numerically wrong if the units are not handled properly.
This calculator converts each variable into base units before evaluating the formula.
\[
x_{\mathrm{base}}
=
x_{\mathrm{entered}}F_{\mathrm{unit}}
\]
After evaluation, the result can be converted into a target unit:
\[
x_{\mathrm{target}}
=
\frac{x_{\mathrm{base}}}{F_{\mathrm{target}}}.
\]
Buckingham Pi theorem preview
Buckingham Pi theorem is used to reduce a physical problem to dimensionless groups. If a problem has
\(n\) variables and the dimension matrix has rank \(r\), then the number of independent dimensionless
groups is
\[
k=n-r.
\]
The calculator gives a preview of \(n\), \(r\), and \(k\). This is useful for seeing whether a physical
model can be simplified by using dimensionless parameters.
Common mistakes
- Adding quantities with different dimensions, such as length plus time.
- Using degrees Celsius directly in formulas that require absolute temperature.
- Forgetting that area and volume conversion factors are squared or cubed.
- Assuming a formula is correct just because the numbers look reasonable.
- Using trigonometric or logarithmic functions with dimensional inputs.
