---
title: "Measuring Mass"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 20
exercises: 11
source: https://one-course.com/books/physics/1/en/chapter/20-measuring-mass
---

# Chapter 20 — Measuring Mass

At the market: “A [kilogram](#def-g3-measuring-mass-units) of apples, please!” The seller pours apples into the pan until the scale is satisfied. What exactly did the customer order — and how does the scale know when to stop? With [levers](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-lever) and [balance scales](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-scale) in our toolbox, we are ready to [measure](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-measure) a new thing: [mass](#def-g3-measuring-mass-mass).

## 20.1 Mass: how much stuff

**Definition 20.1 (Mass).**

The *mass* of a thing tells how much stuff it is made of. The more stuff, the greater the mass — and the harder the Earth pulls it down, which is why things with more mass feel heavier in your hand and press their pan of a [balance scale](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-scale) down.

**Example 20.2 (Big is not heavy).**

A pillow fills your whole arms, yet you carry it easily. A stone smaller than your fist strains your wrist. The pillow is big but holds little stuff — mostly feathers and [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air); the stone is small but packed with stuff. [Mass](#def-g3-measuring-mass-mass) is about *how much stuff*, not about how much room it takes. (These two ideas, and the tricky dance between them, meet again in the floating-and-sinking story one day.)

**Definition 20.3 (Gram and kilogram).**

[Mass](#def-g3-measuring-mass-mass) is measured in *grams* (written $\mathrm{g}$) and *kilograms* (written $\mathrm{kg}$):

$$
1\,\mathrm{kg} = 1000\,\mathrm{g}.
$$

A paperclip has a [mass](#def-g3-measuring-mass-mass) of about $1\,\mathrm{g}$; an apple about $150\,\mathrm{g}$; a carton of milk about $1\,\mathrm{kg}$; you, a few tens of kilograms. Like the [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units), these [units](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-measure) are the same for everyone on Earth.

## 20.2 Weighing with a balance scale

**Method 20.4 (Weighing with blocks).**

To [measure](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-measure) a melon’s [mass](#def-g3-measuring-mass-mass) with a [balance scale](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-scale) and a box of marked metal blocks ($500\,\mathrm{g}$, $200\,\mathrm{g}$, $100\,\mathrm{g}$, $50\,\mathrm{g}$, …):

1. check the empty scale hangs level;
2. put the melon in one pan;
3. add blocks to the other pan, biggest useful ones first, until the pans hang level;
4. add up the blocks: their total is the melon’s [mass](#def-g3-measuring-mass-mass) .

If the pans level with a $500\,\mathrm{g}$, a $200\,\mathrm{g}$ and a $100\,\mathrm{g}$ block, the melon’s [mass](#def-g3-measuring-mass-mass) is $500 + 200 + 100 =
800\,\mathrm{g}$.

![Weighing the melon: blocks are added until the pans level; their sum is the answer.](https://one-course.com/images/onecourse/chapters/physics-1/g3-measuring-mass/fig-af2a2cb6e185.svg)

*Weighing the melon: blocks are added until the pans level; their sum is the answer.*

![A balance scale doing its quiet job: level pans, so the apples’ mass equals the brass weights’ total.](https://one-course.com/images/onecourse/chapters/physics-1/g3-measuring-mass/img-6bfb87389815.jpg)

*A [balance scale](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-scale) doing its quiet job: level pans, so the apples’ [mass](#def-g3-measuring-mass-mass) equals the brass weights’ total.*

**Example 20.5 (Other scales).**

The kitchen scale and the bathroom scale have no second pan: you put the load on, and a needle turns (or a number [lights](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) up) to show the [mass](#def-g3-measuring-mass-mass) directly. Inside, a squeezed spring does the comparing for you. Handy and fast — but the old two-pan balance hides no machinery at all: just the [lever](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-lever) rule you already understand.

