Space · Quantitative
Gravity and weight
Bathroom scales are marked in kilograms and measure newtons. That single piece of everyday dishonesty is behind almost every mistake made in this topic.
Start here
You climb Everest. What has changed about you?
You weigh yourself at sea level, then again on the summit of Everest, 8849 m higher up, using the same scales and wearing the same clothes.
What has changed — your mass, your weight, both or neither?
Your weight changes and your mass does not. Mass is how much matter there is in you, and carrying it up a mountain does not remove any. Weight is the force gravity pulls on that matter with, and gravity is very slightly weaker at the top of Everest — about 0.3% weaker, so a 70 kg climber weighs roughly 2 N less. Take the same climber to the Moon and the mass is still 70 kg while the weight falls from about 700 N to about 112 N.
Mass is how much matter something is made of. It is measured in kilograms, it is the same everywhere in the universe, and nothing but adding or removing material will change it.
Weight is a force: the pull of gravity on that matter. It is measured in newtons, it has a direction — towards the centre of whatever is pulling — and it changes completely depending on where you are. W = m × g, where g is the gravitational field strength of the place you are standing, in newtons for each kilogram.
On Earth g is about 10 N/kg, so every kilogram is pulled with about 10 N. On the Moon it is 1.6, on Mars 3.7, on Jupiter 24.8. The same 70 kg astronaut weighs about 700 N on Earth, 112 N on the Moon and 1740 N on Jupiter, and is made of exactly the same amount of matter throughout.
At the bench · one set of bathroom scales, five places to stand
Same person. Same mass. Five different weights.
Change a control to begin
Choose where to stand and how much mass to put on the scales. The bars are the gravitational field strength at each place — and they are the only thing that changes.
Commit first. You take a 1 kg bag of sugar to the Moon. What happens to it?
Where you are standing
50 kg
Gravitational field strength, in newtons for each kilogram
Mass on the scales
—
—
Field strength here
—
—
So the weight is
—
—
Compared with Earth
—
—
Writing it down · the shape of this relationship
Weight = mass × gravitational field strength
The triangle
Cover the one you want
W = m × g
m = W ÷ g
g = W ÷ m
Two things side by side means multiply. One thing over another means divide.
W · weight, the force gravity pulls with · N
m · mass of the object · kg
g · gravitational field strength where you are · N/kg
The field strength is the one for the place you are standing, never automatically the Earth's.
Worked example · one step at a time
What is the weight of a 6 kg toolbox on Earth?
Step 0 of 5
Convert
6 kg stays 6 kg · 10 N/kg stays 10 N/kg
The mass is already in kilograms, which is what N/kg needs, so there is nothing to convert.
Formula
W = m × g
Cover W on the triangle: m sits beside g, so you multiply.
Insert
W = 6 kg × 10 N/kg
The field strength on Earth is 10 newtons for each kilogram.
Fine-tune
6 × 10 = 60
Kilograms times newtons per kilogram leaves newtons.
Answer
W = 60 N
Sixty newtons, downwards — weight is a force and always has a direction.
Worked example · one step at a time
What is the weight of a 450 g bag of flour on Earth?
Step 0 of 5
Convert
450 g ÷ 1000 = 0.450 kg
× 10 N/kg wants kilograms, and a gram is a thousandth of one, so divide by 1000.
Formula
W = m × g
Cover W on the triangle: m sits beside g, so you multiply.
Insert
W = 0.450 kg × 10 N/kg
The converted mass goes in. The 450 never does.
Fine-tune
0.450 × 10 = 4.5
Kilograms times newtons per kilogram leaves newtons.
Answer
W = 4.5 N
Insert 450 instead of 0.450 and the bag comes out weighing 4500 N — about half a tonne.
Your turn · the same five steps
Your scales: 50 kg, and the place is Earth, where g is 10.0 N/kg.
Write each line out yourself — starting by deciding whether anything needs converting. Then check your working and tick the lines you had.
In deep space the field strength is zero, so every weight comes out as nothing. Stand somewhere with a field and the five lines come back.
The five lines · tick what you had
Convert
50 kg stays 50 kg · 10.0 N/kg stays 10.0 N/kg
The mass is already in kilograms and the field strength already in newtons per kilogram, so there is nothing to convert.
Formula
W = m × g
Cover W on the triangle: m sits beside g, so you multiply.
Insert
W = 50 kg × 10.0 N/kg
The field strength is the one for Earth — the place you are, never a default.
Fine-tune
50 × 10.0 = 500
Kilograms times newtons per kilogram leaves newtons.
Answer
W = 500 N
Downwards, towards the centre of whatever is pulling — and where there is no field there is no pull, so no direction either.
