Problem — Precision of three instruments

Problem

You measure the same length with: a tape measure ($\pm 1\ \text{mm}$), a calliper ($\pm 0{,}05\ \text{mm}$), a micrometer ($\pm 0{,}005\ \text{mm}$). Rank the instruments from most to least precise and comment on why the instrument should still be chosen according to the size of the quantity.

Solution

Ranking by precision (increasing absolute uncertainty $\Rightarrow$ decreasing precision): $\underbrace{\pm 0{,}005\ \text{mm}}_{\text{micrometro}} \ <\ \underbrace{\pm 0{,}05\ \text{mm}}_{\text{calibro}} \ <\ \underbrace{\pm 1\ \text{mm}}_{\text{metro}}$ so micrometer $>$ calliper $>$ tape measure. Between one instrument and the next, the uncertainty changes by roughly a factor of $10$–$20$.

Why the most precise is not always the best. Every instrument has a limited range (measurement interval): the micrometer reaches a few centimetres, the calliper a few tens of centimetres, only the tape measure covers metres. Measuring a room with a micrometer is impossible.

The instrument must be chosen according to the size of the quantity and the relative uncertainty required: a tape measure with $\pm 1\ \text{mm}$ on a $3\ \text{m}$ wall gives a relative error of just $\sim 0{,}03\%$, excellent; the same tape measure on a $5\ \text{mm}$ screw would give a relative error of $20\%$, unacceptable. Absolute precision must always be related to the size of what is being measured.

Links

Topics: Experimental method · Physical quantities and measurement Concepts: Significant figures and uncertainty Skills: Ranking reasoning