Physical quantities and units

Learning objectives

• Resolve vectors into perpendicular components.
• Check equations using base units.
• Estimate combined uncertainties.

The big idea

Base quantities provide a common language for derived quantities such as force and energy. An equation must be dimensionally consistent, although dimensional consistency alone does not prove it is correct.

Vectors require magnitude and direction. Resolve them along chosen perpendicular axes before adding components. Measurement uncertainty describes a defensible range; random spread and systematic offsets need different treatments.

1. What units can establish

Express derived quantities in base units to check whether a proposed equation is possible. Force has units kg m s⁻², so force multiplied by distance has units kg m² s⁻², the units of energy. A unit check can reject an equation, but cannot establish a dimensionless factor such as one half. Convert prefixes before calculation; squaring a millimetre conversion also squares its scale factor.

2. Uncertainty is not a mistake

A measured value has limited resolution and may fluctuate. Random effects create spread, while a systematic offset moves readings consistently. Repetition can reduce uncertainty in an estimated mean but cannot remove a persistent zero error. State absolute uncertainty with the same unit as the measurement. Percentage uncertainty compares it with the measurement size and helps decide which quantity limits a result.

3. Combining measurements

For maximum-uncertainty estimates, add absolute uncertainties when adding or subtracting readings. Add fractional uncertainties for products and quotients; multiply a fractional uncertainty by the magnitude of a power. If a radius has 2% uncertainty, its calculated area has about 4% uncertainty, ignoring uncertainty in exact constants. These are small-uncertainty estimates, not a universal statistical rule for every distribution.

4. Components tell a vector story

Resolve a force into axes chosen for the problem. A 10 N force at 30° above horizontal has horizontal component 10 cos 30° and vertical component 10 sin 30°. Components are not extra forces acting alongside the original arrow: they are an equivalent description. Combine matching components, then use geometry to recover resultant magnitude and direction. Give the reference axis for any quoted angle.

A method you can reuse

1. List quantities, SI units and the required result.
2. Check calibration and distinguish spread from systematic effects.
3. Choose axes and resolve vectors when necessary.
4. Carry uncertainties through the calculation using the stated rule.
5. Check dimensions and round the value consistently with its uncertainty.

Equations and relationships

Investigate and explain

Measure several diameters of a cylinder, check instrument zero and compare the uncertainty in diameter with that in calculated area.

Use vector addition to test components and data tools to examine scatter. Simulations do not reproduce every instrument error; uncertainty estimates need the actual measurement method.

Open the interactive investigation

Check your understanding