Quantities and Units

Physical quantity — something that can be measured, expressed as a number together with a unit. The number alone is meaningless: 5 is not a length.

SI base units — six of the seven are used throughout this course, and every other unit is built from them.

Derived units — combinations of base units. The newton is the clearest example: from F = ma, one newton is one kilogram metre per second squared, so N = kg m s⁻². The joule is a newton metre, kg m² s⁻², and the watt is a joule per second, kg m² s⁻³.

Example: to find the base units of the pascal, start from p = F/A. That is N ÷ m², and a newton is kg m s⁻², so a pascal is kg m s⁻² ÷ m² = kg m⁻¹ s⁻². You never need to memorise these — derive them from an equation you know.

Checking an equation is homogeneous — every term in a physical equation must have the same units. If they do not match, the equation is certainly wrong.

Example: check v² = u² + 2as. The left side is (m s⁻¹)² = m² s⁻². The first term on the right is the same. The second is m s⁻² × m = m² s⁻². All three match, so the equation is homogeneous. Note what this does not prove: v² = u² + 5as is also homogeneous, and wrong. Matching units catches mistakes but never confirms correctness.

Prefixes — a shorthand for powers of ten.

Converting areas and volumes — this is where prefixes trip people up. 1 cm = 10⁻² m, so 1 cm² = (10⁻²)² = 10⁻⁴ m², and 1 cm³ = 10⁻⁶ m³. The prefix is raised to the same power as the unit.

Example: a density of 2.7 g cm⁻³ in SI units is 2.7 × 10⁻³ kg ÷ 10⁻⁶ m³ = 2700 kg m⁻³. Convert both the gram and the cubic centimetre, and remember the cube.