(d) checking the homogeneity of physical equations using S.I. base units

The idea

Homogeneity checking tests whether every term in a physical equation reduces to the same S.I. base units. Write each quantity as its base-unit combination: for example speed is m s⁻¹, acceleration is m s⁻², force is kg m s⁻² and energy is kg m² s⁻². Substitute these into the equation and simplify powers. If both sides match, the equation is homogeneous; if they do not, it cannot be correct. This catches errors such as writing v = u + at², where at² has units m and v has m s⁻¹. It cannot prove an equation is correct, because dimensionless factors such as ½ or 2π are invisible to the check.

What earns the marks

• State that homogeneity requires every additive term in an equation to have the same S.I. base units.
• Express each physical quantity in S.I. base units, for example force as kg m s⁻² and energy as kg m² s⁻².
• Substitute the base units into the equation and simplify the powers of kg, m and s.
• Compare the simplified units on each side and conclude whether the equation is homogeneous.
• Recognise that a homogeneous equation may still be wrong because dimensionless constants cannot be detected.

Common mistakes and how to put them right

• Treating a homogeneous equation as automatically correct.

  Put it right: Homogeneity is a necessary but not sufficient condition, so dimensionless factors must still be checked separately.

• Adding or comparing terms with different base units, such as adding m s⁻¹ to m.

  Put it right: Every additive term must reduce to the same base units before the equation can be homogeneous.

• Forgetting to square or cube the units of a quantity when it is raised to a power.

  Put it right: Raise the whole base-unit combination to that power, for example (m s⁻¹)² = m² s⁻².

Exam tips

• Write the base units of each symbol before substituting, so the substitution step is clear and easy to check.
• Simplify powers of kg, m and s separately, then compare the final combinations on both sides.
• If the units do not match, state clearly that the equation is not homogeneous and therefore cannot be correct.