Units of Mass and Weight
Mathematically, mass and weight are related as follows:
weight = mass x the acceleration of gravity
This means that mass and weight must have different units (because the units of weight are the units of mass times the units of acceleration). However, the metric system and the U.S. unit system handle these units a bit differently, so let’s look at each separately.
Metric Units of Mass and Weight
In the metric system, the standard unit of mass is the kilogram. In other words, if a scale (on Earth) says that you “weigh” 60 kilograms, it is really telling you that you have a mass of 60 kilograms. To calculate your weight, we must multiply by the acceleration of gravity, which for Earth we approximate as 10 m/s2:
The units of weight probably look pretty strange, and you would technically read them as “kilogram-meters per second squared.” Fortunately, this combination of units is given a special name: the “newton.” In other words, if you have a mass of 60 kilograms, your weight is about 600 newtons.
To summarize, when we put an object (on Earth) on a metric scale that reads kilograms, the scale is actually telling us the object’s mass, not its weight. To get its weight, we must multiply by the acceleration of gravity, which will give us a weight in newtons. (You might notice that this also means that metric scales that read kilograms give accurate masses only on Earth; if you wanted to take a metric scale to another planet, you would be better off with one that tells you weight in newtons.)
Example: Metric Mass and Weight on the Moon. Suppose that a scale on Earth tells you that you have a mass of 60 kilograms. What is your mass and your weight on the Moon? Recall from the earlier Do the Math box that the acceleration of gravity on the Moon is about 1/6 that on Earth.
- For mass: An object’s mass is the same everywhere, since it depends only how much matter it contains. Therefore, if you have a mass of 60 kilograms on Earth, you would have the same mass of 60 kilograms on the Moon.
- For weight: Because weight is mass x the acceleration of gravity, and the acceleration of gravity on the Moon is about 1/6 that on Earth, your weight on the Moon would be about 1/6 of your weight on Earth. We’ve already found (above) that your weight on Earth is about 600 newtons. Therefore, your weight on the Moon is 1/6 of that, or about 100 newtons.
U.S. Units of Mass and Weight
In the United States, we commonly talk about pounds instead of kilograms. Technically, however, pounds are a unit of weight, not of mass. In other words, if you get on a scale and it reads 120 pounds, then you have a weight of 120 pounds. So what is your mass? To answer, we start with our earlier definition:
weight = mass x the acceleration of gravity
Finding mass therefore requires a little bit of algebra. First, we first reverse the left and right sides
mass x the acceleration of gravity = weight
Next, we divide both sides by the acceleration of gravity to find:
mass = weight/ the acceleration of gravity
In other words, if you know your weight in pounds, then you would find your mass by dividing by the acceleration of gravity, which in U.S. units is 32 ft/s2. The result would be your mass in units called “slugs.”
So we’ll stop here, because most people don’t like thinking of themselves in terms of slugs. You might also notice that this is yet one more reason why scientists always prefer to work with the metric system.
This optional box covers a subtlety about mass and weight that will be important when students go on to study physics in high school, which is that, as a result of the way the U.S. and metric systems of units developed, they handle mass and weight differently:
- In the metric system, we commonly talk in terms of kilograms, which technically are a unit of mass, not weight.
- In the U.S. system, we commonly talk in terms of pounds, which are a unit of weight, not mass.
As long as we only consider objects on Earth, we can generally ignore this subtlety. Indeed, that is exactly what we do when we use the standard conversion of 1 kilogram = 2.2. pounds. However, for scientific work – or for any situation that is not on the surface of Earth (such as in space or on the Moon) — the distinction becomes very important. As the box demonstrates, it’s easy to work with the distinction in the metric system, where masses in kilograms can be converted to weights in newtons. In contrast, the U.S. system would force you to use the ridiculous-sounding units of “slugs” — which is one more reason why virtually all scientific work is done in metric units.