---
title: "Density: Why Things Float"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 44
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/44-density-why-things-float
---

# Chapter 44 — Density: Why Things Float

In your very first year, a tiny coin sank while a great log floated, and we promised that the “why” was a treasure of this course. You have earned it. The key was forged over two chapters — [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) for the stuff, [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) for the room — and today the two turn together in the lock.

## 44.1 The mass of one cubic centimetre

**Example 44.1 (Fair comparison at last).**

Comparing a whole log with a whole coin was never fair: different sizes, different shapes. The fair contest takes *equal [volumes](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume)*: one [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of each contestant. One $\mathrm{cm}^{3}$ of water has a [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of $1\,\mathrm{g}$. One $\mathrm{cm}^{3}$ of oak: about $0.6\,\mathrm{g}$. Of stone: about $2.5\,\mathrm{g}$. Of iron: $7.9\,\mathrm{g}$. Equal rooms, wildly unequal stuff — this is the number the floating game was waiting for.

**Definition 44.2 (Density).**

The *density* of a substance is the [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of one [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of it, in [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units). Water’s density is $1\,\mathrm{g}$ per $\mathrm{cm}^{3}$; oak’s about $0.6$; iron’s $7.9$. To find a density, no new instrument is needed: [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) a sample’s [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) and its [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume), then divide — share the [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) equally among the [cubic centimetres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume), and the answer is each [centimetre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units)’s share.

![The fair contest: same room for everyone, and each substance shows its signature mass.](https://one-course.com/images/onecourse/chapters/physics-1/g6-density-floating/fig-429398e8e58d.svg)

*The fair contest: same room for everyone, and each substance shows its signature [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass).*

**Proposition 44.3 (Density is a signature).**

The [density](#def-g6-density-floating-density) of a pure substance is the same for every piece of it, large or small: a splinter of oak and a whole [beam](https://one-course.com/books/physics/1/en/chapter/42-rectilinear-propagation-of-light#def-g6-light-propagation-ray) both carry about $0.6\,\mathrm{g}$ in each [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume). [Density](#def-g6-density-floating-density) therefore identifies substances, as [melting points](https://one-course.com/books/physics/1/en/chapter/21-melting-and-boiling-changes-of-state#def-g4-changes-of-state-melting-point) do: [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) a mystery object’s [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) and [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume), divide, and compare with the table — the substance often confesses on the spot.

**Method 44.4 (Measuring a density).**

For a pebble — or a crown:

1. [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) its [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) with the balance: say $75\,\mathrm{g}$ ;
2. [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) its [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) by the rise of water: say $30\,\mathrm{cm}^{3}$ ;
3. divide the [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) by the [cubic centimetres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) : $75 \div 30 =  2.5$ — each [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) carries $2.5\,\mathrm{g}$ ;
4. consult the table: [density](#def-g6-density-floating-density) $2.5$ — our pebble is ordinary stone.

## 44.2 The floating rule

**Proposition 44.5 (Why things float).**

An object [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) in water if its [density](#def-g6-density-floating-density) is *less* than water’s — less than $1\,\mathrm{g}$ per [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) — and [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) if its [density](#def-g6-density-floating-density) is greater. The water, so to speak, compares the intruder with itself, room for room: lighter-than-me-per-room rides on top; heavier-per-room goes down. The same rule governs any [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid), each with its own [density](#def-g6-density-floating-density) as the referee.

![Ice is only a little less dense than water — so an iceberg floats deep, hiding most of its bulk below the waterline.](https://one-course.com/images/onecourse/chapters/physics-1/g6-density-floating/img-028e91187a7c.jpg)

*Ice is only a little less dense than water — so an iceberg [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) deep, hiding most of its bulk below the waterline.*

**Example 44.6 (Six years of puzzles, settled).**

The log ($0.6$): [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words), however huge — every [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of it underbids water. The coin (bronze, near $9$): [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words), however small. The plastic duck, the cork, the pencil: under $1$, floaters all. Ice, at $0.9$ — water’s own swollen [solid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-solid) — [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) on its own [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) by a whisker, which is why ponds wear lids and icebergs sail. And the plasticine boat? Shaped as a bowl, the *boat-plus-air* parcel holds much [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) per room: the parcel’s [density](#def-g6-density-floating-density) drops below $1$, and down at the docks the same trick launches ten-thousand-tonne ships of steel.

