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Unformatted text preview: Unit 9 Gravitation 9.1 Universal Gravitation 9.2 Gravitational field 9.3 Gravitational potential energy 9.4 Satellite orbits 9.5 Launching a satellite 9.6 Mechanical energy of a satellite 9.7 Kepler’s laws 9.1 Gravitation Newton’s law of universal gravitation states that the gravitational force F between any two bodies of mass m 1 and m 2 separated by a distance r , is described by 2 2 1 r m m F ∝ . 1 m 2 m r F F It is a center-to-center attraction between all forms of matter. The force of gravity between any two bodies varies directly in proportion to the product of their masses and inversely with their separation squared. F The proportional constant is referred to as the universal gravitation constant G . Its value is G = 6.67 × 10-11 Nm 2 kg-2 . Now we can write 2 2 1 r m m G F = r Example Twin asteroids X and Y , having the same mass ( M = 3.5 × 10 18 kg) are located 3.00 km apart. Find the net gravitational force of the spaceship when it is located at positions A and B , as shown in figure. Given that the mass of the spaceship is m = 2.50 × 10 7 kg. 1 Answer: 1.50 km 1.50 km B 3.00 km m M M A θ θ Y X When the spaceship is at A : The distance AX = AY = (3000 2 +1500 2 ) 1/2 = 3350 m. The angle o 1 6 . 26 ) 3000 1500 ( tan = = − θ . The attractive force between the spaceship and asteroid X is 2 AX mM G F = along AX . The attractive force between the spaceship and asteroid Y is 2 AY mM G F = along AY . But the vertical components of the two attractive forces are of opposite direction and equal in magnitudes, they counterbalance each other. Hence the net force is summation of the two horizontal forces. θ θ cos cos 2 2 AY mM G AX mM G F net + = θ cos 2 2 AX mM G F net = N F net 8 o 2 18 7 11 10 30 . 9 6 . 26 cos 3350 ) 10 50 . 3 )( 10 50 . 2 ( ) 10 67 . 6 ( 2 × = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ × × × = − When the spaceship is at B : The attractive force between the spaceship and asteroid X is 2 BX mM G F = along BX . The attractive force between the spaceship and asteroid Y is 2 BY mM G F = along BY . Note that they are of the same magnitude but opposite in direction, hence the net force acting on the spaceship is zero. 9.2 Gravitational field Gravitational field is defined as the gravitational force per unit mass at a point, in terms of a test mass, due to the earth. It is a directional interaction between particles....
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This note was uploaded on 09/06/2010 for the course BSC PHY1417 taught by Professor Prof during the Spring '08 term at HKU.
- Spring '08