Math 6

Mass is not the same as weight

An astronaut carries a rock from Earth to the Moon, and there it is far easier to lift. Has the rock changed, and if not, what has?

Where you are 3 in Mass and weight

Measurement12

Mass and weight3

Customary weight units Metric mass units

Why this exists

At the post office where Kerstin works, sacks of letters are weighed all day, and everyone gives the answer in kilograms. Now send one of those sacks to the Moon. An astronaut there lifts it with a fraction of the effort, yet not one letter has been taken out. Whatever “how heavy” means, part of it stayed the same on the journey and part of it changed.

So “how heavy” hides two questions. How much stuff is there? And how hard is that stuff being pulled down? On Earth the two answers move together so closely that everyday speech uses one word for both. Away from Earth they come apart, and so do the instruments that measure them. Keeping them apart is ratio work, and the Rates and unit rates and the The double number line are exactly the tools it needs.

Definition

The mass of an object is how much matter it contains. It is measured in grams and kilograms (see Metric mass units). Mass does not depend on where the object is: a 6 kilogram rock has a mass of 6 kilograms on Earth, on the Moon and on Mars. It changes only when matter is added or taken away.

The weight of an object is how hard gravity pulls it down. A pull is a force, and forces are measured in newtons, written \text{N}, a unit named after Isaac Newton. On Earth an apple of about 100 grams weighs about 1 newton. Weight depends on where the object is, because different worlds pull with different strength.

The strength of the pull at a place is a unit rate: the newtons of pull on each kilogram, written \text{N/kg} and often given the letter g. The values below are rounded.

Place	Pull on each kilogram, g	Compared with Earth’s pull
Earth	about 9.8 N/kg, often rounded to 10	1
Moon	about 1.6 N/kg	about \displaystyle \frac{1}{6}
Mars	about 3.7 N/kg	about \displaystyle \frac{3}{8}

Two instruments go with the two ideas. A pan balance holds the object in one pan and known masses in the other; when the beam is level, both sides hold the same mass. It compares masses. A spring scale hangs the object from a spring, and the harder the pull, the further the spring stretches. It measures weight.

Everyday speech blurs the two. “I weigh 42 kilograms” states a mass. A bathroom scale is a spring scale: it measures a pull, then shows it in kilograms by assuming Earth’s pull, in effect dividing the newtons by about 9.8. Pounds are used in the same double way in the United States (see Customary weight units). In this lesson weight always means the pull, in newtons.

Intuition

Figure 1 shows one rock tested with both instruments, first on Earth and then on the Moon.

Figure 1The same 6 kg rock in two places. The balance is level both times; the spring scale reads 60 N on Earth and only 10 N on the Moon.

On Earth the rock balances against 6 kilograms of known masses, and the spring stretches a long way. On the Moon every kilogram is pulled about a sixth as hard. The rock is pulled less, but so are the masses in the other pan, by the same fraction, so the beam stays level and the balance still says 6 kilograms. The spring scale has nothing to compare against. It feels only the rock’s pull, so it stretches only a sixth as far.

In one place, weight grows with mass at a fixed rate. On Earth each kilogram adds about 9.8 newtons, so mass and weight make a double number line, every weight mark straight below its mass.

Figure 2On Earth each kilogram adds about 9.8 N of weight.

Results

Weight from mass. Where gravity pulls g newtons on each kilogram, an object of mass m kilograms has weight

W = m \times g \text{ newtons}

Mass from weight. Going back divides: m = W \div g.

One place, many objects. Weight is proportional to mass. Twice the mass gives twice the weight, and the ratio of weight to mass is the same unit rate g for every object there.

One object, two places. The mass stays the same. The weights are in the same ratio as the pulls, so an object’s weight on the Moon is about \displaystyle \frac{1}{6} of its weight on Earth, and on Mars about \displaystyle \frac{3}{8} of it.

Figure 3Any object’s Moon weight is about \tfrac{1}{6} of its Earth weight.

The instruments. A pan balance gives the same reading anywhere gravity pulls. A spring scale’s reading changes with the pull.

Why it is true

Mass counts matter. A journey neither adds matter nor takes any away, so mass cannot change with place.

Gravity at one place pulls every kilogram equally hard, g newtons. An object of m kilograms is m such kilograms, so the pulls add up to m lots of g, which is m \times g. That is why weight is mass times a unit rate, and why doubling the mass doubles the weight.

On the Moon each kilogram is pulled about \displaystyle \frac{1}{6} as hard as on Earth. Every kilogram of an object loses the same fraction of its pull, so the whole weight is \displaystyle \frac{1}{6} of the Earth weight.

