The mountain that weighed the Earth

Summary

Suppose you want to find out how much you weigh. It’s easy enough – get a scale, stand on it, and read the number. Now suppose you want to know how much the Earth weighs. You get out a scale and…hmm. What exactly do you do with it? Let’s do a little physics class refresher. When a scale measures weight, that weight is a force, specifically the force of the gravitational attraction between what’s on top of the scale and what’s beneath it. If you stand on the scale and the scale is on the floor, it’s the attraction between you and the Earth. Isaac Newton’s law of universal gravitation tells us that gravitational attraction depends on the mass of the two objects and the distance between them. The mass of the Earth is constant, and the distance from the scale to the center of the Earth is constant. So the only thing that affects the scale’s reading is your own mass – the amount of matter in your body, independent of Earth’s gravity. Isaac Newton’s 3rd law of motion conveniently tells us that every action has an equal and opposite reaction. So if you weigh 180 pounds on the Earth, the Earth also weighs 180 pounds on you. Turn your bathroom scale upside down and try it! As technically correct as this answer for the Earth’s weight is, it’s not super useful for finding the Earth’s mass – at least not by itself. But it could be a good first step. All we need is to know how much something else weighs on you to compare to the Earth. So here’s what we need to know to find the mass the Earth: How much the Earth weighs on you Your distance to the center of the Earth How much some other object weighs on you The distance between you and that object That object’s total mass Some math to compare the results Number 1 is easy to find – just use your upside-down scale. Number 2 is doable as well – we’ve known the radius of the Earth for thousands of years. Number 4 can be found out with a ruler so long as the object is nearby and stays in one place, and number 6 can be done with the mat...