If m1 stands for the mass of one body, m2 for the mass of a second body, d12 for the distance between their centers, F for the mutual force of gravity between them, and G for the constant of gravitation, then the law of gravitation can be stated mathematically as
where the gravitational constant G equals 6.67 x 10-8 centimeter cubed per gram second squared (cm3/g*s2). The gravitation constant G was first measured in the eighteenth century by Henry Cavendish (1731-1810).
Although historical evidence does not appear to support the contention, Galileo is alleged to have demonstrated by dropping balls of different weights from the Leaning Tower of Pisa that they fell with constant acceleration. Why should they have? From the second law of motion, we know that a body of mass m subjected to the Earth's gravitational force of attraction F undergoes an acceleration at the Earth's surface of g = F/m. From the law of gravitation, this force is F = GmM/R, where M is the mass of the Earth and R is the separation between the centers of the two bodies, or the Earth's radius. Assuming that we know G, we have
or
where the mass of the attracted body has canceled out. Thus the acceleration of the attracted body does not depend on its own mass; it depends on the mass of the attracting body, which in this case is the Earth.
Example: The observed acceleration g at Earth's surface is 980 cm/s2. From the known values of the gravitational constant G and the Earth's radius (6.38 x 108 cm), we can find the mass of the Earth by rearranging the preceding equation:
or
And from this the Earth's mean density is