The balance works because both pans are in the same place. If they hold equal masses, each side is pulled with that mass times the same g, so the pulls are equal and the beam is level, whatever g is. A new place changes both pulls by the same factor, and equal pulls stay equal. A spring scale has only one side. Its stretch follows the pull, and the pull follows g.

Worked examples

Example 1

A sack at the post office

A sack of letters at Kerstin’s post office has a mass of 15 kilograms. Find its weight on Earth using 9.8 newtons per kilogram, and again using the rounded value 10.

Solution

Each kilogram is pulled with about 9.8 newtons and the sack holds 15 kilograms, so the weight is 15 lots of 9.8:

15 \times 9.8 = 147 \text{ N}

With the rounded rate the multiplication is easier: 15 \times 10 = 150 N. The answers are only 3 newtons apart, so the rounded rate is good for a quick estimate. The mass is 15 kilograms either way; only the rate used to find the weight was rounded.

Check. 147 \div 15 = 9.8, the rate we started from.

Example 2

Mass from a spring scale

The post office’s spring scale is marked in newtons. A parcel hangs from it and the scale reads 34.3 N. Use the double number line in Figure 2 to find the parcel’s mass.

Solution

The weight line is marked every 9.8 N, one step for each kilogram above it. The reading 34.3 is not a mark: it lies between 29.4 (below 3 kg) and 39.2 (below 4 kg). That gap is 9.8, and 34.3 - 29.4 = 4.9, exactly half of it. The two lines grow at the same rate, so half a gap on the weight line matches half a gap on the mass line, halfway between 3 and 4: the mass is 3.5 kg.

Dividing by the unit rate gives the same answer in one step (see Dividing by a decimal):

34.3 \div 9.8 = 3.5 \text{ kg}

Check. 3.5 \times 9.8 = 34.3.

Example 3

One rock, two instruments, two worlds

On Earth the astronaut Jaromir finds that a rock balances a 5 kg mass and a 1 kg mass together. He takes the rock, the masses, the balance and a spring scale marked in newtons to the Moon. Using 10 N/kg on Earth, and the Moon’s pull as \displaystyle \frac{1}{6} of Earth’s, find the rock’s mass and what each instrument shows in each place.

Solution

The pans balance, so the rock holds as much matter as the masses: 5 + 1 = 6 kg. That is its mass on Earth and on the Moon, because nothing is added or taken away on the trip.

On Earth the spring scale reads the pull on 6 kilograms: 6 \times 10 = 60 N.

On the Moon every kilogram is pulled \displaystyle \frac{1}{6} as hard, so the rock’s pull is

\frac{1}{6} \times 60 = 10 \text{ N}

and the spring scale reads 10 N. The masses in the other pan are also pulled \displaystyle \frac{1}{6} as hard, so both sides still pull equally, the beam is level, and the balance still says 6 kg. This is Figure 1.

Check. On Earth 60 \div 10 = 6 kg. On the Moon each kilogram is pulled \displaystyle \frac{1}{6} of 10 newtons, and \displaystyle 6 \times \frac{10}{6} = 10 N, the Moon reading.

Example 4

An astronaut in a spacesuit

Minna and her spacesuit together have a mass of 150 kg. Find their weight on Earth, using 9.8 N/kg, and on the Moon. Then find the mass that has that Moon weight on Earth.

Solution

On Earth: 150 \times 9.8 = 1{,}470 N.

On the Moon the weight is \displaystyle \frac{1}{6} of the Earth weight:

1{,}470 \div 6 = 245 \text{ N}

The Moon’s own rate gives 150 \times 1.6 = 240 N. The two answers differ a little only because 9.8, 1.6 and \displaystyle \frac{1}{6} are all rounded; either is a fair answer. Here we keep 245 N.

Which mass weighs 245 N on Earth? Going back from weight to mass divides by Earth’s rate: 245 \div 9.8 = 25 kg. So on the Moon, Minna in her suit is pulled down as hard as 25 kg is on Earth. That is 150 \div 6, one sixth of her mass, as the ratio of the pulls says it must be. Her mass on the Moon is still 150 kg.

Example 5

A bathroom scale on the Moon

Ilona’s mass is 42 kg, and her bathroom scale, a spring scale marked in kilograms for use on Earth, shows 42 at home. What does it show if she stands on it on the Moon? What is her mass there?

Solution

The scale measures the pull and turns it into kilograms as though it were on Earth. On the Moon the pull on Ilona is \displaystyle \frac{1}{6} of her Earth weight, so the scale feels \displaystyle \frac{1}{6} of the pull it feels at home, and the dial shows \displaystyle \frac{1}{6} of 42:

42 \div 6 = 7

The scale shows 7 kg. That is not her mass: her mass is still 42 kg. The scale’s number is off because it assumes Earth’s pull. A pan balance on the Moon would still give 42 kg.

