# The Science Guys

### June 2002

How did mankind first determine the mass of the Earth, and what is the known value today?

Too bad we can’t place the Earth on a big bathroom scale! The Earth’s mass must be found by indirect means. The first person credited with "weighing" the Earth was Henry Cavendish (1731-1810.)

But we must first mention one of the greatest physicists who ever lived - Isaac Newton (1642-1727). Newton’s Law of Universal Gravitation states that any two masses are attracted to one another. This attractive force is directly proportional to the masses and inversely proportional to the square of the distance between them and is written F = G M m / d2, where F denotes the force of attraction, M and m are the masses, and d is the distance between the object’s centers. G is a constant that must be determined experimentally.

In 1798 Cavendish measured the force between attracting lead spheres with a torsion balance. He knew the masses of the spheres and how far apart they were. He carefully measured the force between them, which allowed him to calculate G. G is incredibly small, 0.0000000000667310 Nm2/kg2. (G is known only to the six decimal places.)

Assume you are mass m and the Earth is Me.. Your weight is known, and it’s the force F that goes in Newton's equation. We also know how far you are from the Earth’s center, d. Putting numbers in

you solve for the mass of the Earth and get 6,000,000,000,000,000,000,000,000 kilograms. (That’s 6 trillion-trillion kilograms!) Cavendish set out to find G, but he ended up finding the number by which we were able to "weigh" the Earth!

There is another method of finding the mass of the Earth using Newton’s physics. In Newton’s time man was able to measure the distance, d, to the Moon with geometry. It was also possible to time how long it took the Moon to go around the Earth (called the moon's period, T). Further, the Moon orbits around the earth in (approximately) a circle and the physics of circular motion is well-explained by Newton’s physics. Coupling what was known about circular motion to what was known about the gravitational force leads to a mathematical relationship for Earth’s mass: Me = 4 π2 d 3 / G T 2. This equation again reveals the critical importance of knowing G in finding the mass of the Earth and it yields 6 trillion-trillion kilograms once again.

Strictly speaking, we have found the mass of the Earth, not its weight. Weight corresponds to the strength of the pull of earth’s gravity on an object. A kilogram of mass weighs 2.2 pounds, so the Earth’s "weight" is a little over 13 trillion-trillion pounds!

Thus we have Isaac Newton and Henry Cavendish to thank for supplying us with the scales needed to weigh the Earth - those scales are Newton’s equations and the gravitational constant, G.