---
title: "Volume, Mass, Density"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 46
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/46-volume-mass-density
---

# Chapter 46 — Volume, Mass, Density

Last year you unmasked a fraudulent goldsmith by dividing [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) by [cubic centimetres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume). This year, your mathematics course has handed you a new power: letters that stand for numbers. [Density](#def-g7-volume-mass-density-formula) is the perfect place to spend it — one short formula that packs three recipes, a detective’s toolkit, and the whole floating rule into five symbols.

## 46.1 The formula

**Definition 46.1 (Density, by formula).**

The *density* $\rho$ (the Greek letter *rho*) of a substance is its [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) per [unit](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) of [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume): for a sample of [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) $m$ and [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) $V$,

$$
\rho = \frac{m}{V}.
$$

With $m$ in [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) and $V$ in [cubic centimetres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume), $\rho$ comes out 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)* ($\mathrm{g}/\mathrm{cm}^{3}$) — last year’s “[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)”, now wearing its uniform. Water’s density: $\rho = 1.0\,\mathrm{g}/\mathrm{cm}^{3}$. And since a [litre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) is $1000$ [cubic centimetres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume), the same number doubles as [kilograms](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) per [litre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume): water is $1.0\,\mathrm{kg}$ per [litre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume).

**Example 46.2 (Reading the letters).**

A formula is a sentence in shorthand. $\rho = m/V$ reads: “to find the [density](#def-g7-volume-mass-density-formula), divide the sample’s [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) by its [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume).” The letters are placeholders — for *this* pebble, $m$ becomes $75\,\mathrm{g}$ and $V$ becomes $30\,\mathrm{cm}^{3}$:

$$
\rho = \frac{m}{V} = \frac{75}{30} = 2.5\,\mathrm{g}/\mathrm{cm}^{3}.
$$

Same division as last year — but the formula remembers the recipe for every pebble to come.

**Proposition 46.3 (Three recipes in one formula).**

Because [density](#def-g7-volume-mass-density-formula) ties [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) in proportion, one formula serves three needs:

1. *find a [density](#def-g7-volume-mass-density-formula)* : $\rho = m/V$ — divide [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) by [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) ;
2. *find a [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass)* : $m = \rho \times V$ — each [cubic centimetre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) carries $\rho$ [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) , and there are $V$ of them;
3. *find a [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume)* : $V = m/\rho$ — share the [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) out in helpings of $\rho$ [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) ; the number of helpings is the number of [cubic centimetres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) .

**Example 46.4 (The recipes at work).**

*[Mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) from [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume)*: what does $250\,\mathrm{cm}^{3}$ of iron ($\rho = 7.9$) weigh? $m = 7.9 \times 250 = 1975\,\mathrm{g}$ — nearly two [kilograms](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) in a coffee-mug’s bulk. *[Volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) from [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass)*: what room does $540\,\mathrm{g}$ of oak ($\rho = 0.6$) take? $V = 540 \div 0.6 = 900\,\mathrm{cm}^{3}$. *Sanity partner*: iron answers should be heavy-and-small, wood answers light-and-large — the [density](#def-g7-volume-mass-density-formula) table is the judging step’s best friend.

![The density table as a skyline: from cork’s feather-light 0.2 to gold’s monumental 19.3. Water’s 1.0 is the floating frontier.](https://one-course.com/images/onecourse/chapters/physics-1/g7-volume-mass-density/fig-c1df13695c1f.svg)

*The [density](#def-g7-volume-mass-density-formula) table as a skyline: from cork’s [feather-light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) $0.2$ to gold’s monumental $19.3$. Water’s $1.0$ is the floating frontier.*

## 46.2 Measuring densities

**Method 46.5 (Density of a liquid).**

[Liquids](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) will not sit on a balance pan alone — so borrow the tare trick:

1. place an empty measuring cylinder on the balance and tare it to zero;
2. pour in a convenient [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) — say $V = 80\,\mathrm{cm}^{3}$ of the mystery [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) , reading the [meniscus](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#met-g6-mass-and-volume-cylinder) properly;
3. the balance now shows the [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) ’s [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) alone: say $m = 68\,\mathrm{g}$ ;
4. divide: $\rho = 68 \div 80 = 0.85\,\mathrm{g}/\mathrm{cm}^{3}$ — lighter than water: likely an oil.

