# 285lect13 - 1 1.1 Lecture 13 Variable gravitational...

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1 Lecture 13 1.1 Variable gravitational acceleration Newton&s law of gravitation states that the gravitational force between two masses M and m is given by F = GMm r 2 ; where G is an empirical constant: G = 6 : 6726 & 10 & 11 N & m kg 2 ± : Let M E be the mass of Earth. Let a body of mass m is falling toward Earth (the air resistance is neglected). Then F = GM E m ( R E + r ) 2 , where r is the distance from the body to the Earth surface and G = 6 : 6726 & 10 & 11 N ² m kg 2 ³ ; M E = 5 : 97 & 10 24 kg ; R E = 6 : 378 & 10 6 m: 1

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E a r t h s u f c e Center of Earth r R E If the body is close to the surface of Earth, then r & R E and ( R E + r ) 2 ± R 2 E . So, F = mg = & GM E R E 2 ± m , whence g = GM E R E 2 = (6 : 6726 ² 10 & 11 ) (5 : 97 ² 10 24 ) (6 : 378 ² 10 6 ) 2 = 9 : 792 7 ± 9 : 8 m sec 2 : 2
If the body is far away from the surface of the earth, then r cannot be neglected and g ( r ) = GM E ( R E + r ) 2 = (6 : 6726 & 10 & 11 ) (5 : 97 & 10 24 ) (6 : 378 & 10 6 + r ) 2 = 3 : 983 5 ± 10 14 ( r + 6 : 378 ± 10 6 ) 2 is not constant: r g ( r ) 100 m 9 : 792 2 m sec 2 1000 m 9 : 789 5 m sec 2 1 ; 000 ; 000 m = 1000 km 7 : 317 9 m sec 2 100 ; 000 ; 000 m = 100 ; 000 km 3 : 520 2 ± 10 & 2 m sec 2 1.2 Escape velocity What initial velocity is needed to escape the Earth? Here we give the answer

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## This note was uploaded on 01/26/2012 for the course MATH 285 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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285lect13 - 1 1.1 Lecture 13 Variable gravitational...

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