NEWTON DERIVES THE LAW OF GRAVITATION
The recognition that the earth exerts a pull on a body was not original with Newton, for Galileo recognized that the earth exerts a force on a falling body. It was Newton, however, who recognized that the pull of the earth can extend all the way to infinity. Such a pull is a universal phenomenon to which all material bodies are subject.
By analyzing Kepler's second law mathematically, Newton showed that the force acting on a planet must be one directed toward the sun. Only one kind of force would satisfy Kepler's requirement that the sun be at the focus of the ellipse and still be consistent with Kepler's third law, relating the planets' periods to their distances from the sun. The force between the planets and the sun must then be an inverse-square force; that is, the intensity of the force must weaken as the square of the distance between a planet and the sun increases. Several contemporaries of Newton had suspected this relationship, but they could not prove it.
Newton then took into account his third law of motion and assembled his results in one comprehensive statement, the law of gravitation. He showed that it is universal by applying it to a falling apple and the
earth, to the moon's motion around the earth, and to the planets revolving around the sun. He even imagined gravitation at work beyond the solar system, a thought that was verified later.
NEWTON'S LAW OF GRAVITATION: Objects in the universe attract each other with a force that varies directly as the product of their masses and inversely as the square of their distances from each other .
Newton proved that spherical bodies act as if their gravitational mass is concentrated at their centers.
This simplifies the mathematical treatment of such bodies: the distance between their centers is ordinarily used in calculating their mutual gravitational attractions
Orbits Under The Law of Gravitation
Newton used his laws of motion and gravitation to that Kepler's third law was only an approximation to the actual relation. The actual form of the 5 much more useful, for it allows us to determine the masses of other celestial bodies. Newton also showed that the orbit of a body ing around a central force always matches in one of the class of curves called conic sections.
Thsese curves are called conic sections because are formed when we pass a plane (like a knife blade) through a cone at different angles. For the ellipse, of which a planetary orbit is one example, the plane intersects opposite sides of the cone's slant edge. For a circle the plane cuts the cone at right angles to the vertical axis. The other two conic sec- are open at one end: The parabola is formed when the plane passes through the cone parallel to its slant edgei and the hyperbola is formed when the cone is intersected at an angle between that for the parabola and parallel to the vertical axis.
In a parabolic or hyperbolic orbit the body will pass by the attractive central force only once, approaching from and receding toward infinity, never to return.
The different members of the sun's family move around the sun in closed paths, either ellipses or something approaching a circle. An object approaching the sun from outside the solar system would, when attracted by the sun, travel by it in a parabolic or hyperbolic orbit. If its motion is significantly influenced by the gravitational attraction of a planet during a near encounter with one such as Jupiter, it might be forced into an elliptical orbit around the sun, in which case we say it has been "gravitationally captured."
