Measurement, Units, and Uncertainty

Objective: Use base and derived units and factor-of-ten prefixes.

The International System of Units (SI) provides shared measurement conventions. Length is measured in meters (m), mass in kilograms (kg), and time in seconds (s); speed has derived unit m/s, acceleration m/s², and force newtons (N = kg·m/s²). Prefixes scale a unit: kilo- means 10³ and centi- means 10⁻². Dimensional analysis multiplies by conversion factors equal to one, preserving the measured quantity. For instance, 2.5 km = 2.5 × 1000 m = 2500 m. A number without a unit can be ambiguous. A converted value does not gain accuracy merely by adding digits.

Objective: Distinguish closeness to a reference from repeatability.

Accuracy describes closeness to a suitable reference value; precision describes how closely repeated measurements agree. Instrument resolution is the smallest change the tool can distinguish, while calibration and procedure can introduce systematic bias. A series of tightly grouped values can be precise but inaccurate if a scale is miscalibrated. Random variation can reduce precision, and repeated trials alone do not remove a constant bias. Report digits consistent with instrument resolution and method, and label uncertainty rather than implying an exact physical value from a finite measurement.

Objective: Calculate density and give appropriate context to estimates.

Density is mass per volume, ρ = m/V, with common units g/cm³ or kg/m³. For a uniform sample, its average density can help identify or compare materials, but temperature, composition, trapped air, and measurement error matter. If a 30 g sample occupies 10 cm³, density is 3 g/cm³. Volume of a regular rectangular solid is length × width × height; irregular volume may be estimated from fluid displacement when appropriate. Dividing measured quantities does not make the result exact. Qualitative uncertainty can be assessed by seeing how plausible changes in measured mass or volume affect ρ; a small volume error may matter greatly for a small sample.