**Example 46.6 (Density of a solid, start to finish).**

A medal claims to be silver ($\rho = 10.5$). Balance: $m = 84\,\mathrm{g}$. Cylinder: the water climbs from $60.0\,\mathrm{cm}^{3}$ to $68.0\,\mathrm{cm}^{3}$, so $V = 8.0\,\mathrm{cm}^{3}$. Then $\rho = 84 \div 8.0 = 10.5\,\mathrm{g}/\mathrm{cm}^{3}$ — the signature matches: silver it may well be. (A cheaper metal dressed in silver plating would have betrayed itself here — unless chosen with cunning, as the weekend problem will show.)

**Remark 46.7 (Units in formulas).**

A formula is honest only if its [units](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) agree. With $m$ in [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) and $V$ in $\mathrm{cm}^{3}$, $\rho$ speaks $\mathrm{g}/\mathrm{cm}^{3}$; feed it [kilograms](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) and [litres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) and it answers $\mathrm{kg}/\mathrm{L}$ — happily, the same number for any substance. But mix [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) with [litres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) and the formula, uncomplaining, delivers nonsense. Rule of the professionals: before computing, parade the [units](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring); after computing, write them into the answer.

## 46.3 Density thinking

**Example 46.8 (Alloys and in-betweens).**

Blend two substances and the blend’s [density](#def-g7-volume-mass-density-formula) lands *between* their signatures — nearer the more generous ingredient. Bronze (copper $9.0$ with a little tin) signs near $8.8$; the crown’s gold-and-silver blend signed between $10.5$ and $19.3$; sea water, salt dissolved in water, edges up to about $1.03$. Between-ness is itself a clue: a reading of $12$ from a “pure gold” bar is a confession of company.

**Example 46.9 (Density decides the floating world).**

The floating rule, now in uniform: an object [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) in a [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) when $\rho_{\text{object}} < \rho_{\text{liquid}}$. Oak ($0.6$) on water ($1.0$): [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words). Ice ($0.9$) in oil ($0.85$): [sinks](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) — check the table, not your instincts. A swimmer ($\approx 1.0$) in the famous ultra-salty lakes ($\approx 1.2$): [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) like a cork, newspaper in hand. One inequality, the whole harbor.

**Remark 46.10 (Mind the crowd, not the parcel).**

$\rho = m/V$ describes *substances*; ships and swollen life-jackets are *parcels* — substance plus trapped [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) — and it is the parcel’s overall $m/V$ that faces the floating rule. The formula handles both, if you feed it the right $m$ and $V$: the steel’s own [density](#def-g7-volume-mass-density-formula) for the substance, the whole hull’s [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) over the whole hull’s [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) for the parcel. Most floating “paradoxes” are just the two bookkeepings confused.

## 46.4 Exercises

**Exercise 46.1 ★.**

Write the [density](#def-g7-volume-mass-density-formula) formula, name each letter, and give the [unit](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) that goes with [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) and [cubic centimetres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume).

**Solution of Exercise 46.1.**

$\rho = m/V$: $\rho$ the [density](#def-g7-volume-mass-density-formula), $m$ the sample’s [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass), $V$ its [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume). With [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) and [cubic centimetres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume), $\rho$ comes out in $\mathrm{g}/\mathrm{cm}^{3}$.

**Exercise 46.2 ★.**

Compute the [density](#def-g7-volume-mass-density-formula): $m = 270\,\mathrm{g}$, $V = 100\,\mathrm{cm}^{3}$. Which metal of the skyline chart is this?

**Solution of Exercise 46.2.**

$\rho = 270 \div 100 = 2.7\,\mathrm{g}/\mathrm{cm}^{3}$: aluminium.

**Exercise 46.3 ★.**

Use the right recipe: the [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of $40\,\mathrm{cm}^{3}$ of copper ($\rho = 9.0$); the [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of $1930\,\mathrm{g}$ of gold ($\rho = 19.3$).

**Solution of Exercise 46.3.**

$m = 9.0 \times 40 = 360\,\mathrm{g}$ of copper. $V = 1930 \div 19.3 = 100\,\mathrm{cm}^{3}$ of gold — the crown case’s own number.