![Density’s handwriting: liquids stack densest-down, and ice at 0.9 rides with only a tenth of itself above water.](https://one-course.com/images/onecourse/chapters/physics-1/g6-density-floating/fig-5c3abe027ffc.svg)

*[Density](#def-g6-density-floating-density)’s handwriting: [liquids](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) stack densest-down, and ice at $0.9$ rides with only a tenth of itself above water.*

**Example 44.7 (The liquid tower).**

Pour honey, then water tinted with ink, then oil gently down a tall glass’s side: three layers, honey ($1.4$) at the bottom, oil ($0.9$) on top, refusing to mix their ranks. Drop in guests: a grape ($1.05$) [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) through the oil, [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) through the water, and rests on the honey — denser than two floors, lighter than the third. The tower is the floating rule performed as theater.

**Remark 44.8 (Reading the iceberg).**

Ice’s $0.9$ against water’s $1.0$ does more than [float](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) the berg: it sets the waterline. Nine tenths of an iceberg’s bulk rides *below* the surface — the famous hidden [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) that sank proud ships. Exactly *why* the fraction below matches the [density](#def-g6-density-floating-density) ratio is a beautiful law of the ancient bathtub scientist himself, proved honestly in the High School volume; the table of densities already whispers the answer.

## 44.3 The crown, at last

**Example 44.9 (Archimedes closes the case).**

Now finish the story begun two chapters ago. The crown’s [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass): easy, the balance gives it. Its [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume): impossible for a lumpy masterpiece — until the bathtub’s lesson gave the rise method. [Mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) divided by [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume): the crown’s *[density](#def-g6-density-floating-density)* — and [density](#def-g6-density-floating-density) is a signature. Pure gold signs $19.3$; silver signs $10.5$; a secret blend signs something in between. Legend says the goldsmith’s crown confessed a [density](#def-g6-density-floating-density) well short of gold’s, and the fraud was exposed without a scratch on the crown. The weekend problem hands you his numbers.

## 44.4 Exercises

**Exercise 44.1 ★.**

What is the [density](#def-g6-density-floating-density) of a substance? Give water’s, and state the floating rule.

**Solution of Exercise 44.1.**

The [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of one [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of the substance, in [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units); water’s is $1\,\mathrm{g}$ per $\mathrm{cm}^{3}$. The rule: less dense than water [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words), denser [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words).

**Exercise 44.2 ★.**

Predict [float](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) or [sink](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) in water: oak ($0.6$); stone ($2.5$); cork ($0.2$); iron ($7.9$); ice ($0.9$).

**Solution of Exercise 44.2.**

Oak [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words); stone [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words); cork [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words); iron [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words); ice [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) — by a whisker.

**Exercise 44.3 ★.**

A block has [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) $120\,\mathrm{g}$ and [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) $80\,\mathrm{cm}^{3}$. Find its [density](#def-g6-density-floating-density). [Float](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) or [sink](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words)?

**Solution of Exercise 44.3.**

$120 \div 80 = 1.5$: [density](#def-g6-density-floating-density) $1.5$ — denser than water, it [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words).

**Exercise 44.4 ★.**

A mystery lump: [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) $158\,\mathrm{g}$, rise in the cylinder $20\,\mathrm{cm}^{3}$. Its [density](#def-g6-density-floating-density) — and, from the chapter’s numbers, its likely substance?

**Solution of Exercise 44.4.**

$158 \div 20 = 7.9$: the signature of iron.

**Exercise 44.5 ★.**

Why is the [density](#def-g6-density-floating-density) of a splinter of oak the same as of a whole oak [beam](https://one-course.com/books/physics/1/en/chapter/42-rectilinear-propagation-of-light#def-g6-light-propagation-ray)? What is this property good for?

**Solution of Exercise 44.5.**

[Density](#def-g6-density-floating-density) belongs to the substance, not the piece: every [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of oak carries the same share of [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units), splinter or [beam](https://one-course.com/books/physics/1/en/chapter/42-rectilinear-propagation-of-light#def-g6-light-propagation-ray). That makes [density](#def-g6-density-floating-density) an identity card for unknown substances.

