Gravitation Caltec S.6

GRAVITATION Kepler’s laws of planetary motion 1st law (law of elliptical paths) All planets describe ellipses about the sun as one focus. 2nd (Law of equal areas) the imaginary line joining the sun to the planet it sweeps out equal areas in equal time interval 3rd law (law of harmonic) The squares of the periodic times of revolutions of planets about the sun are directly proportional to the cubes of their distance of separation from the sun. i.e Newton’s law of gravitation The force of attraction between two bodies is directly proportional to the product of their masses and inversely proportional to the squares of their distance apart. Consider two bodies of masses M1 and M2 with distance r apart Where G is the universal gravitational constant G = S.1 units for g = = Dimensions of G = = = The other units of G are Proving Kepler’s 3rd law Suppose a planet of mass m kg moves with a constant speed V in a circle of radius r around the sun of mass M. assuming that the planet describes circular motion around the sun, gravitational attraction occurs between the sun and the planet. ………………………..(i) The centripetal force on the planet is; F = ……………..……(ii) Equating Eqn(i) and Eqn(ii) = …………………….. (iii) But period of revolution, T = T= V = …………………………….. (iv) Putting Eqn(iv) into Eqn(iii) = = Since are all constants Mass and density of the earth For a body of amass m, on the surface of the surface of the earth, the force of gravity acting on it is given by F= mg. The earth of mass M and radius re is assumed to be spherical and uniform thus its mass is concentrated at its centre. The force of attraction of the earth on the body …