Gravitation - Kepler and Newton revision notes and calculations.

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               Gravitation

Kepler (1571-1630) has studied for many years the records of observation on planets and summarised three laws.

Kepler’s Law

  1. Each planet moves in an ellipse which has the sun at one focus
  2. The line joining the sun and the moving planet sweeps out equal area in equal time
  3. the square of the time of revolution of any planet (i.e. T) about the sun is proportional to the cube of the planets’ mean distance from the sun

                                       

I.e.   is a constant

Interpretation form Kepler’s laws

  1. Kepler’s second law :

        The area swept out in a very short time interval (Δt), neglecting the    

        small triangular region is A

                         

         The rate of area swept =

         Hence it is a constant.  

         Compare this equation with the angular momentum

                                          = Constant

         This law is in fact an evidence of conservation of angular momentum.

  1. Kepler’s third law

       About 1666, Newton investigated the motion of the moon, and thought that it was the force of gravity to pull the moon and keep it in its orbit

                                                     

                                                       

NB: Time between full moon is 29.5 days but this due to the earth also moving round the sun, the moon is therefore to travel a bit longer

                   

                   

          At the earth surface g=9.81m . This is due to the fact that we are nearer the earth centre than the moon (about 1:60)

    Newton wanted to find out a relation between the distance and the gravitational force   (of acceleration due to gravity g)

 

                                   First trial:

                                   Second trial:

He found that:

                                   

Newton’s Law of universal gravitation

   Every particle of matter in the universe attract every other particle with a force which is directly proportional to the product of this masses and inversely proportional to the square of their distance apart

                                    And              

                               ➔            

G is defined as the gravitational constant = 6.7

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Relation between Newton’s Law of gravitation and Kepler’s Law

                         

                       &    

    ➔          

                     

                  =Constant  

                           

E.g. Given that the period of revolution of ...

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