Homogeneous equation
Physical Quantities & Units - OCR A-Level Physics
Key Definition
Homogeneous equation
An equation in which every term has the same base units. The base units on the left-hand side equal the base units on the right-hand side, and every term inside a sum has matching base units too. A physics equation that is not homogeneous cannot be correct.
An equation in which every term has the same base units. The base units on the left-hand side equal the base units on the right-hand side, and every term inside a sum has matching base units too. A physics equation that is not homogeneous cannot be correct.
Procedure: testing homogeneity
Step 1: take the left-hand side of the equation and write its base units. Step 2: take each term on the right-hand side separately and write its base units. Step 3: compare. If every term has identical base units, the equation is homogeneous. If even one term differs, the equation is not homogeneous and must be wrong.
$$[\text{LHS}] = [\text{term}_{1}] = [\text{term}_{2}] = \ldots$$
- Only quantities with the same base units can be added or subtracted. You cannot add a length to a time.
- Pure numbers (such as $\tfrac{1}{2}$ in $E_{k} = \tfrac{1}{2}mv^{2}$) and angles in radians are dimensionless and play no part in the units check.
- Worked example: in $v = u + at$, $[v] = \text{m s}^{-1}$, $[u] = \text{m s}^{-1}$, $[at] = (\text{m s}^{-2})(\text{s}) = \text{m s}^{-1}$. All three terms share $\text{m s}^{-1}$, so the equation is homogeneous.
- Worked example of failure: $v = u + a$ would give $\text{m s}^{-1} = \text{m s}^{-1} + \text{m s}^{-2}$. The last term has different base units, so the equation cannot be physically correct.
- Homogeneity can be used to derive an unknown power. If $T \propto m^{a} k^{b}$ for a spring, matching base units on both sides fixes $a$ and $b$.
Common Mistake
HIGH
Wrong: Assuming that a homogeneous equation must be physically correct.
Right: Homogeneity is a one-way test. If an equation is not homogeneous it cannot be correct; but if it is homogeneous it may still be wrong. The test cannot detect missing dimensionless constants (such as the $\tfrac{1}{2}$ in $E_{k} = \tfrac{1}{2}mv^{2}$), wrong numerical factors, or sign errors. A homogeneous equation only proves dimensional consistency, not physical truth.
Right: Homogeneity is a one-way test. If an equation is not homogeneous it cannot be correct; but if it is homogeneous it may still be wrong. The test cannot detect missing dimensionless constants (such as the $\tfrac{1}{2}$ in $E_{k} = \tfrac{1}{2}mv^{2}$), wrong numerical factors, or sign errors. A homogeneous equation only proves dimensional consistency, not physical truth.
Examiner Tips and Tricks
- In a "show that this equation is homogeneous" question, write the base units of every term separately, then state that they match.
- Do not just say "units are the same" without working: OCR mark schemes award marks for the substitutions, not the conclusion.
- Set out the working in clear lines: $[\text{LHS}] = \ldots$ on one line, $[\text{RHS}_{1}] = \ldots$ on the next, and so on.