# Newton

## Universal gravitation (1687)

### The gravitation of Newton unifies knowledge on the laws of motion.

*Isaac Newton (1642-1727)*

Newton knew well Galileo's works, especially those related to inertia. He is known as the one who stated the law of universal gravitation. Simultaneously valid for the sublunary world that surrounds us and for the world of stars, this law gave gravitation a major philosophical impact owing to its universal character. The law of gravitation was published in 1687 in "Mathematical Principles of Natural Philosophy". It stipulates that:

- All bodies attract each other by a force pointing along the line intersecting both bodies.
- This force is proportional to the product of the two gravitational masses of the bodies and inversely proportional to the square of the distance between them.

*The gravitational force which the first body exerts on the second is the same one as that which the second body exerts on the first.*

## A "manual" principle (1687)

### To take into account the universality of free fall which he had observed, Newton introduced "by hand" the Equivalence Principle in his equations.

The concept of gravitational mass (mg) was introduced in Newton's theory of gravitation. When he established his mechanics, he followed Galileo and introduced inertia, which means the capability of a body to maintain a state of uniform motion. For a given material, this inertia (or inertial mass), is proportional to the quantity of matter. The inertial mass (mi) has been defined precisely by Newton when he wrote that the force is the product of inertial mass times acceleration: F = mi × a.

A priori, these two masses have no relation, since the gravitational mass means the faculty of being attracted by another gravitational mass. Let us suppose that the considered force is gravitational. Then we have the relation:

mi × a = mg × g where g is the field of gravity.

The acceleration of a body is thus : a = g × mg / mi.

If the ratio of these two masses does not depend on the composition, the acceleration (and thus the movement) of two bodies is identical. As observed from the experiments carried out by Galileo and Newton, the fall of two bodies is identical. Therefore, the ratio of the two masses is a constant (in the system of unit used, this constant is selected as unit).

Thus, Newton introduced "by hand" the principle of equivalence in his laws by stating the equality between gravitational and inertial masses. To understand the mysterious origin of this equality, Newton tried to check the universality of free fall, especially by improving the pendulum method invented by Galileo.

*The law of equal areas, established by Kepler in 1609, is a consequence of the law of universal gravitation. According to this law, while orbiting around the Sun, any planet sweeps an area which is directly proportional to the duration. For very elliptical orbits, as here, the consequence is a significant acceleration of planetary motion when it is close to the Sun.*

## Newton's Pendulums (around 1680)

### Newton improves the test of the Equivalence Principle thanks the pendulum experiments.

Using the innovations brought by Galileo, Newton examined in detail the effects of air friction on motion. It showed that this resistance is proportional to the squared speed (which is the relation that we still know today). He built two identical pendulums. One was made out of wood and served as reference, while the other was made of different materials. Starting the pendulums simultaneously, he measured the time after which one could detect a difference in their motion. Thanks to careful experiments and by taking into account the effect of air resistance, Newton showed that the motion of a pendulum is independent of the material with a relative accuracy of 10^{-3}.

The test of the Equivalence Principle by pendulum experiments has been improved over the centuries. Friedrich Bessel checked it in 1830 with a relative precision of 10^{-5} and, in 1923, Potter even achieved a relative accuracy of 10^{-6}. However, at that time, this test was supplanted by another method which is still used today–the torsion pendulum.

*Friedrich Bessel*