We can predict g if we know the earth’s mass. But how do we know the value of $M_{\text{e}}?$ We cannot place the earth on a giant pan balance, so how is its mass known? Furthermore, how do we know the value of G?

Newton did not know the value of G. He could say that the gravitational force is proportional to the product $m_1{m}_2$ and inversely proportional to $r^2$, but he had no means of knowing the value of the proportionality constant.

Determining G requires a direct measurement of the gravitational force between two known masses at a known separation. The small size of the gravitational force between ordinary-size objects makes this quite a feat. Yet the English scientist Henry Cavendish came up with an ingenious way of doing so with a device called a torsion balance. Two fairly small masses m, typically about 10 g, are placed on the ends of a lightweight rod. The rod is hung from a thin fiber, as shown in FIGURE 13.8a, and allowed to reach equilibrium.

FIGURE 13.8 Cavendish’s experiment to measure G.

If the rod is then rotated slightly and released, a restoring force will return it to equilibrium. This is analogous to displacing a spring from equilibrium, and in fact the restoring force and the angle of displacement obey a version of Hooke’s law: $F_{\text{restore}}=k\text{Δ}\theta$. The “torsion constant” k can be determined by timing the period of oscillations. Once k is known, a force that twists the rod slightly away from equilibrium can be measured by the product $k\text{Δ}\theta$. It is possible to measure very small angular deflections, so this device can be used to determine very small forces.

Two larger masses M (typically lead spheres with $M\approx 10\text{kg}$) are then brought close to the torsion balance, as shown in FIGURE 13.8b. The gravitational attraction that they exert on the smaller hanging masses causes a very small but measurable twisting of the balance, enough to measure $F_{M\text{on }m}$. Because m, M, and r are all known, Cavendish was able to determine G from

His first results were not highly accurate, but improvements over the years in this and similar experiments have produced the value of G accepted today.

With an independently determined value of G, we can return to Equation 13.7 to find

We have weighed the earth! The value of $g_{\text{earth}}$ at the earth’s surface is known with great accuracy from kinematics experiments. The earth’s radius $R_{\text{e}}$ is determined by surveying techniques. Combining our knowledge from these very different measurements has given us a way to determine the mass of the earth.

The gravitational constant G is what we call a universal constant. Its value establishes the strength of one of the fundamental forces of nature. As far as we know, the gravitational force between two masses would be the same anywhere in the universe. Universal constants tell us something about the most basic and fundamental properties of nature. You will soon meet other universal constants.