**Exercise 44.6 ★.**

In the [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) tower, why does honey sit at the bottom and oil on top? Where does a grape of [density](#def-g6-density-floating-density) $1.05$ come to rest?

**Solution of Exercise 44.6.**

[Liquids](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) stack by [density](#def-g6-density-floating-density), densest at the bottom: honey $1.4$ below water $1.0$ below oil $0.9$. The grape ($1.05$) [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) through oil and water and rests on the honey floor — denser than both upper floors, less dense than honey.

**Exercise 44.7 ★.**

Ice [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) on water — why is this odd for a [solid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-solid), and which old proposition about freezing water explains the $0.9$?

**Solution of Exercise 44.7.**

Most [solids](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-solid) are denser than their own [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) and [sink](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) in it. Ice [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) because water swells on freezing — the same stuff in a tenth more room gives [density](#def-g6-density-floating-density) $0.9 < 1$.

**Exercise 44.8 ★.**

The great log and the little coin of your first year: give the final, complete answer, with numbers from the chapter.

**Solution of Exercise 44.8.**

Wood signs about $0.6$: every [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of the log underbids water’s $1\,\mathrm{g}$, so the log [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) however huge. Bronze signs near $9$: every [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of the coin outbids water, so it [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) however small. Size never mattered; the per-room [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) always did.

**Exercise 44.9 ★★.**

One [litre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of a certain [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) has a [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of $800\,\mathrm{g}$. What is its [density](#def-g6-density-floating-density) in [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) per [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume)? Will an ice cube ($0.9$) [float](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) or [sink](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) in it?

**Solution of Exercise 44.9.**

$1\,\mathrm{L} = 1000\,\mathrm{cm}^{3}$, so $800 \div 1000 = 0.8$ [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) per [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume). Ice at $0.9$ is *denser* than this [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid): the ice cube [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) in it.

**Exercise 44.10 ★★.**

Steel signs $7.9$, yet steel ships [float](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words). Resolve the paradox with the parcel idea of [Example 44.6](#ex-g6-density-floating-settled) — what belongs to the ship-parcel besides steel, and what must its overall [density](#def-g6-density-floating-density) be? When does a holed ship stop satisfying the rule?

**Solution of Exercise 44.10.**

The floating parcel is the whole hull: steel shell *plus* the great [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) it encloses. Averaged over all that room, the parcel’s [density](#def-g6-density-floating-density) falls below $1$, and the rule says [float](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words). A hole lets water replace the [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air): the parcel’s [density](#def-g6-density-floating-density) climbs toward steel’s own, passes $1$ — and the rule, unmoved, says [sink](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words).

**Exercise 44.11 ★★.**

Sea water is a little denser than fresh water — about $1.03$. Explain why swimmers [float](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) noticeably better in the sea, and predict what happens to a ship’s waterline as it sails from the salty sea into a fresh river.

**Solution of Exercise 44.11.**

The referee changed: against sea water’s $1.03$, a swimmer’s near-$1$ body is comfortably on the floating side, so the sea carries you higher. Entering fresh water ($1.0$), the ship loses that margin and settles deeper: its waterline rises up the hull.

**Exercise 44.12 ★★★.**

A hollow metal sphere [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) exactly half-submerged in water. Reason out the [density](#def-g6-density-floating-density) of the *whole sphere-parcel* (metal shell plus the [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) inside). Then argue which way it moves if water slowly leaks inside, and at what moment it begins to [sink](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) — rule and parcel, working together.

**Solution of Exercise 44.12.**

Half-submerged, the sphere sits exactly as deep in the water as a parcel half water’s [density](#def-g6-density-floating-density) should: shell plus [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) average out to about $0.5$ [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) per [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) — the hidden [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) doing most of the lightening. As water leaks in it replaces [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air), the parcel’s average [density](#def-g6-density-floating-density) climbs, and the sphere settles deeper and deeper; the moment the average passes water’s $1.0$, the floating rule [switches](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) sides and the sphere goes down.

