Units & Measurement

Base vs Derived Quantities

Base quantities are the fundamental building blocks that can't be broken down further (e.g. length, mass, time, electric current, amount of substance). Derived quantities are built by combining base quantities (e.g. speed = length/time).

Order of Magnitude

The order of magnitude of a number is its nearest power of 10. Worked example: 30 is closest to $10^1$ (=10), so its order of magnitude is $10^1$.

SI Prefixes

Common prefixes represent powers of 10 — e.g. kilo $= \times 10^3$. Knowing these lets you quickly convert between units like g/cm³ and kg/m³.

Worked example: a density of 93.2 g/cm³ converts to SI units (kg/m³) as $9.32\times10^4 \text{ kg/m}^3$ — this involves both a unit change (g→kg divides by 1000) and a volume change (cm³→m³ divides by $10^{-6}$, i.e. multiplies by $10^6$), which combine to an overall $\times 100$ ($10^4$) factor here.

Dimensional Analysis

Every physical quantity can be broken into the dimensions Mass (M), Length (L), and Time (T). This is useful for checking or deriving a formula's units.

Worked example: Pressure = Force/Area. Force has dimensions $MLT^{-2}$ (from F=ma), and Area has dimensions $L^2$. So Pressure $= MLT^{-2}/L^2 = ML^{-1}T^{-2}$.

Converting Between Unit Systems

Worked example: g = 9.8 m/s² converts to imperial units as 32.2 ft/s² (using 1 m ≈ 3.28 ft).

For very large distances, the same conversion principle applies at a bigger scale: the Milky Way and Andromeda galaxy are 2.5 million light-years apart, where 1 light-year = 9.5×10¹⁵ m. Converting: 2.5×10⁶ × 9.5×10¹⁵ m = 2.375×10²² m, then m→km divides by 1000, giving 2.375×10¹⁹ km.

Why This Matters for Your Exams

This topic is mostly about careful, methodical unit tracking — write out every conversion factor explicitly rather than trying to do it in your head, especially when a question chains multiple conversions together (like the galaxy-distance example).