The five lines give 500 N for a mass of 50 kg — and that 50 kg is the same everywhere, whatever the scales say.
A rover of mass 185 kg is landed on Mars, where g is 3.7 N/kg. What does it weigh there?
Write each line out yourself — starting by deciding whether anything needs converting. Then check your working and tick the lines you had.
The five lines · tick what you had
Convert
185 kg stays 185 kg · 3.7 N/kg stays 3.7 N/kg
Both quantities are already in the units the formula wants, so there is nothing to convert.
Formula
W = m × g
Cover W on the triangle: m sits beside g, so you multiply.
Insert
W = 185 kg × 3.7 N/kg
Use the Martian field strength, not the Earth one.
Fine-tune
185 × 3.7 = 684.5
Kilograms times newtons per kilogram leaves newtons.
Answer
W = 685 N
Reach for 10 N/kg out of habit and the rover comes out weighing 1850 N — the mistake is using the wrong planet, not the wrong unit.
The five lines give 685 N on Mars. On Earth the same rover would weigh 1850 N, which is why the landing legs could be built lighter.
Key fact
Weight = mass × gravitational field strength, W = m × g. Mass is in kilograms and never changes; weight is a force in newtons and changes with where you are. On Earth g is about 10 N/kg, so a 1 kg bag of sugar weighs about 10 N.
Think again
“Astronauts float because there is no gravity in space.”
The International Space Station orbits about 400 km up, where the Earth’s gravitational field is still around 90% as strong as it is at the surface. If gravity were absent the station would fly off in a straight line instead of going round. What the astronauts have lost is not the pull but the push — the floor is falling at exactly the same rate they are, so nothing presses on them. That is free fall, not the absence of gravity, and the correct name for the sensation is weightlessness.
“Weight is measured in kilograms.”
Weight is a force, and forces are measured in newtons. The scales in your bathroom really do measure a force, then quietly divide it by 10 N/kg and print a mass, because a mass is what you wanted to know and the assumption is that you are on Earth. It is a reasonable shortcut and a bad habit: the moment the question moves off this planet, the difference between the 70 kg and the 700 N is the whole answer.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
A crate has a mass of 24 kg. What is its weight on Earth, where g = 10 N/kg?
Rung 2 · The one that catches people
An astronaut floats inside the International Space Station. What is true of them?
Rung 3 · Explain
A set of bathroom scales is marked in kilograms. Explain what it is actually measuring, and why the reading would be wrong if you took it to the Moon.
Rung 4 · Take it somewhere new
Jupiter’s field strength is about 24.8 N/kg. Explain what would be hard about walking there, and what would be exactly as hard as on Earth.
Key note
Mass is the amount of matter in an object, measured in kilograms and unchanged by location. Weight is the force of gravity acting on that mass, measured in newtons and given by W = m × g. Gravitational field strength g is the pull in newtons on each kilogram: about 10 N/kg on Earth, 1.6 on the Moon, 3.7 on Mars and 24.8 on Jupiter. Bathroom scales measure a force and display a mass by assuming Earth’s field strength.
Going further
Field strength depends on the mass of the body you are standing on and on how far you are from its centre. Jupiter is over three hundred times the mass of the Earth and pulls only about two and a half times as hard at its cloud tops, because those cloud tops are eleven times further from the middle than the Earth’s surface is from ours. Distance is doing as much work in that comparison as mass.
Because g varies slightly across the Earth — with altitude, latitude and the density of the rock underneath — a laboratory balance that compares an unknown mass against a known one gives the same answer anywhere, while a spring scale does not. That is why the kilogram was defined by a physical object for 130 years, and why gravimeters sensitive enough to detect a few parts per billion are used to hunt for oil and for buried voids.
Before this lesson
- Nothing — this is where the unit starts.
Connects to
At GCSE this becomes
- Gravitational field strength in N/kg, weight measured with a calibrated spring balance, and gravitational potential energy as mass × g × height.
Where to next
- Next: Mass vs weight
- Previous: Why ice floats
Matter and the particle model
Ask Mr Badmus AI
Got a mass in one unit and want the weight in newtons?
The bench is a teaching model. Gravitational field strengths are surface values rounded to one decimal place: Earth 10.0, the Moon 1.6, Mars 3.7 and Jupiter 24.8 N/kg. Earth’s true mean value is 9.81 N/kg and varies by about 0.5% between the poles and the equator; 10 is the figure used throughout KS3. Jupiter has no solid surface, so its figure is quoted at the top of the cloud layer. Deep space is treated as zero field, which is an idealisation — no point in the universe is entirely free of gravity.
Lesson content © MrBadmusAI.