Physical quantities, units and dimensions
Every physical quantity has a magnitude and a unit. The SI system has seven base quantities: length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), temperature (kelvin, K), amount of substance (mole, mol) and luminous intensity (candela, cd). Derived units are combinations: newton N = kg m s⁻², joule J = kg m² s⁻², watt W = kg m² s⁻³, pascal Pa = kg m⁻¹ s⁻².
The dimensions of a quantity show how it is built from the base quantities: mass [M], length [L], time [T]. Speed has dimensions [L T⁻¹], force [M L T⁻²] and energy [M L² T⁻²]. Dimensions are used to check equations (both sides must have the same dimensions) and to derive relationships. A homogeneous equation has the same dimensions on both sides, but this does not prove it is correct: numerical constants such as ½ are dimensionless.
| Quantity | SI unit | Dimensions |
|---|---|---|
| Velocity | m s⁻¹ | L T⁻¹ |
| Acceleration | m s⁻² | L T⁻² |
| Force | N | M L T⁻² |
| Work, energy | J | M L² T⁻² |
| Power | W | M L² T⁻³ |
| Pressure | Pa | M L⁻¹ T⁻² |
| Frequency | Hz | T⁻¹ |
Show that the equation v² = u² + 2as is dimensionally consistent.
- Left side: v² has dimensions (L T⁻¹)² = L² T⁻².
- Right side: u² is L² T⁻²; as is (L T⁻²)(L) = L² T⁻².
Answer: Every term has dimensions L² T⁻², so the equation is consistent.
Prefixes and significant figures
| Prefix | Symbol | Factor |
|---|---|---|
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| milli | m | 10⁻³ |
| micro | µ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
The number of significant figures shows the precision of a measurement. In calculations the answer should not have more significant figures than the least precise data. Use standard form for very large or small numbers: 0.000 045 = 4.5 × 10⁻⁵.
Errors and uncertainties
An error is the difference between a measured value and the true value. Random errors vary unpredictably and are reduced by repeating and averaging. Systematic errors shift all readings the same way (a zero error, a wrongly calibrated instrument) and are not reduced by repeating; correct them by calibration. Parallax and reaction time cause errors too.
| Instrument | Typical precision |
|---|---|
| Metre rule | ±1 mm |
| Vernier calipers | ±0.1 mm |
| Micrometer screw gauge | ±0.01 mm |
| Stopwatch | ±0.1 s (reaction time) |
| Digital balance | ±0.01 g |
A rectangle has a length and a width each measured with an uncertainty. Find the area and its percentage and absolute uncertainty.
Key terms
- SI unit
- A unit of the International System of Units.
- Dimension
- The base quantities that make up a physical quantity.
- Systematic error
- An error that shifts all readings in the same direction.
- Random error
- An unpredictable variation in repeated readings.
Summary
- Seven SI base units; derived units combine them; dimensions test equations.
- Use prefixes, standard form and sensible significant figures.
- Random errors are reduced by averaging; systematic errors by calibration.
- Uncertainties: add percentage uncertainties for products and quotients.