Grade 11 - Physics - Gravitation - Concept of Gravitation

Kepler’s Laws of planetary motion:

1. Law of Orbits: The orbits of the planets are elliptical with the sun at one focus.

2. Law of Area: A line joining from the sun to the planet sweeps out equal areas in equal time.

Explanation:If a planet P moves from A to B in a certain interval of time and again moves from A` to B` in the same interval of time, then according to Kepler second law the area ASB and A`SB` must be equal where S represents the position of the sun.

3. Law of Periods:
The squares of the periods of revolution of the planets are proportional to the cubes of the semi-major axis of the elliptical orbit is a, then

Where K is a constant. For circular orbits since a=b=r
Where r is the radius of the orbit.

Significance of Kepler Laws
The first law of Kepler clearly suggests that there must be some force of attraction between a planet and the Sun.The force of attraction between two material bodies is known as gravitational force.Now the question arises regarding the direction and magnitude of this gravitational force.The second law of Kepler gives information regarding the direction of the gravitational force that should act between the planet and sun.

Since, it follows that t=0.On the basis of Kepler’s second law it can be proved that the angular momentum L of the planet round the sun remains constant.In other words, The gravitational force acting on the planet is such that it has no perpendicular component i.e., it acts along the line joining the planet with the Sun. The third law of Kepler gives the information about the factors on which the magnitude of this force depends.
Let the mass of the planet m Revolving in an orbit of radius r And the velocity v, As the gravitational force must provide the centripetal force, it must be given by

Where T is the period of revolution. But according to Kepler’s third law, ,

therefore replacing

we get from the above relations

The gravitational force depends directly on the mass of the planet and inversely on the square of the distance between the planet and the sun.Since the force is mutual and acts on the sun as well, it is not unreasonable to suppose that is also proportional to the mass M of the sun. Therefore,

Where G is a constant.

Newton’s Law of Universal Gravitation
The universal law of gravitation states that-
Every particle in the universe attracts every other particle with a force which directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

If m1 and m2 are the masses of two particles separated by a distance r then the force of attraction F between them is given by

Where G is the universal gravitational constant.
In the equation, if m1=m2=1 kg and r=1 m then F=G and hence the universal gravitational constant is numerically equal to the force of attraction between two masses of 1 kg each placed at 1 m apart.

Note: Cavendish, in the year 1798, first successfully determined the value of G and its latest accurate value is-

In the language of vectors, the force exerted on m1 by m2 is given by

Things to remember:

(i) As G s very small, the gravitational force is a weak force.It becomes appreciable only when at least one of the particles is massive.
(ii) The gravitational force acts along the line joining the two interacting particles – it is an example of central force.
(iii) The gravitational force between a pair of particles is independent of the presence of other particles and properties of the intervening medium.
(iv) If there are more than two interacting particles, then the total force on any particle can be obtained by the principle of superposition.For example, the net force on particle 1 is the vector sum of the contributing forces i.e.,

In other words,