Dimensions and Dimensional Analysis: Quick Revision

A brief note on Dimensions and Dimensional Analysis. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• The dimensions of a quantity are the powers of $[\text{M}]$, $[\text{L}]$, $[\text{T}]$ in it: a force is $[\text{M}\,\text{L}\,\text{T}^{-2}]$.
• Find them from any formula: multiplying adds powers, dividing subtracts them, pure numbers have none.
• Homogeneity: every term of a correct equation has the same dimensions; the argument of sin, cos, log or $e$ is a pure number.
• Deriving: write the answer as $k\,a^{x}b^{y}c^{z}$ and match M, L and T.
• Changing units: $n_2 = n_1 (M_1/M_2)^{a} (L_1/L_2)^{b} (T_1/T_2)^{c}$.
• Limits: no pure numbers, no sums, no functions, at most three unknown powers in mechanics.

Dimensions at a glance

A worked example

The pressure at a depth $h$ in a liquid might depend on its density $\rho$, $g$ and $h$. Find the form.

1. $p = k\,\rho^{x}g^{y}h^{z}$: $[\text{M}\,\text{L}^{-1}\,\text{T}^{-2}] = [\text{M}\,\text{L}^{-3}]^{x}[\text{L}\,\text{T}^{-2}]^{y}[\text{L}]^{z}$.
2. M: $x = 1$. T: $-2 = -2y$, so $y = 1$. L: $-1 = -3 + 1 + z$, so $z = 1$.
3. So $p = k\rho g h$; experiment gives $k = 1$.

Practice with new numbers

A program to try

The program checks an equation term by term, and catches a wrong one.

Mistakes to avoid

• Thinking every constant is a pure number. $G$ and $h$ have dimensions.
• Forgetting that only like terms add. In $A + Bt$, $Bt$ must have the dimensions of $A$.
• Trusting a passed check. A right-looking formula can still have a wrong number or sign.
• Changing units the wrong way round. A bigger unit gives a smaller number.

Check yourself

1. What are the dimensions of pressure times volume?

2. A distance follows $x = At^2 + \dfrac{B}{t}$. What are the dimensions of $B$?

3. Is $E = mc$ possible for an energy?

4. The speed of a falling raindrop might depend on $g$ and its radius $r$. Find the form.

5. Why can dimensions not find the $2\pi$ in $T = 2\pi\sqrt{l/g}$?