Mechanics

# Gravitation:examples of gravitational force

Newton’s law of universal gravitation

Mass of earth derivation

Gravitational field strength

## What is Gravitation (force of gravity)?

Gravitation or gravity is defined as:”The force of attraction between any two objects in the universe.”

History:

From at least the time of the ancient Greeks,two problems were the subjects of searching inquiry:

1. The tendency of objects such as stones to  fall to earth when released.
2. The motions of the planets,including the sun and the moon,which were classified with the planets in those times.

In early days these problems were thought of as completely separate.One of newton’s great achievements is that he saw them clearly as aspect of a single problem and subject to the same laws.

Geocentric theory:

The earliest serious attempts to explain the kinematics of the solar system were made by ancient Greeks. Ptolemy (Claudius ptolemaeus, 2nd century AD ) developed a geocentric (earth centered )  scheme for the solar system in which ,as the name implies,the earth remains stationary at the center while the planets,including the sun and the moon,revolve around it.This should not be a surprising deduction.The earth seems to us to be a substantial body. Shakespeare referred to it as “this goodly friend,the earth…”even today,in navigational astronomy we use a geocentric reference frame,and in ordinary conversations we use terms such as “sunrise” which implies such a frame.

Because simple circular orbits cannot account for the complicated motions of the planets ,Ptolemy had to use the concept of epicycles,in which a planet moves around a circle whose center moves around another circle centered on the earth.

Heliocentric theory:

In the 16th century Nicolas Copernicus (1477 – 1543) proposed a heliocentric scheme,in which the earth (along with other planets) moves about the sun.Even though the Copernican scheme much simpler than Ptolemy,it was not immediately accepted.Copernicus still believed in the sanctity of circles.And his use of epicycles and other arrangements was about as great as that of Ptolemy. However,by putting the sun at the center of things ,Copernicus provided the correct reference frame ,from which our modern view of the solar system could develop.

To resolve the conflict between the Copernican and Ptolemaic schemes ,more accurate observational data were needed.Such data were compiled by Tycho Brahe (1546-1601) who was the last great astronomer to make observations without the use of telescope.His data on planetary notions were analyzed and interpreted by Johannes Kepler (1571 -1630),who had been Brahe’s assistant.Kepler found important regularities in the motion of the planets,which lead him to develop three laws that govern the motion of the planets.

Kepler’s laws showed the great simplicity with which planetary motion could be described when the sun was taken as the central body,if we give up the notion of perfect circles on which both the Ptolemaic and Copernican systems were based.However ,Kepler’s laws were empirical ,they simply described the observed motions of the planets without any basis in terms of forces.It was therefore a great triumph when newton was later able to derive Kepler’s laws, from his laws of motion and his law of gravitation,which specified the force that acts between each planet and the sun.

In this way newton was able to account for the motion of the planets in the solar system and of bodies falling near the surface of the earth with one common concept.He thereby unified into one theory the previously separate sciences of terrestrial mechanics and celestial mechanics.The real scientific significance of Copernicus work lies in the fact that the heliocentric theory opened the way for this synthesis.Subsequently ,on the assumption that the earth rotates and revolves about the sun.It become possible to explain such diverse phenomenon as the daily and the annual apparent motion of the stars,the flattening of the earth from a spherical shape the behavior of the trade winds ,and many other observations that could not have been explained to easily in a geocentric theory.

The historical development of gravitational theory can be viewed as a model example of the way the method of scientific inquiry leads to insight.Copernicus provided and Brahe supplied systematic and precise experimental data.Kepler used the data to proposed the some empirical laws,and newton proposed a universal force law from which Kepler’s laws could be derived. Finally,Einstein was led to a new theory which explained certain small discrepancies in the Newtonian theory.

There are three overlapping realms in which we can discuss gravitation.

1. The gravitational attraction between two bowling balls,for example,although measurable by sensitive techniques,is too weak to fall within our ordinary sense perceptions.
2. The attraction of ourselves and objects around us by the earth is a controlling feature of our lives from which we can escape only by extreme measures.The designers of our space program have the gravitational force constantly in mind.
3. On the scale of the solar system and of the interaction of stars and galaxies ,gravitation is by far the dominant force.It is remarkable that all three situations can be described by the same force law.

The first man who came up with the idea of gravity was Isaac newton.It was an evening of 1665  when he was trying to solve the mystery why planets revolve around the sun.Suddenly an apple fell from the tree under which he was sitting.The idea of gravity flashed in his mind.He discovered not only the cause of falling apple but also the cause that makes the planets to revolve around the sun and the moon around the earth.This deals with the concepts related to gravitation.

### The force of gravitation:

On the basis of his observations,newton concluded that the force which keeps the moon in its orbit are of the same nature.He further concluded that there exists a force due to which everybody of the universe attracts every other body.He named this force the gravitation.

## Gravitational potential energy equation

The gravitational potential is defined as”potential energy per unit test mass.”

The gravitational potential V at a point is given by:

From the above equation (3) ,it is clear that potential V (r) is independent of the value of the test mass m0.