**Exercise 46.4 ★.**

In [Method 46.5](#met-g7-volume-mass-density-liquid), why is the cylinder tared first? What two readings then feed the formula?

**Solution of Exercise 46.4.**

Taring removes the cylinder’s own [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) so the balance reports the [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) alone — the honest-zero rule. The formula is then fed the balance’s [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) $m$ and the [meniscus](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#met-g6-mass-and-volume-cylinder) reading $V$.

**Exercise 46.5 ★.**

A [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid): $V = 50\,\mathrm{cm}^{3}$, $m = 70\,\mathrm{g}$. Its [density](#def-g7-volume-mass-density-formula)? Does 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 46.5.**

$\rho = 70 \div 50 = 1.4\,\mathrm{g}/\mathrm{cm}^{3}$ — honey-like. Ice at $0.9$ is far less dense: it [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) high in it.

**Exercise 46.6 ★.**

Why is water’s [density](#def-g7-volume-mass-density-formula) the same number in $\mathrm{g}/\mathrm{cm}^{3}$ and in [kilograms](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) per [litre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume)? What warning does [Remark 46.7](#rem-g7-volume-mass-density-units) attach to mixing [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) with [litres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume)?

**Solution of Exercise 46.6.**

Both [units](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) scale together: a [litre](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) is $1000$ $\mathrm{cm}^{3}$ and a [kilogram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) is $1000$ [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units), so the two thousands cancel — one number serves both. Mixing [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) with [litres](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) skips that cancellation and delivers a number a thousandfold wrong.

**Exercise 46.7 ★.**

A bracelet marked “pure gold” has $m = 58\,\mathrm{g}$ and $V = 5.0\,\mathrm{cm}^{3}$. Compute its [density](#def-g7-volume-mass-density-formula) and give your verdict, with the table as witness.

**Solution of Exercise 46.7.**

$\rho = 58 \div 5.0 = 11.6\,\mathrm{g}/\mathrm{cm}^{3}$ — near lead’s $11.3$, nowhere near gold’s $19.3$. Verdict: not pure gold; likely a lead-hearted impostor in gold clothing.

**Exercise 46.8 ★★.**

One cubic [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) is a cube of $100 \times 100 \times 100$ [centimetres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units). How many $\mathrm{cm}^{3}$ is that — and what is the [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of a cubic [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) of water, in [kilograms](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units)? (The answer explains why waterbeds worry landlords.)

**Solution of Exercise 46.8.**

$100 \times 100 \times 100 = 1000000$ $\mathrm{cm}^{3}$ — a million. At $1\,\mathrm{g}$ each, that is $1\,000\,000\,\mathrm{g}$ $=$ $1000\,\mathrm{kg}$: a full tonne per cubic [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) of water — and why a large waterbed is furniture for the ground floor.

**Exercise 46.9 ★★.**

A “silver” trophy: $m = 420\,\mathrm{g}$, water rise from $200\,\mathrm{cm}^{3}$ to $260\,\mathrm{cm}^{3}$. [Density](#def-g7-volume-mass-density-formula)? Between which two table substances does it fall — and what does between-ness whisper?

**Solution of Exercise 46.9.**

$V = 260 - 200 = 60\,\mathrm{cm}^{3}$; $\rho = 420 \div 60 = 7.0\,\mathrm{g}/\mathrm{cm}^{3}$ — between aluminium ($2.7$) and iron ($7.9$), closest below iron. Between-ness whispers: a blend or a plated impostor, certainly not silver’s $10.5$.

**Exercise 46.10 ★★.**

An empty bottle weighs $380\,\mathrm{g}$; filled to its $75\,\mathrm{cL}$ mark with a mystery [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) it weighs $1010\,\mathrm{g}$. Find the [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid)’s [density](#def-g7-volume-mass-density-formula) in $\mathrm{g}/\mathrm{cm}^{3}$ and propose its identity.

**Solution of Exercise 46.10.**

[Liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass): $1010 - 380 = 630\,\mathrm{g}$; [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume): $75\,\mathrm{cL} = 750\,\mathrm{cm}^{3}$; so $\rho = 630 \div 750 =
0.84\,\mathrm{g}/\mathrm{cm}^{3}$ — an oil.

