---
title: "Measurement in Science: Units and Instruments"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 37
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments
---

# Chapter 37 — Measurement in Science: Units and Instruments

A carpenter, a doctor and an astronomer walk into their workdays: one [measures](#def-g6-measurement-in-science-measuring) a shelf, one a fever, one the wanderings of Mars. All three trust the same quiet machinery — agreed [units](#def-g6-measurement-in-science-measuring), honest instruments, careful reading. This year physics turns properly quantitative, so we begin by polishing the tools every later chapter will lean on.

## 37.1 From seeming to number

Five years of experiments have taught us a rhythm: eyes and hands guess, instruments decide. Ruler against seeming-longer; balance against seeming-heavier; [thermometer](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) against feeling-cold. Each time, the instrument’s verdict is a *number with a [unit](#def-g6-measurement-in-science-measuring)* — and that pair travels, compares, and can be checked by anyone, which no feeling ever could.

**Definition 37.1 (Measuring a quantity).**

To *measure* a quantity — a length, a [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass), a [duration](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#def-g2-measuring-time-duration), a [temperature](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) — is to count how many times its agreed *unit* fits into it, using an instrument built for the job. The result is always written as a number *with* its unit: $28.5\,\mathrm{cm}$, $450\,\mathrm{g}$, $37.2\,{}^{\circ}\mathrm{C}$. A number alone is a rumor; number-plus-unit is a measurement.

**Example 37.2 (The price of a missing unit).**

“Add 3 of flour.” [Grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units)? Spoonfuls? Cups? The cake’s fate hangs on the missing word. Real disasters have followed missing or mixed-up [units](#def-g6-measurement-in-science-measuring) — pilots have run out of fuel mid-flight because [kilograms](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) were confused with pounds. Professionals write the [unit](#def-g6-measurement-in-science-measuring) every time, without exception; from this chapter on, so do we.

## 37.2 The world’s units

**Definition 37.3 (The scientist’s starter kit).**

The world has agreed on common [units](#def-g6-measurement-in-science-measuring), one family per quantity:

1. length: the *[metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units)* ( $\mathrm{m}$ ), with its offspring the kilometre ( $1\,\mathrm{km} = 1000\,\mathrm{m}$ ), [centimetre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) ( $100\,\mathrm{cm} = 1\,\mathrm{m}$ ) and millimetre ( $1000\,\mathrm{mm} = 1\,\mathrm{m}$ );
2. [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) : the *[kilogram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units)* ( $\mathrm{kg}$ ), with the [gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) ( $1\,\mathrm{kg} = 1000\,\mathrm{g}$ ) and milligram ( $1000\,\mathrm{mg} = 1\,\mathrm{g}$ );
3. time: the *second* ( $\mathrm{s}$ ), with the minute ( $1\,\mathrm{min} = 60\,\mathrm{s}$ ) and hour ( $1\,\mathrm{h} = 60\,\mathrm{min} = 3600\,\mathrm{s}$ );
4. [temperature](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) : the [degree Celsius](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) ( ${}^{\circ}\mathrm{C}$ );
5. volume of [liquids](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) : the litre ( $\mathrm{L}$ ), which we [measure](#def-g6-measurement-in-science-measuring) out in the next chapter.

**Example 37.4 (Converting like a scientist).**

Conversions are rides on the decimal staircase. $2.35\,\mathrm{m}$ in [centimetres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units): multiply by $100$ — the point slides two places right: $235\,\mathrm{cm}$. $4500\,\mathrm{g}$ in [kilograms](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units): divide by $1000$: $4.5\,\mathrm{kg}$. Time is the rebel of the family — its steps are $60$, not $10$: $2.5\,\mathrm{min}$ is not $250$ seconds but $2 \times 60 + 30 = 150\,\mathrm{s}$. Respect the rebel.

**Remark 37.5 (Who guards the metre?).**

For two centuries, the world’s [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) was a bar of precious metal in a guarded vault near a great European capital, and the [kilogram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) a gleaming cylinder beside it: every nation’s rulers and [weights](https://one-course.com/books/physics/1/en/chapter/24-weight-and-mass#def-g4-weight-and-mass-weight) were copies of copies of those. Today the guards have been retired: the [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) and its friends are now defined from unchangeable facts of nature itself — [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) and the inner clocks of atoms — so that any well-equipped laboratory on Earth (or beyond it!) can rebuild them exactly. [Units](#def-g6-measurement-in-science-measuring) are humankind’s longest-running peaceful agreement.

![The measurer’s toolkit: length, volume, mass, temperature and time, each with its own instrument and its own unit.](https://one-course.com/images/onecourse/chapters/physics-1/g6-measurement-in-science/img-96f75c66438c.jpg)

*The measurer’s toolkit: length, volume, [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass), [temperature](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) and time, each with its own instrument and its own [unit](#def-g6-measurement-in-science-measuring).*

## 37.3 Using instruments well

**Method 37.6 (Reading any graduated scale).**

Rulers, [thermometers](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature), measuring jugs, dials: one skill reads them all.

