Dimensional Analysis

forces

interactions between objects (either pushes or pulls)

model

simplified mental image of an object or process

law

mathematical expression of a pattern in nature supported by evidence and repetition

theory

consistent explanation of a collection of natural phenomena expressed in terms of one or more laws

scientific method

observation (reproducible by multiple observers, either qualitative or quantitative) → hypothesis → predictions → experiment (ultimate test of validity, and if it fails, then go back to the hypothesis)

value

ratio between the observed quantity and a known constant quantity of the same kind (standard)

unit

specific standard (object) that the measured quantity is compared to, which implies that measurable and standard are of the same type to be comparable

uncertainty

gives lower and upper limit for the value, the true value being unknown

• needs to be both careful (clear of predictable mistakes) and reproducible (repeatable by others)

SI units

• time (t, seconds)

• length (l, meters)

• mass (m, kilograms)

• temperature (T, Kelvin)

• electrical current (A, Amperes)

• amount of substance (mol, mole)

• luminous intensity (cd, candela)

derived SI units

• area = m²

• volume = m³

• density = kg/m²

• speed = m/s

dimensional analysis

all sides of the equation must be dimensionally homogeneous

• addition, subtraction, and comparisons (< > =) are only possible among quantities with the same dimensions

• mathematical functions sin, cos, tan, exp, ln and angles are all dimensionless

significant figures

figures that can be trusted, given the measurement uncertainty

• the number of significant figures must be consistent with the actual uncertainty on the value (cannot imply a smaller uncertainty than the actual one)

dimension

type of physical observable

derived dimensions

• density: mass units per volume = m/V = kg/m³

• speed: distance per units time = m/s

• frequency: events per units time = number/s = 1/s = Hz

dimensionless quantities

• numerical factors

• ratios of homogenous dimensions

• angle by their definition are the ratio between the arc and radius, which is expressed in radians which is dimensionless