Sir Isaac Newton

Newton's Law of gravitation

Sir Isaac Newton (1642-1727) was a very sickly boy, and was very small at birth. He attended Trinity College at Cambridge University. In 1666 an outbreak of Bubonic Plague forced the University to shut down. So Newton returned to his birthplace to study on his own and do research. This period in Newton's life is considered his most productive, for he invented differential calculus and made discoveries about light and gravitation. He nearly blinded himself by staring at the image of the sun in a looking glass.
      Though it's been legendary that Newton was struck by the falling of an apple, he did wonder if the force acting on the moon was the same force of falling objects on the earth. Five years after returning to Trinity, his professor turned over the chair of the mathematics department to Newton.
      When Edmund Halley requested his advice on the elliptical orbits in Kepler's Laws and the nature of the force between the sun and planets, he was surprised to find that Newton had solved the problem in exact detail. Newton admitted that he had solved the problem during his time away from Trinity with the help of Kepler's Laws. Newton, however, had misplaced his notes and could not find them for Halley immediately. With Halley's support Newton published the Philosophiae Naturalis Principa Mathematica or Principa as it is often referred in 1687.
      In Principa Newton defines mass, velocity, and acceleration and his three laws of motion. From this and Kepler's Laws he attacked the problem of the planets devising the law of gravitation. He knew that the moon should fly away from the earth but somehow was held in orbit around the earth. Just like someone swinging something tied to a rope over their head. But there was nothing connecting the moon to the earth. He showed that the force that causes kepler's elliptical orbits is a central force, directed to the center of motion. He also demonstrated that planets under the influence of this central force follow Kepler's second law. The gravitational law means all masses attract all other masses but there is no repulsion as in magnetism. And it is directly proportional to their masses & inversely proportional to the square of their distance.

      Newton used the thought experiment shown in the figures. Suppose we fire a cannon horizontally from a high mountain; the projectile will eventually fall to earth, as indicated by the shortest trajectory in the figure, because of the gravitational force directed toward the center of the Earth and the associated acceleration. But as we increase the muzzle velocity for our imaginary cannon, the projectile will travel further and further before returning to earth. Finally, Newton reasoned that if the cannon projected the cannon ball with exactly the right velocity, the projectile would travel completely around the Earth, always falling in the gravitational field but never reaching the Earth, which is curving away at the same rate that the projectile falls. That is, the cannon ball would have been put into orbit around the Earth. Newton concluded that the orbit of the Moon was of exactly the same nature: the Moon continuously "fell" in its path around the Earth because of the acceleration due to gravity, thus producing its orbit.

      Utilizing Kepler's third law he derived the law of gravitation.

Newton realized that the moon travels in almost a circular orbit so the equation of force must be related to centripetal force which can be written in this form.


He also knew that the velocity could be expressed in the form of circular variables.



Substituting in for the velocity,



Newton then took the equation for Kepler's Third Law,



and substituted for the square of the period.



Then he combined the constants into one called k'.


Newton further realized that the mass of the sun must also be a factor, so he adjusted his constant k' to allow for the mass of the sun.


Newton further generalized his theory to apply to any two masses.










      Also by combining gravitational constant G, the mass of the sun, and the distance squared into one constant, you have the familiar form of Newton's Second Law for objects on the surface of the earth.
















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