## 44.5 Problem: The King’s Crown

**Problem 44.1.**

Weekend problem — the case of the king’s crown, reopened with your instruments; the goldsmith’s trial, solved by division

Reopen antiquity’s most famous fraud case with modern numbers. Signatures: gold $19.3$, silver $10.5$ ([grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) per [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume)).

**Part I — The evidence.** The king gave the goldsmith $1930\,\mathrm{g}$ of pure gold. The finished crown, on the balance: $1930\,\mathrm{g}$ exactly.

1. The [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) matches perfectly. Why does this alone prove nothing about the fraud? (What could the goldsmith have done and still matched it?)
2. What [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) should $1930\,\mathrm{g}$ of *pure gold* occupy? (Share the [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) : how many [cubic centimetres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) at $19.3\,\mathrm{g}$ each?)
3. Archimedes lowers the crown into a brim-full vessel and collects the overflow: $130\,\mathrm{cm}^{3}$ . What is the crown’s true [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) ?
4. Compare the two [volumes](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) . What does the comparison already shout?

**Part II — The signature.**

5. Compute the crown’s [density](#def-g6-density-floating-density) from its [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) and its measured [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) (one decimal is enough).
6. Set the three signatures side by side: gold, the crown, silver. Where does the crown fall?
7. Explain, in one sentence to the king, why the crown cannot be pure gold.
8. The goldsmith protests: “Perhaps the balance erred!” Grant him a generous error of $30\,\mathrm{g}$ either way. Does any [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) between $1900\,\mathrm{g}$ and $1960\,\mathrm{g}$ rescue a pure-gold crown of [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) $130\,\mathrm{cm}^{3}$ ? (Check the [density](#def-g6-density-floating-density) range.)

**Part III — How much was stolen?** Suppose the goldsmith kept some gold and replaced it, [gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) for [gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units), with silver.

9. Why does replacing gold by silver, [gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) for [gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) , keep the [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) right but *swell* the [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) ?
10. Test a guess: a crown of $965\,\mathrm{g}$ gold $+$ $965\,\mathrm{g}$ silver. [Volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of the gold share? Of the silver share? (One decimal.)
11. Total the guessed crown’s [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) , and judge the guess against the measured $130\,\mathrm{cm}^{3}$ : too much silver, or too little?
12. The court needs a verdict sentence: state what was proved beyond doubt (pure gold or not), what the method was, and why the crown never needed to be harmed — Archimedes’ true triumph.

**Solution of Problem 44.1.**

**1.** Nothing: swapping gold for an equal *[mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass)* of silver keeps the balance happy — [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) alone cannot see the substitution. **2.** $1930 \div 19.3 = 100\,\mathrm{cm}^{3}$. **3.** $130\,\mathrm{cm}^{3}$ — the overflow is the crown’s [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume), by displacement. **4.** The crown takes $30\,\mathrm{cm}^{3}$ more room than pure gold of its [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) should: something bulkier than gold hides inside. **5.** $1930 \div 130 \approx 14.8$. **6.** Gold $19.3$ — crown $14.8$ — silver $10.5$: the crown falls between the two signatures. **7.** “Sire, every [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of pure gold carries $19.3\,\mathrm{g}$; your crown carries only about $15$ — pure gold it is not.” **8.** The range gives densities $1900 \div 130 \approx
14.6$ up to $1960 \div 130 \approx 15.1$ — nowhere near $19.3$: no plausible balance error rescues the goldsmith. **9.** Silver is less dense: each stolen [gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) of gold gave back a [gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) of silver that needs *more room* — gram-for-gram substitution preserves [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) and inflates [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume). **10.** Gold share: $965 \div 19.3 = 50.0\,\mathrm{cm}^{3}$; silver share: $965 \div 10.5 \approx 91.9\,\mathrm{cm}^{3}$. **11.** Total $\approx 142\,\mathrm{cm}^{3}$ — more than the measured $130$: the half-and-half guess used too much silver; the true theft was smaller (the measured [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) sits between $100$ and $142$). **12.** “Proved: the crown is not pure gold. Method: its [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) shared over its water-measured [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) gives a [density](#def-g6-density-floating-density) far below gold’s unchangeable signature. And the crown never needed so much as a scratch: the water read its secret from the outside — that is the triumph.”