Summary — What This Chapter Covers
- A physical quantity = numerical magnitude × unit. A number with no unit is not a measurement.
- The five SI base quantities you need: mass (kg), length (m), time (s), current (A), temperature (K).
- Derived units can be written as products and quotients of base units; use them to check that an equation is homogeneous.
- Prefixes from pico (10−12) to tera (1012).
- Systematic errors shift every reading the same way; random errors scatter readings. Accuracy is closeness to the true value, precision is closeness of readings to each other.
- Add absolute uncertainties when quantities are added or subtracted; add percentage uncertainties when they are multiplied, divided or raised to a power.
- Scalars have magnitude only; vectors have magnitude and direction. Add vectors head to tail, and resolve a vector into two perpendicular components.
1. Physical Quantities
Physical quantities
A number and a unit
Writing "the length is 3.2" says nothing: 3.2 m, 3.2 mm and 3.2 km are different lengths. Every physical quantity is a numerical magnitude multiplied by a unit.
You are also expected to make reasonable estimates of quantities in the syllabus. Learn a few anchors: mass of an adult ≈ 70 kg, height of a door ≈ 2 m, walking speed ≈ 1.5 m s−1, a car ≈ 1000 kg, mass of an apple ≈ 0.1 kg, atmospheric pressure ≈ 1 × 105 Pa, wavelength of visible light ≈ 5 × 10−7 m, diameter of an atom ≈ 10−10 m.
Worked Example
Estimate the kinetic energy of a sprinter running a 100 m race.
Practice Question
Which is the best estimate of the weight of a 1 litre carton of milk: 0.1 N, 1 N, 10 N or 100 N?
2. SI Units
SI units
Base units, derived units and homogeneity
The SI base quantities on this syllabus are mass (kg), length (m), time (s), current (A) and temperature (K). Every other unit is derived from these through a defining equation. For example force: F = ma, so 1 N = 1 kg m s−2.
An equation is homogeneous when every term has the same base units. A homogeneous equation may still be wrong (a missing ½ does not show up), but an inhomogeneous equation is certainly wrong.
Useful derived units in base units
N = kg m s−2 · J = kg m2 s−2 · W = kg m2 s−3 · Pa = kg m−1 s−2 · C = A s · V = kg m2 s−3 A−1 · Ω = kg m2 s−3 A−2
Prefixes
p 10−12 · n 10−9 · μ 10−6 · m 10−3 · c 10−2 · d 10−1 · k 103 · M 106 · G 109 · T 1012
Worked Example
Show that the equation v2 = u2 + 2as is homogeneous.
Practice Question
Resistivity ρ is defined by R = ρL/A. Express the unit of ρ in SI base units.
3. Errors and Uncertainties
Errors and uncertainties
Systematic, random, precise, accurate
systematic error
zero error
A random error makes readings scatter unpredictably either side of the true value (reaction time, judging a moving pointer). Repeating readings and averaging reduces its effect.
Accuracy
precision