S.I. units: (b) derived units of S.I. base units

The idea

Derived units are combinations of S.I. base units. For example, the newton (N) is kg m s⁻², the joule (J) is kg m² s⁻², and the watt (W) is kg m² s⁻³. You can check whether an equation is dimensionally consistent by replacing each quantity with its base units and seeing whether both sides match. For instance, in F = ma, the right-hand side has units kg × m s⁻² = kg m s⁻², which is the newton. Derived units may have special names, but they can always be broken down into base units. This skill helps you identify errors in equations and convert between units.

What earns the marks

• Derived units are formed from combinations of S.I. base units.
• The newton (N) is equivalent to kg m s⁻².
• The joule (J) is equivalent to kg m² s⁻².
• The watt (W) is equivalent to kg m² s⁻³.
• Dimensional analysis checks whether an equation is consistent by comparing base units on both sides.
• Special names for derived units can be replaced by their base-unit combinations.

Common mistakes and how to put them right

• Writing the base units of the newton as kg m s⁻¹.

  Put it right: The correct combination is kg m s⁻² because acceleration is m s⁻².

• Forgetting that the joule includes a squared metre.

  Put it right: J = kg m² s⁻², not kg m s⁻².

• Treating the watt as kg m² s⁻².

  Put it right: Power is energy per second, so W = kg m² s⁻³.

• Assuming that any combination of base units is a valid derived unit.

  Put it right: The combination must match the physical quantity.

Exam tips

• In multiple-choice questions, substitute base units into each option and eliminate those that do not match the required quantity.
• Learn the base-unit forms of common derived units such as N, J, W, Pa and C to save time.
• When checking an equation, cancel units algebraically and compare the final combination on each side.