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08%20-%20Gravitation - 8 GRAVITATION Page 1 Introduction...

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8 - GRAVITATION Page 1 Introduction Ptolemy, in second century, gave geo-centric theory of planetary motion in which the Earth is considered stationary at the centre of the universe and all the stars and the planets including the Sun revolving round it. Nicolaus Copernicus, in sixteenth century, gave helio-centric theory in which the Sun is fixed at the centre of the universe and all the planets moved in perfect circles around it. Tycho Brahe had collected a lot of data on the motion of planets but died before analyzing them. Johannes Kepler analyzed Brahe’s data and gave three laws of planetary motion known as Kepler’s laws. 8.1 Kepler’s Laws First Law: “The orbits of planets are elliptical with the Sun at one of their two foci.” Second Law: “The area swept by a line, joining the Sun to a planet, per unit time ( known as areal velocity of the planet ) is constant.” Third Law: “The square of the periodic time ( T ) of any planet is directly proportional to the cube of the semi-major axis ( a ) of its elliptical orbit. 8.2 Newton’s universal law of gravitation “Every particle in the universe attracts towards it every other particle with a force directly proportional to the product of their masses and inversely proportional to the distance between them. This is the statement of Newton’s universal law of gravitation. Two particles of masses m 1 and m 2 having position vectors 1 r and 2 r respectively are shown in the figure. By Newton’s law of gravitation, the force exerted on particle 1 by particle 2 is given by 12 F = G ^ 12 2 2 1 r r m m , where ^ 12 r = r r 12 12 l l = r r r 1 2 - , where r = distance between the particles and G = universal constant of gravitation.
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8 - GRAVITATION Page 2 S I unit of G is Nm 2 /kg 2 and its dimensional formula is M - 1 L 3 T - 2 . Its value is the same everywhere in the universe at all times and is 6.673 × 10 - 11 Nm 2 /kg 2 . It was Cavendish who first determined its value experimentally. The force exerted on particle 2 by particle 1, 21 F , is the same in magnitude but opposite in direction to 12 F . Thus, 21 F = - G ^ 12 2 2 1 r r m m . The forces 12 F and 21 F are as shown in the figure. 8.3 Gravitational acceleration and variations in it The acceleration of a body produced by the gravitational force of the Earth is denoted by g. The gravitational force, F, exerted by the Earth having mass, M e and radius R e , on an object having mass m and situated at a distance r ( r Re ) from the centre of the Earth, is F = G 2 e r mM m F = g = 2 e r GM ( 1 ) For an object on the surface of the
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