Derivation of the Law of Universal Gravitation

In the following we try to understand how Newton came up with his law by considering some basic principles and observations involving mass of an object depicted by small letter m, the mass of the Earth depicted by capital M and the moon:

• g gravitational acceleration independent of m (F_G=mg):

Galileo demonstrated that on Earth (mass M) all objects fall at the same rate (neglecting air resistance), i.e. the gravitational acceleration g is independent of m mass of an object, small letter m. Thus the gravitational force must be proportional to m and can be written as F_G=mg F G is equal to mass of an object multiplied by gravitational acceleration g(Newton's second law of motion). The gravitational acceleration g is independent of m but might depend on M mass of the Earthand r, i.e. g is a function of M capital M and r.

• g gravitational acceleration proportional to mass of the Earth (g~M):

The dependence of g on M capital M is hard to measure (except you are in a virtual lab where you can change the mass of the Earth M) but since Newton's third law of motion holds we know that the force the Earth (mass M) is exerting on the object with mass m must be of the same magnitude than the force the object (mass m) is exerting on the Earth (mass M). Thus the F_g F small g and also g must be proportional to M capital M.

• g gravitational acceleration proportional to inverse square of the distance:

Newton's great idea was to generalize the gravitational force and to apply the concept not only to objects on Earth but to all objects, including the moon.

He identified the gravitational acceleration as centripetal acceleration which keeps the moon in orbit and he was therefore able to estimate the gravitational acceleration at a distance (Earth - moon) which is roughly 60 times the radius of the Earth. Assuming a circular orbit the acceleration is approximately 3,600 times smaller than on the surface of the Earth. From that he 'guessed' the inverse square law.

Combining the considerations above and adding the gravitational constant G capital G we arrive at the expression for gravitational force F_G F capital G depicted on the right upper side of the figure 1.

Applying the law, Newton was able to calculate the orbits of planets and indeed found that the most general, bound trajectory (orbit) is an ellipse - in agreement with Kepler's first law of planetary motion. Additionally, Newton could show mathematically that all possible trajectories of an object in a gravitational field can be described by conic sections.

Figure 1: Newton assumed that not only the apple but also the moon should be attracted to the Earth. By deducing the gravitational acceleration of the moon, Newton came up with the inverse square law.

Vector form:

Until now we simplified our discussion and considered the magnitude of the gravitational force only. In vector form the gravitational force acting on mass m due to the attraction of mass M capital M is given by the equation on the lower right side of the figure 1, where r_mM is given by r_M-r_m and r_mM = |r_M-r_m|the distance between small m and capital M in bold is given by the difference between distance of capital M and distance small m in bold, and the distance between small m and capital M is given by the module of the difference between distance of capital M and distance small m in bold.