How much does the Earth Weigh?

I actually heard this question in a smartphone commercial and it bothered me. They were showing off how easy it is to do a Google search from the newest tablet or whatever; just ask your question and it speaks the answer to you. In this case 5.97E24kg.

Only that’s not the answer. That’s what bothered me. And the truth is there’s a lot of interesting physics in that question and a very common misconception. Being a physics teacher, I feel like I should address it. So let’s discuss.

First of all, the question that Google answered isn’t how much the Earth weighs, it’s how massive the Earth is. This is a common misconception, that mass and weight are the same thing. They’re related, but separate, just how you and your parents share some of the same genes yet aren’t the same person. An object’s mass is just a measure of how much stuff it contains, its density multiplied by its volume. An object’s weight is the gravitational force pulling it toward the nearest massive object, usually the Earth. The mass of the Earth isn’t all that interesting to consider, even though it is interesting to think about how exactly we figured it out.

So now that we know 5.97E24kg isn’t how much the Earth weighs, what is? Here’s where things start to get fun. An object’s weight is just the force with which the Earth pulls on that object though gravity. But forces can only exist between two separate objects. By that logic, if we consider the Earth to be a single, solid object, it doesn’t weigh anything.

But that doesn’t really make sense, does it? It can’t weigh nothing. So what if we could take the Earth and put it on top of a big bathroom scale. If we then took the reading on that scale, wouldn’t that tell us how much the Earth weighs?

Believe it or not, you can do this yourself if you have a flat scale. All you need to do is flip it over and put it on the ground. You now have a scale with the world resting on top of it. So what does that scale read? Whatever the scale weighs. So if you’re using a 10 pound scale, the scale would read 10 pounds, a 15 pound scale would read 15 pounds and so on. Scales work by measuring the force used to compress their pressure-sensitive plates. If you flip the scale over, its own weight is the force pressing that plate down, so that’s the force it reads.

So if we take this definition to be our measurement of weight, then the Earth only weighs as much as whatever you’re using to measure it.

But that still can’t be right. Let’s assume we take all of the matter that makes up the Earth, duplicate it, and then put that on a scale. What would be the weight of all that material? Surely we can call this the weight of the world.

Sadly, this also gets us into some trouble, in that we need to consider exactly how big we make our pile of matter. The bigger the pile is, the farther away from the Earth the top of it would be. The farther away the top of the pile is from the Earth, the less it is pulled by the Earth’s gravity. The less gravity it feels, the less it weighs. If we assume our material is molded into a sphere the size of the Earth, then everything halfway up the pile would weight 4 times less than it would if it were at the bottom of the pile (gravity is an Inverse Square Law, so if we double the distance between two objects we reduce the force between them by a factor of 4). The material all the way at the top of the pile weighs 9 times less than it would on the bottom of the pile, since it’s three Earth-radii away from the Earth’s center as opposed to only one. So whatever our scale reads in this case will be much less than the Earth’s true weight.

We can fix this by taking that big ball of matter and compressing it to a size where we can measure it without noticing the changes in the Earth’s own gravitational field–say shrink it down to the size of a baseball (fortunately that’s still big enough that it doesn’t turn into a black hole and doom us all). If we plop this baseball with all the mass of the Earth crammed into it onto our scale, what would it read? What would be the weight of the world?

If you do the math: 5.85E25 Newtons or about 13,000,000,000,000,000,000,000,000 pounds.

With that out of the way, there’s really only one question left: Why am I even bothering to write about this?

The answer is two-fold. First, because this is my blog and I can write about whatever I want. But more importantly because these are the kinds of exercises that make teaching and learning and knowing physics worthwhile. In my opinion this is true physics. Notice I didn’t need to do a single calculation until the very end. Sure there are still equations, but they’re not nearly as important as the relationships they imply between the different variables. And this is what so often gets lost in the increasingly “plug and chug” nature of physics education. With both our kids’ worth as students and ours as teachers determined by the outcome of a single test at the end of the year, fewer and fewer teachers are willing to venture off and teach kids what they should be learning rather than simply what they “need” to know. To me, this is one of the most depressing things to happen to education in recent history.

So I’ll post here whatever I want to. Because no matter how they try, no one can ever stop you from learning.