Newton knew that the force of gravity causes falling objects near the Earth’s surface (such as the famous apple) to accelerate toward the Earth at a rate of 9.8 m/s2. He also knew that the Moon accelerated toward the Earth at a rate of 0.00272 m/s2. If it was the same force that was acting in both instances, Newton had to come up with a plausible explanation for the fact that the acceleration of the Moon was so much less than that of the apple. What characteristic of the force of gravity caused the more distant Moon’s rate of acceleration to be a mere 1/3600th of the acceleration of the apple?

It seemed obvious that the force of gravity was weakened by distance. But what was the formula for determining it? An object near the Earth’s surface is approximately 60 times closer to the center of the Earth than the Moon is. It is roughly 6,350, km from the surface to the center of the Earth and the Moon orbits at a distance of 384,000, \text{km}$ from the Earth. The Moon experiences a force of gravity that is 1/3600 that of the apple. Newton realized that the force of gravity follows an inverse square law \(6,350 \times 60 \approx 384,000\)).

In 1798, by careful experiment, Henry Cavendish succeeded in making an accurate determination of *G*, the gravitational constant, as \(6.67 \times 10^{-11}\). This meant that the mass of the Earth could now be determined. A 1-kg mass at the Earth’s surface is approximately 6.3 Mm from the center of the Earth, and the force acting on it is approximately 10 N. So, by using these values into the gravity equation, we can find that the mass of the Earth is roughly \(6 \times 10^{24} , \text{kg}\).

See Cavendish, H., Experiments to Determine the Density of the Earth. Philosophical Transactions of the Royal Society of London, 1798.