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PHY183-Lecture39 - Gravitational Force The general...

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April 4, 2006 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Spring Semester 2006 Lecture 39 April 4, 2006 Physics for Scientists&Engineers 1 2 Gravitational Force Gravitational Force ! The general expression for the magnitude of the attraction between two masses m 1 and m 2 a distance r apart is ! G is the universal gravitation const. F ( r ) = G m 1 m 2 r 2 G = 6.67 ! 10 " 11 m 3 kg -1 s " 2 1 2 21 2 1 2 2 2 21 1 1 ( ) | | ( ) ( ) m m F r G r r F r F r e = ! ! = ! r r r r r r r r r 2 1 r r r = ! r r r 2 1 21 2 1 r r e r r ! = ! r r r r r Near surface: g ( h ) = g 1 ! 2 h R E + ... " # $ % & April 4, 2006 Physics for Scientists&Engineers 1 3 Gravitation inside the Earth Gravitation inside the Earth ! We can treat the gravitational force from a spherical object as if all the mass is concentrated at the center of mass, as long as we are located a distance larger than the radius of the spherical object ! Let’s consider the Earth to be composed of many concentric spherical shells ! We can write the gravitational force on the mass m due to all the shells with radius less than r Note: the outer shells do not contribute (we omit the proof here) ! This formula is similar to what we have been using except we have written M(r) instead of M F ( r ) = G M ( r ) m r 2 April 4, 2006 Physics for Scientists&Engineers 1 4 Gravitation inside the Earth (2) ! If we assume that the Earth has a constant density we can write ! We can calculate the density of the Earth as ! Substituting this result back into the above equation we get !
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