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How can the weight of Earth be determined?

Michael Wysession, an associate professor of earth and planetary sciences at Washington University, explains.

There is an easy answer to this question, and it involves some good news and some bad news. Imagine this scenario: step on the scale in your bathroom and weigh yourself. (Hopefully this isn't bad news.) Now suppose you take the scale, travel to the moon, and stand on it again. What will you weigh there? The new number will be about 1/6 of what you weighed on Earth. Finally, imagine traveling out into deep space and weighing yourself once more. You will weigh nothing. (This is the good news.) Your weight is variable because weight is a force that depends on something pulling on you. Specifically, it is the force of gravity, which depends on the mass of the object that is attracting you. If you pushed Earth out into deep space, it, too, would weigh nothing. (So the bad news is that you would weigh as much as Earth.)

That said, I think that this reader was really asking how the mass of Earth can be determined. This is a bit more complicated, though only slightly so. What is more, unlike the previous question, it does have a straightforward answer, and you can still get the answer using your bathroom scale. As is often the case in physics, fairly complicated things can be described very well with a simple equation. In the case of gravitational attraction, the equation is as follows:

The value G is called the gravitational constant, and it has been ascertained through many decades of careful experiments. We know how far a person standing on the surface is from the planet¿s center (about 6,371 kilometers), so all we need to know is his mass, and then we can calculate Earth's mass. Finding a person¿s mass will only involve counting all of the atoms in his body. (But don¿t forget to take into account that burrito he ate last night, too.) This is quickly turning out to be a complicated problem. As an alternative, perhaps we could use a block of some material for which it is easier to make an estimate of the number of atoms it contained. Maybe, but this is not the sort of experiment you can carry out in your bathroom.

Fortunately, a bathroom scale can still aid in solving this problem (albeit in an unexpected way) by using a simpler equation. It turns out that the rate at which an object accelerates due to the force of gravity, called "g," depends of the mass of the object doing the pulling. In the case of Earth, we have:

So, you can open your bathroom window, hurl your scale out the window, and count how many seconds it takes to hit the sidewalk. Then measure the distance from your window to the ground, and you can compute the acceleration of the scale. The answer you will get is 9.8 m s-2. Knowing this value of g for Earth's surface, along with the constant G and the 6,731-kilometer distance to Earth's center, you can then calculate Earth's mass to be 6 x 1024 kilograms. (You also won't be bothered by bad news from your scale anymore.)

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