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Newton’s law of universal gravitation formula

What is Gravitation?

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.

Newton’s law of gravitation:

Newton’s law of gravitation can be stated as:

“Everybody in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.”

Explanation:Newton's law of gravitation

Consider two bodies of masses m1 and m2 .The distance between the centers of masses is r.According to the law of gravitation ,the gravitational force of attraction F with which the two masses m1 and m2 separated by a distance r attract each other is given by:

DERIVATION OF NEWTON'S GRAVITATION LAW

Here G is the proportionality constant.It is called the universal constant of gravitation.Its value is same everywhere.In SI units its value is 6.673 × 10-11 Nm2kg-2. The universal constant G must not be confused with the g that is the acceleration of a body arising from the earth’s gravity.The constant G has the dimensions L3/MT2 and is a scalar quantity,While g is the magnitude of a vector ,has the dimensions L/T2,and is neither universal nor constant.Due to small value of G ,the gravitational force of attraction between objects around us is very small and we do not feel it.Since the mass of earth is very large,it attracts nearby objects with a significant force.The weight of an object on the earth is the result of gravitational force of attraction between the earth and the object.

It is important to note that the gravitational forces between two particles are an action -reaction pair.The first particle exerts a force on the second particle that is directed toward the first particle along the line joining them.Likewise,the second particle exerts a force on the first particle that is directed toward the second particle along the line joining them.These forces are equal in magnitude but opposite in direction.

What is the mass of  earth?
diagram of mass of earth

Consider a body of mass m on the surface of the earth.Let the mass of the earth be  M and radius of the earth be R,The distance of the body from the center of the earth will also be equal to the radius R of the earth.According to the law of gravitation,the gravitational force F of the earth acting on a body is given by:derivation of mass of earth

  =6 × 1024 kg

Thus mass of the earth is 6 × 1024 kg.

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.

What is Gravitational field?

“The space around the earth in which earth can exert a force of attraction on other bodies is called gravitational field.”A basic fact of the gravitation is that to particles exerts forces on one another.This is a direct interaction between the two particles and is called action at a distance because the two particles interact even though they are not in contact.

Another point of view is the field concept.The gravitational field is a force field.The gravitational field exists around a particle.Whenever another particle enters in the gravitational field ,it will experience an attractive force.The field therefore plays an important role in our thinking about the forces between particles.According to this view we  first determine the gravitation field established by the given distribution of particles.Secondly we calculate the gravitational field exerted on another particle placed in its field.When a body is placed in the vicinity of earth,it will experience an attractive force equal to the weight of the body m0 g. The  direction of the gravitational force is towards the center of the earth.The gravitational field becomes weaker and weaker as we go farther and farther away from the earth.

The gravitational field is an example of a vector field ,each point in this field having a vector associated with it.There are also scalar fields,such as temperature field in a heat conducting solid.The gravitational field arising from a fixed distribution of matter is also an example of a static field,because the value of the field at a given point does not change with time.

What is Gravitational field strength?

The gravitational field strength at a point is defined as ” Gravitational force per unit mass at that point.”At any place its value is equal to the value of g at that point.Near the surface of the earth,the gravitational field strength is 10 N kg-1. Its formula is given as:

g = F/mo

Where mis the test mass.By moving the test mass to various positions,we cam make a map showing the gravitational field at any point in space.We can then find the force on a particle at any point in that field by multiplying the mass m of the particle by the value of the gravitational field g at that point:

F = mg

Gravitational potential:

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

The gravitational potential V at a point is given by:

gravitational potential

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

See also:

  • Satellites

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