Chapter 8: Measurement, Units, Errors, and Vectors
The International System of Units
Physics uses the SI (Système International). Its seven base units are the metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd). Since 2019 all seven are defined by fixing the numerical values of fundamental constants: the speed of light , the Planck constant , the elementary charge , the Boltzmann constant , the Avogadro constant , the caesium hyperfine frequency, and the luminous efficacy (Bureau International des Poids et Mesures [BIPM], 2019). For example, exactly, and the kilogram is defined through .
Derived units combine base units: newton (), joule (), watt (), pascal (), hertz, coulomb (), volt (), ohm (), farad, henry, tesla, weber. Prefixes: nano , micro , milli , kilo , mega , giga .
Dimensional analysis
Every physical equation must be dimensionally homogeneous. Writing dimensions as allows checking equations and deriving relationships. Example: suppose the period of a pendulum depends on length , mass , and . Writing and equating dimensions gives , , : so , , , giving , . Hence (the constant needs a full analysis).
Errors and uncertainty
No measurement is exact. Random errors scatter readings around a true value and are reduced by repeating and averaging; systematic errors shift all readings the same way (for example, a zero error) and are found by calibration and good method. Accuracy is closeness to the true value; precision is the closeness of repeated readings.
Write results as . Rules for combining uncertainties:
Sum or difference: add the absolute uncertainties.
Product or quotient: add the fractional (or percentage) uncertainties.
Power : multiply the fractional uncertainty by .
Worked example. A pendulum has m and period s. Then . Fractional uncertainty: . So , and .
Graphs
Plot so as to linearize the relationship. For the pendulum, , so a graph of against is a straight line through the origin with gradient . Draw a best-fit line (not joining points), choose a large triangle to find the gradient, and estimate uncertainty from the maximum and minimum acceptable gradients.
Vectors in physics
A scalar has magnitude only (mass, energy, temperature); a vector also has direction (displacement, velocity, force, field). Vectors add by the triangle or parallelogram rule. A vector of magnitude at angle to the -axis has components , . The resultant of perpendicular vectors of magnitudes and is at an angle to the first. The vector methods of Chapter 5 apply directly.
Common mistakes
Mixing units (cm with m, or hours with seconds).
Reporting too many significant figures: the answer cannot be more precise than the data.
Adding percentage uncertainties for sums, or absolute uncertainties for products.
Joining plotted points instead of drawing a best-fit line.
Practice questions
Show that the equation is dimensionally consistent.
A rectangle measures cm by cm. Find the area with its uncertainty. [ cm²]
Find the resultant of forces 6 N and 8 N at right angles. [10 N]