Units and Measurements

Topic-by-topic explanations

1. Introduction

Physics describes natural phenomena with measurable quantities. Every measurement must include a number, a unit and an understanding of its uncertainty.

2. The international system of units

The seven SI base quantities are length (metre, m), mass (kilogram, kg), time (second, s), current (ampere, A), temperature (kelvin, K), amount of substance (mole, mol) and luminous intensity (candela, cd). Derived units are combinations of these, such as newton = kg m s⁻².

3. Significant figures

Significant figures indicate measurement precision. Non-zero digits and zeros between non-zero digits count; leading zeros do not; trailing zeros after a decimal point count. For multiplication/division use the fewest significant figures; for addition/subtraction round to the least precise decimal place.

4. Dimensions of physical quantities

Dimensions express dependence on base dimensions such as [M], [L] and [T]. For example, velocity is [LT⁻¹], force [MLT⁻²] and energy [ML²T⁻²]. Dimensions are distinct from units: metres and centimetres are different units of length.

5. Dimensional formulae and dimensional equations

A dimensional formula expresses a quantity as powers of base dimensions. A dimensional equation is homogeneous when each side and all added terms have the same dimensions. Pure numerical constants have no dimensions.

6. Dimensional analysis and its applications

Dimensional analysis checks homogeneity, helps infer the powers in a proposed relationship and converts between unit systems. It cannot determine constants such as 2 or π, distinguish work from torque, or prove a formula is physically correct.

Formula sheet

Use consistent SI units, identify the system and assumptions, and show the steps of each calculation. Check units and whether the result is physically reasonable.

Worked example

Common errors and revision checklist

Do not count leading zeros as significant. Keep guard digits until the final step. Dimensional correctness is necessary but not sufficient for a valid physical law.

• Write the relevant law or relation before substituting values.
• Show units, signs and directions.
• Check assumptions and round only the final result.