**Exercise 46.11 ★★.**

A hollow aluminium buoy has a total [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of $4000\,\mathrm{cm}^{3}$ and a total [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of $1200\,\mathrm{g}$. Compute the *parcel’s* [density](#def-g7-volume-mass-density-formula) and 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) — then explain, with [Remark 46.10](#rem-g7-volume-mass-density-parcel), why aluminium’s own $2.7$ was the wrong number to consult.

**Solution of Exercise 46.11.**

Parcel [density](#def-g7-volume-mass-density-formula): $1200 \div 4000 = 0.3\,\mathrm{g}/\mathrm{cm}^{3}$ — far below water’s $1.0$: the buoy [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) high. Aluminium’s $2.7$ describes only the metal skin; the parcel is mostly enclosed [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air), and it is the parcel — total [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) over total [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) — that faces the floating rule.

**Exercise 46.12 ★★★.**

Design a complete protocol to decide whether a chain is pure copper: list every measurement, the recipe applied, the expected number for purity, and *two* honest reasons your verdict could still be wrong (think of hollow links, and of cunning blends whose [density](#def-g7-volume-mass-density-formula) lands near copper’s). What extra test would tighten the case?

**Solution of Exercise 46.12.**

Protocol: weigh the chain ($m$); [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 method, fully submerged, no bubbles ($V$); compute $\rho = m/V$; purity expects about $9.0\,\mathrm{g}/\mathrm{cm}^{3}$. Honest doubts: hollow links trap [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) and swell $V$, faking a low [density](#def-g7-volume-mass-density-formula); and a cunning blend (or a plated core) can land near $9.0$ while containing no pure copper at all. Tightening test: repeat the [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) measurement after flooding the links (shake out bubbles), and add an independent signature — for instance the [magnet](https://one-course.com/books/physics/1/en/chapter/6-magnets#def-g1-magnets-magnet) (a steel core betrays itself instantly) or, in a workshop, a measured [melting](https://one-course.com/books/physics/1/en/chapter/39-water-in-all-its-states#def-g6-water-states-changes) behavior.

## 46.5 Problem: The Scrapyard Detective

**Problem 46.1.**

Weekend problem — an afternoon with the scrapyard’s metal detective; four mystery lots, one impostor ingot; the limits of the density test

The scrapyard buys metal by what it *is*, not what it looks like, and the yard’s detective works with a balance, a big graduated vessel, and the skyline table. You are the apprentice. (Table extract: aluminium $2.7$; iron $7.9$; copper $9.0$; lead $11.3$; gold $19.3$; tungsten $19.3$.)

**Part I — Four lots.**

1. Lot A, a gray ingot: $m = 5400\,\mathrm{g}$ , displacement $V = 2000\,\mathrm{cm}^{3}$ . [Density](#def-g7-volume-mass-density-formula) and identity?
2. Lot B, a coil of wire: $m = 1800\,\mathrm{g}$ , $V = 200\,\mathrm{cm}^{3}$ . [Density](#def-g7-volume-mass-density-formula) and identity?
3. Lot C, a dull heavy plate: $m = 4520\,\mathrm{g}$ , $V = 400\,\mathrm{cm}^{3}$ . [Density](#def-g7-volume-mass-density-formula) and identity — and why must lot C be handled with gloves and respect?
4. Lot D, a sack of mixed pale scrap: $m = 8100\,\mathrm{g}$ , $V = 1500\,\mathrm{cm}^{3}$ . Show that lot D’s [density](#def-g7-volume-mass-density-formula) lands between two table metals, and say what the sack most likely contains.

**Part II — Prices and predictions.** The yard pays by [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) but plans transport by [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume).

5. A buyer wants $540\,\mathrm{g}$ of aluminium cut from lot A. What [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume) of ingot is that?
6. The copper coil of lot B is to be melted into cubes of $25\,\mathrm{cm}^{3}$ each. What is the [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of one cube, and how many full cubes does the coil yield?
7. A crate can carry at most $20\,\mathrm{kg}$ . How many of those copper cubes may it legally hold?
8. The truck’s tank-well holds $3000\,\mathrm{cm}^{3}$ more of iron scrap. What extra [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) , in [kilograms](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) , is the detective allowed to load?