Example 6

A Moon rover and a Mars rover

Orsola’s team builds two rovers. The Moon rover has a mass of 300 kg and the Mars rover a mass of 160 kg. Which has the greater mass? Which weighs more on Earth, using 9.8 N/kg? And once each is at work on its own world, using 1.6 N/kg for the Moon and 3.7 N/kg for Mars, which presses down harder on the ground beneath it?

Solution

Mass: the Moon rover, since 300 > 160, and that stays true wherever they go.

On Earth both rovers are in the same place, so the greater mass has the greater weight: 300 \times 9.8 = 2{,}940 N against 160 \times 9.8 = 1{,}568 N.

At work, each is on a different world:

300 \times 1.6 = 480 \text{ N} \qquad 160 \times 3.7 = 592 \text{ N}

Now the Mars rover has the greater weight, with only a little more than half the mass, because Mars pulls each kilogram more than twice as hard as the Moon (3.7 against 1.6). Comparing weights in two different places needs both the masses and the pulls.

Boundaries

Mass stays the same only while no matter is added or removed. Moving an object does not change its mass; a rover burning fuel, or an astronaut drinking water, does.
The pulls in the table are rounded values at the surface; even on Earth the pull varies a little from place to place. Answers found with these values are approximate, and two rounded routes can give slightly different answers, as in Example 4.
In a spacecraft coasting in orbit, where everything floats, neither instrument works: a balance needs gravity to pull its pans, and a spring needs a pull to stretch. The object’s mass is still all there.
Weight is proportional to mass only within one place. Across two places a greater mass can have the smaller weight, as in Example 6.
A weight “in kilograms”, on a bathroom scale or in everyday speech, is a mass worked out from a pull using Earth’s g. It is right on Earth and wrong anywhere else, as in Example 5.

Where this goes next

When science wants a number that belongs to the object itself, wherever it is, it uses mass. That is why Density divides mass, not weight, by volume: a block of iron has the same density on the Moon as at home. The rule W = m \times g is a proportional relationship with g as its The constant of proportionality and proportional tables, and later physics draws weight as a force with a direction, an arrow pointing down (Forces and velocities as vectors).

Exercises

Unless an exercise says otherwise, use 9.8 N/kg on Earth, 1.6 N/kg on the Moon and 3.7 N/kg on Mars.

Mass or weight?

Say whether each describes a mass or a weight: (a) a bag holding 3 kg of flour; (b) the 12 N a spring scale shows; (c) a stone that balances 500 g on a pan balance; (d) the 2{,}960 N with which a rover presses on the ground of Mars.
A crate has a mass of 20 kg on Earth. What is its mass on the Moon, and on Mars? Explain.
Which would give the same reading on Earth and on Mars, a pan balance or a spring scale? Explain.

Weight on Earth

A box of stamps has a mass of 13 kg. Find its weight on Earth.
A sack of parcels has a mass of 12 kg. Find its weight using 9.8 N/kg, then using the rounded 10 N/kg.
A spring scale on Earth reads 88.2 N for a parcel. Find the parcel’s mass.
A packet of envelopes has a mass of 500 g, which is 0.5 kg. Find its weight on Earth.
A double number line has mass on top, marked 0, 2, 4, 6 and 8 kg. Write the Earth weights that go below the marks, then find the weight of 7 kg.
A parcel on a spring scale on Earth reads 343 N. Find its mass.

The Moon and Mars

On Earth an astronaut in her suit weighs about 840 N. Using the \displaystyle \frac{1}{6} ratio, about how much does she weigh on the Moon?
A tool kit has a mass of 11 kg. Find its weight on the Moon, and its mass there.
On the Moon a spring scale reads 32 N for a rock. Find the rock’s mass, then its weight on Earth.
Using 10 N/kg, a rover weighs 4{,}000 N on Earth. Find its mass, then its weight on Mars using the \displaystyle \frac{3}{8} ratio. What is its mass on Mars?
Make a table with columns for mass, Earth weight and Mars weight, and rows for 1, 8, 40 and 200 kg. Use 10 N/kg on Earth, and take each Mars weight as \displaystyle \frac{3}{8} of the Earth weight.
A Mars rover has a mass of 900 kg. Find its weight on Mars and on Earth.
(a) On the Moon an astronaut stands on a bathroom scale made for Earth and it shows 11 kg. Using the \displaystyle \frac{1}{6} ratio, find his mass. (b) A person of mass 64 kg stands on the same kind of scale on Mars. Using the \displaystyle \frac{3}{8} ratio, what does it show?
Rock A, of mass 30 kg, lies on the Moon. Rock B, of mass 14 kg, lies on Mars. Which has the greater mass? Which has the greater weight where it lies?
On the Moon a rock balances three 2 kg masses and a 500 g mass on a pan balance. Find its mass, its weight on the Moon and its weight on Earth.

