§ 1.1 Physical quantities
Key ideas
- Every physical quantity is a magnitude with a unit; a number on its own answers no physics question.
- Be ready to make reasonable estimates: a student's mass ~60 kg, a door ~2 m, walking speed ~1.5 m s⁻¹, mains lamp ~60 W.
Watch out: sanity-check estimates by order of magnitude. An answer of 10⁶ m s⁻¹ for a runner is wrong by five powers of ten, however neat the algebra looked.
§ 1.2 SI units
Key ideas
- Five SI base quantities to recall: mass (kg), length (m), time (s), current (A), temperature (K).
- Derived units are built by multiplying and dividing base units: the newton is kg m s⁻².
- An equation must be homogeneous: the base units of every term must match.
- Prefixes from pico to tera: p (10⁻¹²), n (10⁻⁹), µ (10⁻⁶), m (10⁻³), c (10⁻²), k (10³), M (10⁶), G (10⁹), T (10¹²).
Watch out: homogeneity is necessary but not sufficient. s = ut is homogeneous yet wrong for accelerated motion; dimensionless constants like the ½ are invisible to a unit check.
§ 1.3 Errors and uncertainties
Key ideas
- Systematic error shifts every reading the same way (including zero errors); repeating does not reduce it.
- Random error scatters readings; reduce it by repeating and averaging.
- Precision is the closeness of repeats to each other; accuracy is the closeness of the mean to the true value.
- Adding or subtracting quantities: add absolute uncertainties. Multiplying or dividing: add percentage uncertainties. Raising to a power n: multiply the percentage uncertainty by n.
Equations
% unc. = Δx/x × 100percentage uncertainty from absolute uncertainty%
y = aⁿ: %Δy = n × %Δaa power multiplies the percentage uncertainty%
Fig. 1 · Precision and accuracy are independent: the left cluster repeats well but carries a systematic shift; the right set averages to the truth but scatters randomly.
Watch out: repeating and averaging attacks random error only. A systematic error (a zero error, a miscalibrated scale) survives any number of repeats; it must be found and subtracted.
§ 1.4 Scalars and vectors
Key ideas
- Scalars (magnitude only): distance, speed, mass, energy, time, temperature. Vectors (magnitude and direction): displacement, velocity, acceleration, force, momentum, weight.
- Add vectors nose-to-tail; the resultant runs from the start of the first to the end of the last. Subtracting a vector means adding it reversed.
- Resolve a vector at angle θ to an axis into F cos θ along the axis and F sin θ perpendicular to it.
Equations
Fx = F cos θ, Fy = F sin θcomponents along and perpendicular to the axisN
R² = Fx² + Fy²resultant of two perpendicular componentsN
Fig. 2 · Resolving: the component along the axis carries cos θ, the perpendicular one sin θ; Pythagoras rebuilds the original magnitude from the pair.
Watch out: cos θ goes with the side the angle touches. If θ is measured from the vertical instead of the horizontal, the components swap: always check which axis the angle is drawn from.