**Example 20.6 (Reading the market).**

“A [kilogram](#def-g3-measuring-mass-units) of apples” means: apples until the scale balances a $1\,\mathrm{kg}$ block — about $6$ or $7$ apples. Flour comes in $1\,\mathrm{kg}$ bags, chocolate bars are often $100\,\mathrm{g}$, a letter needs about $20\,\mathrm{g}$ of paper, and the postman’s rule “under $20\,\mathrm{g}$, one stamp” is a [mass](#def-g3-measuring-mass-mass) rule.

## 20.3 Mass follows the stuff

**Proposition 20.7 (Same stuff, same mass).**

Changing a thing’s *shape* does not change its [mass](#def-g3-measuring-mass-mass). Roll a ball of dough into a snake, flatten it into a pancake, or cut it in four and weigh all the pieces together: the scale reports the same [mass](#def-g3-measuring-mass-mass) every time. [Mass](#def-g3-measuring-mass-mass) counts the stuff — and squashing, rolling and cutting neither add stuff nor take any away.

![One lump of dough, three shapes — and one mass. The scale is never fooled by a change of shape.](https://one-course.com/images/onecourse/chapters/physics-1/g3-measuring-mass/fig-589cf6b3dfaf.svg)

*One lump of dough, three shapes — and one [mass](#def-g3-measuring-mass-mass). The scale is never fooled by a change of shape.*

**Remark 20.8 (The scale ignores appearances).**

This is the quiet power of measuring: eyes judge a flattened pancake “bigger” than the ball it came from, and a fluffed pillow “heavier” than a stone — but the [balance scale](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-scale), like the [thermometer](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) and the ruler before it, reports the stuff and nothing else. One by one, our instruments are replacing “it seems” with “it is”.

## 20.4 Exercises

**Exercise 20.1 ★.**

What does the [mass](#def-g3-measuring-mass-mass) of a thing tell? Which two [units](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-measure) [measure](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-measure) it, and how many [grams](#def-g3-measuring-mass-units) make one [kilogram](#def-g3-measuring-mass-units)?

**Solution of Exercise 20.1.**

How much stuff a thing is made of. [Units](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-measure): the [gram](#def-g3-measuring-mass-units) and the [kilogram](#def-g3-measuring-mass-units), with $1\,\mathrm{kg} = 1000\,\mathrm{g}$.

**Exercise 20.2 ★.**

Choose the likelier [mass](#def-g3-measuring-mass-mass): a paperclip — $1\,\mathrm{g}$ or $1\,\mathrm{kg}$? An apple — $150\,\mathrm{g}$ or $15\,\mathrm{kg}$? A carton of milk — $1\,\mathrm{g}$ or $1\,\mathrm{kg}$? A cat — $4\,\mathrm{kg}$ or $400\,\mathrm{kg}$?

**Solution of Exercise 20.2.**

Paperclip: $1\,\mathrm{g}$. Apple: $150\,\mathrm{g}$. Milk carton: $1\,\mathrm{kg}$. Cat: $4\,\mathrm{kg}$.

**Exercise 20.3 ★.**

The pans level when a pumpkin faces a $500\,\mathrm{g}$ block and two $200\,\mathrm{g}$ blocks. What is the pumpkin’s [mass](#def-g3-measuring-mass-mass)?

**Solution of Exercise 20.3.**

$500 + 200 + 200 = 900$: the pumpkin’s [mass](#def-g3-measuring-mass-mass) is $900\,\mathrm{g}$.

**Exercise 20.4 ★.**

Why must the empty scale hang level before any weighing? (Remember the cheating merchant.)

**Solution of Exercise 20.4.**

If the empty scale already tips, every weighing lies in favor of one side — like the cheating merchant’s long arm. Level when empty is the proof of a fair start.

**Exercise 20.5 ★.**

A beach ball is much bigger than a can of soup, yet the can presses its pan down. Explain with [Example 20.2](#ex-g3-measuring-mass-bignotcheavy).

