# Weighing the Earth

Now that might seem like a ludicrous idea! How on Earth can we weigh something as big as, well, the whole planet? There is no scale huge enough to support the whole world so, really, just how can we determine the weight of the Earth?

Atlas carrying the Earth

Before we proceed, however, let me just say that this very question itself is wrong. You see, it’s not that we can’t determine the mass of the Earth but it’s the very idea of “weight” that’s confusing here. To weigh something, any thing, means to determine the force being exerted on that thing by another body. Typically, then, when we speak of the weight of a bag, what we’re really talking about is how much force is the Earth exerting on the bag by pulling it towards its centre. The Earth pulls on every other body with a force proportional to the mass of the body. This is simply stating the law of gravitation. The more massive an object is, the more the Earth will pull on it. An empty bag contains less mass than a full one and, according to the law of gravitation, will experience a smaller force of attraction towards the Earth. Simple.

So, to speak of weighing the Earth, we have to have at least one other body relative to which we’ll determine the Earth’s weight. You can’t have a body in isolation, in complete isolation, away from any other planet, star or galaxy, in the depths of vacuous space and talk about its weight. That just doesn’t make sense. Weight has to be relative to some other body. However, that isolated body will have a mass, regardless of it being on its own or not. The mass of an object, generally speaking, does not change whether you place it on top of a mountain, at the bottom of the ocean or on a planet in the Andromeda galaxy. The weight, on the other hand, depends on 2 things. First, it depends on the relative mass of the object compared to the other. Second, it depends on the distance between the objects. The further apart they are, the smaller the force of attraction between them and, as a result, the less the object will weigh relative to the other.

We therefore cannot speak of the weight of the Earth unless we specify what the weight is relative to. We can certainly determine the weight of the Earth relative to Jupiter, for example. All we need to know is their masses, the distance between them and some number called the Gravitational Constant. With these values already determined, we can easily calculate the weight of the Earth relative to Jupiter, or, conversely, the weight of Jupiter relative to the Earth. It’s a simple arithmetic problem of multiplying and dividing some numbers. Nothing fancy going on there.

But here’s the catch. How do you find out the mass of the Earth to begin with? Unless we know the mass, we can’t work out its weight relative to Jupiter or another celestial object for that matter. That is the real problem here. Not weighing the Earth but determining its mass! How can we do that? How do you figure out the mass of such a huge body? Are there any instrument suitable enough to measure this quantity? Do we need some funky gadgets or sophisticated apparatus to figure out the Earth’s mass? Well, it depends what you mean by sophisticated but a reliable watch and a long enough ruler would do the trick. Some patience and perseverance might come in handy, too.

A watch and a ruler to measure the Earth’s mass? Seriously? Well, it’s one way to do it. So, what can we do with the watch then? We can use it to count the number of seconds a pendulum takes to swing. That’s a simple enough experiment, you have to admit. There is a relationship between how long a pendulum takes to complete a swing and the acceleration due to gravity. This relationship also depends on the length of the pendulum, which can be measured using the ruler. So, we can put all this together and determine the acceleration due to gravity. Again, some simple arithmetic will do.

But the story doesn’t end here. Again, using the law of gravitation, there is a connection between the acceleration due to gravity and the mass of the Earth. What connects these quantities together are the radius of the Earth and that special number called the Gravitational Constant. The Gravitational Constant, represented by G, is some number which is invariant no matter where in the universe you happen to be. It’s a universal constant. We can determine this number using 2 known masses and measuring, very accurately, the force they exert on each other. Through some clever arrangement of these masses, tied to some thin strings, we can determine how much they twist the strings as they pull on each other with the force of their mutual gravity. Again, a little bit measurement and a little bit of arithmetic, we put it all together and out comes, rather conveniently, this neat little number G. (Check out what Henry Cavendish did, apparently, to figure out G.)

The other part of the puzzle is the radius of the Earth. Now, how can we find this out? One way to do it is to take a long, really long walk along the equator until we’ve covered, let’s say, a tenth of the way round the world. A tenth of the way is simply a tenth of 360 degrees, which is 36 degrees. Easy. While walking, however, we use our old trusted ruler to measure how long we’ve travelled. Throw in some mathematics in the mix and out pops the radius of the Earth. You see, there is a rather clever equation which brings together the distance S we’ve walked along the equator, the angle θ we’ve covered and the radius r: r = S/θ.

With all the pieces now in place we can finally determine the mass of the Earth. We’ve timed the pendulum and measured its length; we’ve calculated G and the radius of the Earth; we’ve rested after this long and exhausting walk; now all we need is a chilled drink, some mathematics and some painkillers for the sore legs. Churning all those numbers in our calculator will give us the much sought after mass of the Earth. If we’ve done all the measurements and calculations right then we should end up with something like this: 5972200000000000000000000 kg. That’s about 6 million billion billion kilograms! I challenge you to find anything more massive than that on this planet! There are other ways, of course, to determine the mass of the Earth or that of any other celestial object but we can talk about that in another blog.

So there you have it, the mass of the Earth. But I’m sure Atlas already knew that. He’s been carrying this load around for some time now…