Units, Measurement and Error Analysis
What the exam tests
Questions test SI base units, derived units, dimensions, dimensional consistency, significant figures, least count, absolute and relative error, and combination of errors in sums, products and powers.
Core rules
- SI base units include metre, kilogram, second, ampere, kelvin, mole and candela.
- Dimensional formulae use M, L, T, A, K, mol and cd. Dimensionally consistent equations have the same dimensions on both sides.
- Dimensionless quantities include strain, refractive index, relative density, angle and efficiency.
- Absolute error is the magnitude of uncertainty; relative error = absolute error/measured value; percentage error = relative error×100%.
- For addition or subtraction, absolute errors add. For multiplication or division, relative errors add. For a power a^n, the relative error is |n| times the relative error in a.
- Significant figures in a result should match the least precise measurement.
Worked example
Mass is 4.00 g ± 0.01 g and volume is 2.00 cm^3 ± 0.02 cm^3. Density is 4.00/2.00 = 2.00 g/cm^3. Relative error in mass is 0.01/4.00 = 0.0025; in volume 0.02/2.00 = 0.0100. For division, add relative errors: 0.0125, or 1.25%. Absolute error in density is 2.00×0.0125 = 0.025, so the result is about 2.00 ± 0.03 g/cm^3.
Traps
Add absolute errors only for sums and differences; add relative errors for products and quotients. Do not count zero error and least count as the same thing. Dimensional analysis cannot determine dimensionless constants. Rounding too early changes the final error. A quantity can be dimensionless but still have units such as radians.