 # How do they know the Earth's weight?

In a recent column, Cecil mentioned that the mass of the earth was “reliably estimated at 6 sextillion, 588 quintillion tons”.

How did they figger that out? Like how you weigh your dog; first step on the scale holding it, then without, then subtract??

It’s calculated backward from the laws of gravitation.

Newton figured out that, if you know the mass of two objects and the distance between their centers, you can determine the gravitational force. Taking it the other way around, if you know the distance and the force, and the mass of one object, you can solve for the mass of the other.

We can measure how fast an object falls when dropped on Earth’s surface, and we can measure the radius of the planet. So it’s fairly simple to work out the mass of the Earth.

If you want some more details, I can write out all the equations and numbers and everything.

I’m not a warlock. I’m a witch with a Y chromosome.

First, you’ve got to figure out how strong a gravitational field the Earth generates. This is the easy part - just calculate the distance to the Moon, and also its orbital period. Period is one month. Distance is a bit tricky, but you can probably do it with a sort of triangulation. These days you’d use radar or laser. The orbital period depends only on gravitational field and radius of the orbit, so you can calculate the gravitational field.

Note that you do NOT need to know the mass of the Moon for this. It should be easy to imagine that if you split the moon in half and left them next to each other, they would not suddenly start orbiting around the Earth any faster or slower.

OK, so now you’ve measured the gravitational field of the Earth. The next question is, how much mass does it take to generate this field? The conversion factor you need here is the gravitational constant.

This number is rather tricky to measure. You’ve got to take two objects of known masses and measure the gravitational force between them.
(or possibly one object of known mass, and a second object that is much ligher than the first) You can’t use Earth as one of the objects, because you don’t know the mass of the Earth yet. You have to take two big weights into your lab and do your best to measure the almost infinitesimal force between them.

You don’t need to screw around with laser triangulation or any of that crap. In fact, tracking the lunar orbit is a hideously over-complicated way of measuring gravity.

Drop a golf ball from a given height (s), and measure the time (t) it takes to hit the ground. Those two numbers are all you need to calculate the acceleration due to gravity.

I’m not a warlock. I’m a witch with a Y chromosome.

Bewary of circular reasoning. If you calculated the acceleration due to gravity based on the falling times for dropping a golf ball, then you can’t go backwards and calculate the mass of the earth based on the gravitational constant you derived from the golf ball drop. That’s circular, methinks.

This is true. Dropping an object only tells you the acceleration due to gravity, which is nice but not sufficient.

To find the value of the gravitational constant, scr4 is correct about the experiment that needs to be performed. Take two massive objects, put them in as close to a frictionless environment as you can get, and measure the teensy attractive force between them.

I’m not a warlock. I’m a witch with a Y chromosome.

I think AuraSeer and I agree, but to make it clear:

As I said, measuring the mass of the Earth is a two-step process: 1) measure the gravitational field, and 2) know how to convert the field strength to mass.

To measure the gravitational field, you need the field strength at some point, and also know how far that point is from the center of the Earth. One method is to measure the acceleration of a falling object on the surface of Earth, and also measuring the radius of the Earth. Another method is to measure the orbital radius and period of the Moon. I do concede that the first method is probably easier.

To measure the conversion factor of gravitational field to mass, you need to do some elaborate experiments as I said in my first message.

Hope this is a bit more clear…

If we agreed any louder, we’d be fighting.

“Hey, buddy! Disagree with me right now or I’ll kick yer ass!” They used to two lead balls on either end of a stick hung balanced from a very thin wire to measure the gravitational constant. Next to each of the hanging lead balls was placed a large lead ball and the gravitational force between the lead balls was measured by the torsin of the wire.

Actually, Headless, a whole sloo of scientists are using a whole sloo of different techniques to figure out G. It’s one of the harder physical constants to compute.

AFAIK, any Cavendish-like experiment designed to determine the value of G should ideally be done in an environment where gravity only acted between the balls and not the wire, the container, etc… This enivornment doesn’t exist. Which is why G is a relatively imprecise value. Physicists know the charge of an electron to seven decimal places, G is only known to three.

Wow. Cool. A whole buncha smart guys politely dukin’ it out over my question. Thanks for the answer(s), and feel free at any point to use colorful language.

Aura, no need to write out the figures for me. A) I wouldn’t want you to go through the trouble, and B) I wouldn’t understand them anyway.

-nmd

Any a you’se care to post a link to the column?

I think this is the column being referred to.

Actually, it was the column: “Is the Earth Getting Lighter?” 23rd June.

nmd

That would be
http://www.straightdope.com/classics/a3_355.html