1. find the *graduation value* : pick two neighboring numbered marks, subtract, and divide by the number of little steps between them — on a ruler numbered every [centimetre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) with ten steps, each step is $1 \div 10 = 0.1\,\mathrm{cm}$ , one millimetre;
2. put your eye squarely in front of the mark (a slanted view shifts the reading);
3. count on from the last numbered mark: $7$ and $3$ little steps is $7.3$ ;
4. write the number *with its [unit](#def-g6-measurement-in-science-measuring)* .

![Reading a graduated scale: first learn what one little step is worth, then count on from the last numbered mark.](https://one-course.com/images/onecourse/chapters/physics-1/g6-measurement-in-science/fig-0a3d8e02d82d.svg)

*Reading a graduated scale: first learn what one little step is worth, then count on from the last numbered mark.*

**Method 37.7 (Estimate, measure, judge).**

Professionals never [measure](#def-g6-measurement-in-science-measuring) cold:

1. *estimate* first — “this table is about two [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) ”;
2. *choose* the instrument whose range and steps fit the job (a ruler for a stamp, a tape for a corridor, kitchen scale for flour, bathroom scale for people);
3. *[measure](#def-g6-measurement-in-science-measuring)* , by the reading rules;
4. *judge* : does the result agree roughly with the estimate? A table measured at $21.3\,\mathrm{m}$ did not grow — you misread by a factor of ten, and the estimate just caught it.

The estimate is your safety net: it catches slipped decimal points before they catch you.

**Example 37.8 (Choosing wisely).**

To weigh a letter for posting, the bathroom scale is useless — the letter’s $18\,\mathrm{g}$ vanish between its steps; the kitchen scale, stepping by the [gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units), answers perfectly. To time a $100$-metre sprint, the kitchen clock’s minute steps are hopeless; a stopwatch counts in hundredths of a second. Neither instrument is “better”: each has a territory, set by its range and its smallest step.

## 37.4 Honest numbers

**Remark 37.9 (Between the marks).**

Real needles love to land between marks. Honesty then has two levels: at least, name the two neighbors — “between $7.3$ and $7.4$, closer to $7.3$”; better, estimate one digit beyond the marks — “about $7.32\,\mathrm{cm}$”. What honesty forbids is inventing precision: reading $7.3214\,\mathrm{cm}$ off millimetre marks is fiction wearing a lab coat. An instrument’s marks set the limit of what it can promise.

**Example 37.10 (Measure twice, trust more).**

Three friends time the same toy car run with the same stopwatch: $4.6\,\mathrm{s}$, $4.8\,\mathrm{s}$, $4.7\,\mathrm{s}$. Nobody blundered — fingers simply start and stop a hair apart. Small scatter is the normal breath of real measurement. The wise habits: repeat the measurement, watch the readings huddle, and distrust any single lonely number — especially your own.

## 37.5 Exercises

**Exercise 37.1 ★.**

Complete each measurement with a sensible [unit](#def-g6-measurement-in-science-measuring): a door is $2$ … tall; a coin has [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) $8$ …; a school lesson lasts $55$ …; a fever is $39$ …

**Solution of Exercise 37.1.**

$2$ $\mathrm{m}$; $8$ $\mathrm{g}$; $55$ $\mathrm{min}$; $39$ ${}^{\circ}\mathrm{C}$.

**Exercise 37.2 ★.**

Convert: $3.2\,\mathrm{m}$ to $\mathrm{cm}$; $750\,\mathrm{g}$ to $\mathrm{kg}$; $4\,\mathrm{min}$ to seconds; $5600\,\mathrm{m}$ to $\mathrm{km}$.

**Solution of Exercise 37.2.**

$320\,\mathrm{cm}$; $0.75\,\mathrm{kg}$; $240\,\mathrm{s}$; $5.6\,\mathrm{km}$.

**Exercise 37.3 ★.**

Why is time the rebel of the [unit](#def-g6-measurement-in-science-measuring) family? Convert $3.5\,\mathrm{min}$ to seconds — and say what wrong answer the decimal staircase would have given.

**Solution of Exercise 37.3.**

Its steps go by $60$, not by $10$, so the decimal staircase lies: $3.5\,\mathrm{min} = 3 \times 60 + 30 = 210\,\mathrm{s}$, while the staircase would have wrongly given $350\,\mathrm{s}$.

**Exercise 37.4 ★.**

A jug is numbered every $100\,\mathrm{mL}$ with five little steps between numbers. What is one step worth? The juice stands two steps above the $300$ mark — how much juice?

