Two different quantities with one everyday name
Mass — The quantity of matter in a body, and a measure of its resistance to a change in motion. A scalar, measured in kilograms.
In ordinary speech mass and weight are used interchangeably. In physics they are different quantities with different units, and a good part of this topic is simply keeping them apart.
Mass is how much matter there is. It does not depend on where the object is: take a 2 kg bag of rice to the Moon and it is still 2 kg. Mass is a scalar and is measured in kilograms.
Weight is the gravitational force acting on that mass. It is a vector, measured in newtons, and it changes with location because gravitational field strength changes. The same 2 kg bag weighs about 20 N on Earth and about 3 N on the Moon.
| Mass | Weight | |
|---|---|---|
| What it is | quantity of matter | gravitational force on that matter |
| Unit | kilogram (kg) | newton (N) |
| Scalar or vector | scalar | vector — acts downwards |
| Changes with location? | no | yes, with g |
| Measured with | a balance | a newtonmeter (spring balance) |
Bathroom scales lie, usefully
A set of bathroom scales measures the force you press down with and then divides by g to display a mass. Take the same scales to the Moon and they would read about one sixth of your Earth value — although your mass has not changed at all.
Gravitational field strength
Gravitational field strength — The force per unit mass acting on a body placed in the field, g = W/m. Measured in newtons per kilogram.
A gravitational field is a region in which a mass feels a force. Its strength at a point tells you how many newtons act on each kilogram placed there, which is exactly what the equation W = mg says.
On the Earth's surface g is about 9.8 N kg⁻¹, often rounded to 10 N kg⁻¹ in problems. On the Moon it is about 1.6 N kg⁻¹, and on Jupiter about 25 N kg⁻¹. The value falls as you move further from a planet, which is why weight decreases with altitude while mass does not.
The unit N kg⁻¹ is numerically the same as the acceleration of free fall in m s⁻², and that is no coincidence: a freely falling body of mass m has a resultant force mg on it, so by F = ma its acceleration is g. Both descriptions of g are correct, but in a question about weight the unit N kg⁻¹ is the one to quote.
- W
- weightN
- m
- masskg
- g
- gravitational field strengthN kg⁻¹
An astronaut and equipment have a combined mass of 120 kg. Calculate the weight on Earth (g = 9.8 N kg⁻¹). On another planet the weight is 444 N — find g there, and state the astronaut's mass on that planet.
W = mg = 120 × 9.8.W = 1176 N, about 1200 N.- Rearrange:
g = W/m. g = 444 / 120 = 3.7 N kg⁻¹.Roughly Mars. The mass used is unchanged.- The mass is still 120 kg.Mass never changes with location — this is the mark the question is really testing.
1176 N; g = 3.7 N kg⁻¹; mass still 120 kg
A gravitational field behaves much like the electric field drawn here, with one difference: gravity only ever attracts. Notice how the lines spread out with distance — that is why g falls as you move away from a planet.
Inertia: mass in its other role
Inertia — The tendency of a body to resist a change in its state of rest or uniform motion. Mass is the measure of inertia.
Mass turns up in two apparently unrelated equations. In W = mg it tells you how strongly gravity pulls. In F = ma it tells you how hard the body is to accelerate. That second role is called inertia.
Inertia has nothing to do with gravity, which is why it is worth separating. A loaded supermarket trolley is hard to get moving and hard to stop, and it would be exactly as hard in the weightlessness of deep space, where it has no weight at all.
This is what seat belts are for. In a crash the car stops abruptly but the passenger, having inertia, continues forward at the original speed until something applies a force. The belt supplies that force over a longer time and across a wider area of the body than the windscreen would.
Key points
- Mass is in kilograms and never changes; weight is in newtons and changes with
g. W = mg, andgis force per unit mass in N kg⁻¹.- Weight is a vector acting vertically downwards.
- Inertia is resistance to a change in motion, and mass measures it.
- In any calculation, carry the mass across unchanged between locations.
Falling, and why a feather is the exception
Drop a heavy stone and a light one from the same height and they land together. This astonishes people, because heavier objects are pulled harder — and they are. A 2 kg mass has twice the weight of a 1 kg mass.
But it also has twice the inertia, so it needs twice the force to give it the same acceleration. The extra pull and the extra reluctance cancel exactly. Putting it algebraically, a = F/m = mg/m = g: the mass cancels, and every object accelerates at g regardless of how heavy it is.
A feather falls slowly for a completely different reason — air resistance, which is large compared with its tiny weight. In an evacuated tube a feather and a coin fall side by side, a demonstration famously repeated on the Moon by an Apollo astronaut with a hammer and a falling feather.
So "heavier objects fall faster" is not a statement about gravity at all. It is a statement about air resistance, and it disappears the moment the air does.
Explain why a 10 kg mass and a 1 kg mass, released together in a vacuum, reach the ground at the same instant.
- The 10 kg mass has ten times the weight, so ten times the force acts on it.
W = mg. - It also has ten times the mass, so ten times the resistance to acceleration.
a = F/m. - The mass cancels:
a = mg/m = gfor both.Both accelerate at the same rate, so both land together.
both accelerate at g, because the mass cancels
Key points
- Mass is in kilograms and never changes; weight is in newtons and changes with
g. W = mg, andgis force per unit mass in N kg⁻¹.- Inertia is resistance to a change in motion, measured by mass.
- In a vacuum, everything falls at the same rate whatever its mass.
- Air resistance, not gravity, is why a feather falls slowly.