**Solution of Exercise 20.5.**

The beach ball is big but mostly [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) — little stuff, small [mass](#def-g3-measuring-mass-mass). The can is small but packed with stuff — more [mass](#def-g3-measuring-mass-mass), so it presses its pan down. [Mass](#def-g3-measuring-mass-mass) counts stuff, not size.

**Exercise 20.6 ★.**

How many [grams](#def-g3-measuring-mass-units) are in $2\,\mathrm{kg}$? In half a [kilogram](#def-g3-measuring-mass-units)? Which is more: $1200\,\mathrm{g}$ or $1\,\mathrm{kg}$?

**Solution of Exercise 20.6.**

$2\,\mathrm{kg} = 2000\,\mathrm{g}$; half a [kilogram](#def-g3-measuring-mass-units) is $500\,\mathrm{g}$; $1200\,\mathrm{g}$ is more than $1\,\mathrm{kg} = 1000\,\mathrm{g}$.

**Exercise 20.7 ★.**

A chocolate bar of $100\,\mathrm{g}$ is broken into all its little squares, and every square goes on the scale together. What [mass](#def-g3-measuring-mass-mass) does the scale show, and which law says so?

**Solution of Exercise 20.7.**

Still $100\,\mathrm{g}$: breaking changes the shape, not the stuff — the law “same stuff, same [mass](#def-g3-measuring-mass-mass)”, with all the pieces on the pan together.

**Exercise 20.8 ★.**

The kitchen scale and the two-pan balance both weigh a bag of flour. What does each use to do the comparing — and which one works by a rule you learned three chapters ago?

**Solution of Exercise 20.8.**

The kitchen scale compares with a squeezed spring inside; the two-pan balance compares with blocks across equal arms — the [lever](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-lever) rule from the [levers](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-lever) chapter.

**Exercise 20.9 ★.**

The baker needs $800\,\mathrm{g}$ of flour and has blocks of $500\,\mathrm{g}$, $200\,\mathrm{g}$, $200\,\mathrm{g}$, $100\,\mathrm{g}$ and $50\,\mathrm{g}$. Find two different ways to weigh out exactly $800\,\mathrm{g}$ — knowing that blocks are allowed to sit in *either* pan.

**Solution of Exercise 20.9.**

First way: $500 + 200 + 100 = 800\,\mathrm{g}$ of blocks opposite the flour. Second way: put the $100\,\mathrm{g}$ block *with the flour* and $500 + 200 + 200 = 900\,\mathrm{g}$ opposite; level pans then mean flour $+ 100\,\mathrm{g}$ equals $900\,\mathrm{g}$, so the flour is $900 - 100 = 800\,\mathrm{g}$.

**Exercise 20.10 ★★.**

Nina weighs a whole orange: $200\,\mathrm{g}$. She peels it and weighs the peeled orange alone: $160\,\mathrm{g}$. Has [Proposition 20.7](#prop-g3-measuring-mass-conserved) failed? Where are the missing [grams](#def-g3-measuring-mass-units), and how could Nina prove it with one more weighing?

**Solution of Exercise 20.10.**

The law has not failed: no stuff vanished — it was only separated. The missing $40\,\mathrm{g}$ are the peel. One more weighing proves it: the peel alone should read $40\,\mathrm{g}$, and $160 + 40 =
200\,\mathrm{g}$.

**Exercise 20.11 ★★.**

A riddle, at last easy for you: which has more [mass](#def-g3-measuring-mass-mass), a [kilogram](#def-g3-measuring-mass-units) of feathers or a [kilogram](#def-g3-measuring-mass-units) of iron? And which takes more room? Explain why so many people answer the first question wrongly.

**Solution of Exercise 20.11.**

They have the *same* [mass](#def-g3-measuring-mass-mass) — one [kilogram](#def-g3-measuring-mass-units) each. The feathers take far more room. People answer wrongly because hands and eyes judge iron “heavy stuff”; but the question fixed the [mass](#def-g3-measuring-mass-mass) at one [kilogram](#def-g3-measuring-mass-units) each — only the room differs.