Problem — The digital scale

Problem

A kitchen scale shows "$157\;\text{g}$" for whatever object you place on it, within its range. How can you estimate the sensitivity of the display without dismantling the scale, using only objects of known mass (coins, packets of pasta)? Discuss an operational procedure and the associated significant figures.

Solution

Step 1 — What we want to measure. The sensitivity (or resolution) of the display is the smallest change in mass that produces a change in the digit shown. If the scale shows whole numbers of grams, the nominal resolution is $1\;\text{g}$, but this must be checked experimentally: it might only respond to steps of $2\;\text{g}$ or $5\;\text{g}$.

Step 2 — Objects of known mass. Small, known masses are needed. Euro coins are excellent because they are standardised: for instance, the $1$-cent coin weighs $2{,}30\;\text{g}$, the $2$-cent coin $3{,}06\;\text{g}$, the $5$-cent coin $3{,}92\;\text{g}$. A sealed packet of pasta shows the nominal mass ($500\;\text{g}$).

Step 3 — Operational procedure.

1. Place a base object (a tare) on the pan and note the reading $L_0$.
2. Add coins of increasing known mass one at a time, recording after each addition the reading and the actual total mass added.
3. Identify the smallest added mass that changes the digit on the display: that change is of the order of the sensitivity.
4. Repeat starting from different tares, to check that the reading step is constant across the range (linearity).

Step 4 — Quantitative estimate. If adding a $2{,}30\;\text{g}$ coin does not change the reading, but adding two ($4{,}60\;\text{g}$) makes the reading go up by one unit, the sensitivity lies between $2{,}3\;\text{g}$ and $4{,}6\;\text{g}$: we conclude the display “steps” by about $\ev{\sim 5\;\text{g}}$. Narrowing down with intermediate-value coins refines the limit.

Step 5 — Significant figures. If the resolution is $\sim 5\;\text{g}$, it makes no sense to attribute more than three significant figures to the measurement around $150\;\text{g}$: writing "$157\;\text{g}$" falsely suggests precision to the gram. Consistently the result should be expressed as $(157 \pm 5)\;\text{g}$, i.e. with an uncertainty of about half the reading step. The display’s sensitivity thus fixes the number of physically justified significant figures.

Linked notes

Topics: Grandezze fisiche e misura Concepts: Cifre significative e incertezza Skills: Uso delle cifre significative