Units & Measurements - Quick Revision
Units
- Measurement = number + unit. 7 SI base units: metre (length, via c), kilogram (mass, via Planck h), second (time, via Cs frequency), ampere (current), kelvin (temperature, via Boltzmann k), mole (amount), candela (luminous intensity).
- Supplementary: radian (plane angle), steradian (solid angle) - both dimensionless.
- Systems: SI, CGS (cm,g,s), FPS (foot,pound,s).
Dimensions
- Dimensions = powers of base quantities, written in [ ]. e.g. force [M L T^-2], work/energy [M L^2 T^-2], pressure [M L^-1 T^-2], momentum [M L T^-1], density [M L^-3].
- Principle of homogeneity: every term must have the same dimensions. Used to (i) check equations, (ii) derive relations, (iii) convert units.
- Limitations: can't find dimensionless constants; can't handle trig/log/exp (arguments must be dimensionless); can't distinguish quantities with the same dimensions. A dimensionally correct equation may still be wrong; a dimensionally wrong one is definitely wrong.
Errors
- Systematic (one-directional: instrumental/method/personal) vs random (irregular). Accuracy = closeness to true value; precision = resolution/least count.
- Absolute error, relative error = ฮa/a, percentage error = relative x 100.
- Combination: sum/difference -> add absolute errors; product/quotient -> add relative errors; power a^n -> n x relative error.
Significant figures
- Non-zero digits significant; zeros between non-zero digits significant; leading zeros NOT significant; trailing zeros significant only with a decimal point.
- Multiply/divide: keep least significant figures. Add/subtract: keep least decimal places.
- Rounding: drop <5 (down), >5 (up); exactly 5 -> make preceding digit even.