**Part III — The impostor.** A seller arrives with a gleaming “gold” ingot: $m = 3860\,\mathrm{g}$, displacement $V = 200\,\mathrm{cm}^{3}$.

9. Compute the ingot’s [density](#def-g7-volume-mass-density-formula) . Does it match gold’s signature?
10. The detective, unmoved, consults the table’s last line and sighs. Which cheaper metal wears exactly gold’s [density](#def-g7-volume-mass-density-formula) — and what does this teach about the [density](#def-g7-volume-mass-density-formula) test’s limits?
11. Which of the yard’s two instruments has been defeated here: the balance, the vessel, both, or neither? Say precisely what the [density](#def-g7-volume-mass-density-formula) test *did* honestly establish about the ingot.
12. Suggest a further physical test from earlier years of this course that tells gold from its double without harming the ingot. (Their [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) differ enormously — gold near $1064\,{}^{\circ}\mathrm{C}$ , the double far above every furnace here — but no yard [melts](https://one-course.com/books/physics/1/en/chapter/9-ice-water-steam#def-g2-ice-water-steam-meltfreeze) a maybe-treasure: find gentler evidence, perhaps the [magnet](https://one-course.com/books/physics/1/en/chapter/6-magnets#def-g1-magnets-magnet) ’s verdict on iron cores, or the ring of a struck bar, and defend your choice honestly.)

**Part IV — The detective’s craft.**

13. Write the detective’s three-line creed: what [density](#def-g7-volume-mass-density-formula) can prove, what it can only suggest, and what it can never do alone.

**Solution of Problem 46.1.**

**1.** $5400 \div 2000 = 2.7\,\mathrm{g}/\mathrm{cm}^{3}$: aluminium. **2.** $1800 \div 200 = 9.0\,\mathrm{g}/\mathrm{cm}^{3}$: copper. **3.** $4520 \div 400 = 11.3\,\mathrm{g}/\mathrm{cm}^{3}$: lead — dense, soft, and poisonous to handle carelessly: gloves. **4.** $8100 \div 1500 = 5.4\,\mathrm{g}/\mathrm{cm}^{3}$ — between aluminium ($2.7$) and iron ($7.9$): a mixed sack of the two, roughly half and half by [volume](https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume#def-g6-mass-and-volume-volume). **5.** $V = 540 \div 2.7 = 200\,\mathrm{cm}^{3}$. **6.** One cube: $m = 9.0 \times 25 = 225\,\mathrm{g}$. The coil’s $1800\,\mathrm{g}$ yield $1800 \div 225 = 8$ full cubes. **7.** $20\,\mathrm{kg} = 20\,000\,\mathrm{g}$; $20000 \div 225 =
88.9$: $88$ cubes. **8.** $m = 7.9 \times 3000 = 23\,700\,\mathrm{g} \approx
23.7\,\mathrm{kg}$. **9.** $3860 \div 200 = 19.3\,\mathrm{g}/\mathrm{cm}^{3}$ — a perfect match for gold. **10.** Tungsten — [density](#def-g7-volume-mass-density-formula) $19.3$, the same signature to the decimal. The [density](#def-g7-volume-mass-density-formula) test identifies candidates; it cannot distinguish substances that happen to share a signature. **11.** Neither instrument failed: [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) are correct, and so is the division. The test honestly established that the ingot is *either* gold or something of gold’s exact [density](#def-g7-volume-mass-density-formula) — it narrowed the suspects to two. **12.** Defensible choices: the struck bar’s ring and feel (tungsten is far harder — a file or hardness test on a hidden corner tells them apart quickly), or an accepted expert [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) of how the bar conducts [heat](https://one-course.com/books/physics/1/en/chapter/34-heat-and-insulation#def-g5-heat-and-insulation-heat) or current (gold is among the best [conductors](https://one-course.com/books/physics/1/en/chapter/22-conductors-and-insulators#def-g4-conductors-insulators-conductor), tungsten far behind) — gentler than any furnace, and decisive together with [density](#def-g7-volume-mass-density-formula). **13.** For example: “[Density](#def-g7-volume-mass-density-formula) can prove a substance is *not* what it claims. It can only suggest what it is — signatures narrow the suspects. And alone it can never convict: identity wants two independent witnesses.”