Checking equations for homogeneity

Overview

Recall questions on reducing every term of an equation to base units and comparing the two sides, worked checks on named equations, why a homogeneous equation need not be correct, and what a mismatch tells you before any number is substituted.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What does it mean to say that an equation is homogeneous?

   Model answer

   An equation is homogeneous when every term in it reduces to the same combination of SI base units.

2. 2

   If an equation is found not to be homogeneous, what can be concluded about it?

   Model answer

   The equation is certainly wrong, because quantities with different base units cannot be equated or added.

3. 3

   A homogeneous equation is not necessarily a correct one. Explain why.

   Model answer

   A homogeneous check cannot detect a wrong dimensionless factor, so an error such as writing 2π in place of ½, or omitting a pure number altogether, survives the test.

4. 4

   Why do pure numbers such as ½ and 2π have no effect on a homogeneity check?

   Model answer

   Pure numbers carry no units at all, so multiplying a term by one leaves that term's base units unchanged.

5. 5

   Show that v = u + at is homogeneous.

   Model answer

   Both u and v are measured in m s⁻¹, and at has units m s⁻² × s = m s⁻¹, so all three terms share the base units m s⁻¹.

6. 6

   Show that s = ut + ½at² is homogeneous.

   Model answer

   The term ut has units m s⁻¹ × s = m and ½at² has units m s⁻² × s² = m, which match the metres of s.

Other ways to work this

The same content, four other ways.

What is inside Levo

Six ways to work the same material.