Why units matter
A measurement is a number together with a unit. "The table is 2 long" means nothing until you say 2 metres. The International System of Units (SI) is used worldwide in science.
The seven SI base units
Quantity | Unit | Symbol |
|---|---|---|
Length | metre | m |
Mass | kilogram | kg |
Time | second | s |
Electric current | ampere | A |
Thermodynamic temperature | kelvin | K |
Amount of substance | mole | mol |
Luminous intensity | candela | cd |
All other units, such as the newton (kg m s⁻²) and the joule (kg m² s⁻²), are derived units built from these.
Common prefixes: kilo (10³), mega (10⁶), giga (10⁹), centi (10⁻²), milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹).
Significant figures
The significant figures in a measurement are the digits that are reliably known, plus the first uncertain digit. The rules:
All non-zero digits are significant.
Zeros between non-zero digits are significant: 2.05 has 3.
Leading zeros are not significant: 0.0032 has 2.
Trailing zeros after a decimal point are significant: 2.500 has 4.
Trailing zeros in a whole number without a decimal point are ambiguous. Use scientific notation: 4.20 × 10³ has 3.
Changing units does not change the number of significant figures: 4.700 m = 470.0 cm = 4.700 × 10³ mm, all with 4 significant figures.
Arithmetic with significant figures
Multiplication and division: the result has as many significant figures as the least precise factor. 4.237 × 2.51 = 10.63487 → 10.6 (3 s.f.).
Addition and subtraction: the result has as many decimal places as the number with the fewest decimal places. 436.32 + 227.2 + 0.301 = 663.821 → 663.8.
Rounding
If the digit to be dropped is more than 5, round up. If it is less than 5, leave the previous digit unchanged. If it is exactly 5 (followed by nothing or zeros), round so that the preceding digit is even: 2.745 → 2.74, but 2.735 → 2.74.
Errors in measurement
No measurement is perfect.
Systematic errors push readings consistently in one direction, for example a zero error in a vernier caliper.
Random errors vary unpredictably from one reading to the next. Taking the mean of many readings reduces them.
For a set of readings, the mean absolute error Δa is the average of |aᵢ − a_mean|, and the relative error is Δa / a_mean (×100 for percentage error).
How errors combine:
For a sum or difference Z = A ± B, the absolute errors add: ΔZ = ΔA + ΔB.
For a product or quotient Z = AB or A/B, the relative errors add: ΔZ/Z = ΔA/A + ΔB/B.
For a power Z = Aⁿ, the relative error is multiplied by n: ΔZ/Z = n ΔA/A.
Dimensions
The dimensions of a quantity show how it depends on the base quantities, written with square brackets: mass [M], length [L], time [T].
Quantity | Dimensional formula |
|---|---|
Velocity | [L T⁻¹] |
Acceleration | [L T⁻²] |
Force | [M L T⁻²] |
Work / energy | [M L² T⁻²] |
Power | [M L² T⁻³] |
Pressure | [M L⁻¹ T⁻²] |
Uses of dimensional analysis
1. Checking equations. By the principle of homogeneity, every term in a correct equation must have the same dimensions.
Check s = ut + ½at²: [s] = [L], [ut] = [L T⁻¹][T] = [L], [at²] = [L T⁻²][T²] = [L]. All the terms match, so the equation is dimensionally consistent.
2. Deriving relations. Suppose the period T of a simple pendulum depends on its length l and on g. Write T = k lᵃ gᵇ, so [T] = [L]ᵃ [L T⁻²]ᵇ. Matching powers: for T, −2b = 1, so b = −½. For L, a + b = 0, so a = ½. This gives T = k √(l/g). Experiment shows k = 2π.
Limitations. Dimensional analysis cannot find pure-number constants (such as the 2π above), cannot handle sums of terms or trigonometric, exponential and logarithmic functions, and a dimensionally correct equation is not necessarily a physically correct one.