Reasoning

Explain why a pan balance gives the same result on the Moon as on Earth, while a spring scale does not.
A poster says “On the Moon you would weigh only one sixth as much.” Say what changes on the Moon and what does not, and why a bathroom scale showing kilograms would give a misleading number there.
Using a double number line or a ratio table, explain why doubling an object’s mass on Earth doubles its weight.
Two packs balance each other on a pan balance on Mars. Must they have the same weight on Mars? On Earth? Explain.

Answers

(a) mass; (b) weight; (c) mass; (d) weight. Kilograms, grams and a balance describe the amount of matter; newtons and a spring scale’s pull describe weight.
20 kg on both. Moving the crate adds or removes no matter, so its mass does not change; only its weight does.
The pan balance. Both its pans are pulled by the same world, so equal masses still balance. A spring scale’s stretch follows the pull, which is weaker on Mars.
127.4 N. 13 \times 9.8 = 127.4.
117.6 N, or about 120 N. 12 \times 9.8 = 117.6 and 12 \times 10 = 120.
9 kg. 88.2 \div 9.8 = 9.
4.9 N. 0.5 \times 9.8 = 4.9.
0, 19.6, 39.2, 58.8 and 78.4 N; 7 kg weighs 68.6 N. Each 2 kg adds 2 \times 9.8 = 19.6 N, and 7 \times 9.8 = 68.6, halfway between the marks for 6 and 8 kg.
35 kg. 343 \div 9.8 = 35.
About 140 N. 840 \div 6 = 140.
17.6 N; its mass is still 11 kg. 11 \times 1.6 = 17.6.
20 kg; 196 N. 32 \div 1.6 = 20 and 20 \times 9.8 = 196.
400 kg; 1{,}500 N; still 400 kg. 4{,}000 \div 10 = 400 and \displaystyle \frac{3}{8} \times 4{,}000 = 1{,}500.
Earth weights 10, 80, 400 and 2{,}000 N; Mars weights 3.75, 30, 150 and 750 N. Each Earth weight is the mass times 10, and each Mars weight is \displaystyle \frac{3}{8} of it, so \displaystyle \frac{3}{8} \times 10 = 3.75 and \displaystyle \frac{3}{8} \times 2{,}000 = 750.
3{,}330 N on Mars and 8{,}820 N on Earth. 900 \times 3.7 = 3{,}330 and 900 \times 9.8 = 8{,}820.
(a) 66 kg. The scale shows \displaystyle \frac{1}{6} of his mass, so his mass is 11 \times 6 = 66. (b) 24 kg. \displaystyle \frac{3}{8} \times 64 = 24.
Rock A has the greater mass, since 30 > 14. Rock B has the greater weight: A weighs 30 \times 1.6 = 48 N and B weighs 14 \times 3.7 = 51.8 N.
6.5 kg; 10.4 N; 63.7 N. The mass is 3 \times 2 + 0.5 = 6.5 kg, then 6.5 \times 1.6 = 10.4 and 6.5 \times 9.8 = 63.7.
Both pans of a balance are pulled by the same world, each kilogram with the same g. Equal masses get equal pulls, so the beam is level whatever g is, and a new place changes both pulls by the same factor. A spring scale has nothing on the other side: its stretch follows the object’s pull alone, which is mass times the local g.
The weight changes: on the Moon each kilogram is pulled about \displaystyle \frac{1}{6} as hard, so the whole pull is about \displaystyle \frac{1}{6} as big. The mass does not: no matter is lost. A bathroom scale measures the pull and divides by Earth’s g, so on the Moon it shows about \displaystyle \frac{1}{6} of the true mass in kilograms.
On Earth each kilogram adds the same 9.8 N, so weight is mass times one unit rate. In a ratio table, doubling the mass column doubles the weight column; for example 2 kg gives 19.6 N and 4 kg gives 39.2 N, which is 2 \times 19.6. Doubling the mass doubles the number of kilograms being pulled, and so doubles the pull.
Yes, both times. Balancing shows the packs have equal mass. Wherever they are together, each kilogram of each pack gets the same pull, so equal masses have equal weights, on Mars and on Earth alike (though the Earth weight is bigger than the Mars weight).