**Solution of Exercise 37.4.**

Each step: $200 \div 5 = 40\,\mathrm{mL}$. Two steps above $300$: $300 + 80 = 380\,\mathrm{mL}$.

**Exercise 37.5 ★.**

Which instrument for each job, and why: weighing a pinch of yeast; timing a heartbeat count; the width of this page; the length of the schoolyard?

**Solution of Exercise 37.5.**

Yeast: the kitchen scale ([gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) steps — the bathroom scale would not notice a pinch). Heartbeats: a stopwatch (second steps and finer). Page width: a ruler (millimetre steps). Schoolyard: a long tape [measure](#def-g6-measurement-in-science-measuring) or surveyor’s wheel (metre-scale range).

**Exercise 37.6 ★.**

Nadia [measures](#def-g6-measurement-in-science-measuring) her pencil: $17.4\,\mathrm{m}$. Which safety net of [Method 37.7](#met-g6-measurement-in-science-estimate) should have caught this, and what is the likely true value?

**Solution of Exercise 37.6.**

The judging step: a pencil is estimated at under $20\,\mathrm{cm}$, so $17.4\,\mathrm{m}$ fails the sanity check at once. She read [centimetres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) and wrote [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units): the true value is $17.4\,\mathrm{cm}$.

**Exercise 37.7 ★.**

A [thermometer](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature)’s column stops between the $18$ and $19$ marks, nearer $19$. Give one honest way to report it — and one dishonest report to avoid.

**Solution of Exercise 37.7.**

Honest: “between $18$ and $19\,{}^{\circ}\mathrm{C}$, nearer $19$” (or “about $18.8\,{}^{\circ}\mathrm{C}$”). Dishonest: “$18.7643\,{}^{\circ}\mathrm{C}$” — precision the marks cannot promise.

**Exercise 37.8 ★.**

Three stopwatch readings of one run: $6.1\,\mathrm{s}$, $6.3\,\mathrm{s}$, $6.2\,\mathrm{s}$. Is someone at fault? What is the wise report?

**Solution of Exercise 37.8.**

No one is at fault: fingers start and stop a hair apart, and small scatter is measurement’s normal breath. Wise report: the readings huddle around $6.2\,\mathrm{s}$ — “about $6.2\,\mathrm{s}$”.

**Exercise 37.9 ★★.**

A recipe abroad asks for “a quarter pound of butter”. Your kitchen scale speaks [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units), and a full pound is about $450\,\mathrm{g}$. How many [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) do you weigh out? What does this errand teach about shared [units](#def-g6-measurement-in-science-measuring)?

**Solution of Exercise 37.9.**

A quarter of $450\,\mathrm{g}$: about $112\,\mathrm{g}$ ($113\,\mathrm{g}$ is fine). The errand shows why shared [units](#def-g6-measurement-in-science-measuring) matter: recipes written in one country’s [units](#def-g6-measurement-in-science-measuring) must be translated before another country’s instruments can obey them.

**Exercise 37.10 ★★.**

A ruler’s first [centimetre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) is worn away, so its edge starts at the $1$ mark. A stamp laid from the edge reads $3.4$ at its far end. How wide is the stamp really? What habit defeats worn rulers?

**Solution of Exercise 37.10.**

The stamp spans from the $1$ mark to the $3.4$ mark: $3.4 - 1 =
2.4\,\mathrm{cm}$. The habit: never trust an instrument’s edge — start from a chosen mark and subtract, and worn rulers lose their power.

**Exercise 37.11 ★★.**

The class [measures](#def-g6-measurement-in-science-measuring) the corridor with a $20\,\mathrm{m}$ tape twice: $18.25\,\mathrm{m}$ and $18.35\,\mathrm{m}$. The teacher writes “$18.3\,\mathrm{m}$”. Explain her choice, using [Remark 37.9](#rem-g6-measurement-in-science-between) and [Example 37.10](#ex-g6-measurement-in-science-repeat).

**Solution of Exercise 37.11.**

The two readings scatter by $0.10\,\mathrm{m}$, so the last digit of either is not to be trusted; they huddle around $18.3$. Writing “$18.3\,\mathrm{m}$” keeps every digit the tape can honestly promise and no more.

**Exercise 37.12 ★★★.**

An old tale: a king decreed his own foot the kingdom’s [unit](#def-g6-measurement-in-science-measuring) of length. List two practical disasters that follow (think of carpenters in distant towns, and of the king’s growing son as heir), and explain how the retired metal bar of [Remark 37.5](#rem-g6-measurement-in-science-guard) solved both — and what problem even the bar left unsolved that nature’s own constants finally fixed.

**Solution of Exercise 37.12.**

Disasters: no two towns’ “feet” agree, so beams cut in one town never fit houses in another; and when the son inherits, the [unit](#def-g6-measurement-in-science-measuring) itself changes length — every measurement in the kingdom expires with the king. The metal bar fixed both: one unchanging length, copied for all towns, outliving all kings. Its own flaw: a single object can be scratched, stolen or slowly altered, and the world had to trek to its vault to check copies — until [units](#def-g6-measurement-in-science-measuring) were rebuilt on nature’s constants, which cannot wear out and are on duty everywhere at once.

## 37.6 Problem: The Instrument Inspector

**Problem 37.1.**

Weekend problem — a day with the town’s instrument inspector; scales, jugs and clocks on trial; the apprentice’s notebook

Every year the town’s inspector tests the instruments of shops and schools. Today you are the apprentice, notebook in hand.

**Part I — The bakery’s scale.** The inspector carries certified blocks: $500\,\mathrm{g}$, $200\,\mathrm{g}$, $200\,\mathrm{g}$, $100\,\mathrm{g}$, $50\,\mathrm{g}$.

1. She checks that the empty scale’s needle stands exactly on zero. Why must this be the first test of all?
2. She loads the $500\,\mathrm{g}$ and one $200\,\mathrm{g}$ block. What should an honest scale read?
3. The needle shows $680\,\mathrm{g}$ instead. Does this scale flatter the baker or the customers? Explain who loses money on every sale.
4. Which combinations of her blocks let her test the reading $750\,\mathrm{g}$ ? Give one.
5. The bakery’s scale steps by $5\,\mathrm{g}$ . Could the inspector use it to check a $2\,\mathrm{g}$ letter stamp? Why not?

**Part II — The school’s measuring jugs.**

6. A jug is numbered every $200\,\mathrm{mL}$ , with four steps between numbers. What is each step worth?
7. Water stands one step above the $600\,\mathrm{mL}$ mark. The teacher’s log says “about $700\,\mathrm{mL}$ ”. Grade that report: honest, too vague, or fiction?
8. A pupil reads the jug from a low chair, eye well below the water line, and gets a different number than the inspector. Which reading rule was broken?
9. The inspector pours the jug’s declared $650\,\mathrm{mL}$ into a certified jug, which shows $640\,\mathrm{mL}$ . By how much is the school jug off, and is that worse or better than one of its own steps?

**Part III — Clocks, and the apprentice’s verdict.**

10. The station clock is checked against the radio time signal: it shows the signal’s noon at $12$ h $2$ min. Fast or slow, and by how much?
11. If nobody corrects it, how far ahead will it be after a week? (It gains the same each day.)
12. Three shops’ clocks show $12$ h $2$ min, $11$ h $58$ min, and $12$ h exactly at the signal’s noon. The inspector says: “Averaging would be silly here — fix each clock.” What repair does each need?
13. Evening. In your notebook, write the inspector’s three golden rules — one about zero, one about reading, one about [units](#def-g6-measurement-in-science-measuring) — each in a single sentence of your own.

**Solution of Problem 37.1.**

**1.** Every weighing adds the load to whatever the needle already showed; a non-zero start poisons all the day’s readings — the fair-start check must come first. **2.** $500 + 200 = 700\,\mathrm{g}$. **3.** It reads $20\,\mathrm{g}$ [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source). Pouring flour until the needle says $700\,\mathrm{g}$, the baker has really poured $720\,\mathrm{g}$: the scale makes the baker hand over more goods than the customer pays for — it flatters the customers, and the baker loses on every sale. **4.** $500 + 200 + 50 = 750\,\mathrm{g}$. **5.** No: a $2\,\mathrm{g}$ stamp vanishes below the scale’s $5\,\mathrm{g}$ steps — outside its territory. **6.** $200 \div 4 = 50\,\mathrm{mL}$ per step. **7.** Water at $600 + 50 = 650\,\mathrm{mL}$; “about $700\,\mathrm{mL}$” rounds away a full step — too vague for an instrument this good. **8.** Eye level with the mark: reading from below shifts the apparent water line — the squarely-in-front rule was broken. **9.** Off by $10\,\mathrm{mL}$ — better (smaller) than its own $50\,\mathrm{mL}$ step: acceptable for a school jug. **10.** Fast, by $2$ minutes. **11.** $2 \times 7 = 14$ minutes ahead after a week. **12.** First clock: set back $2$ min; second: forward $2$ min; third: nothing — it is right. (Averaging the three would “fix” nothing and falsify the one honest clock.) **13.** For example: *Zero*: check the empty instrument reads zero before trusting anything it says. *Reading*: learn the step’s worth and read squarely in front, between marks with honest modesty. *[Units](#def-g6-measurement-in-science-measuring)*: a number without its [unit](#def-g6-measurement-in-science-measuring) is not a measurement — write the [unit](#def-g6-measurement-in-science-measuring), always.