13 1 Model Model the sun s and the earth e as spherical masses Due to the large difference between your size and mass and that of either the sun or the earth a human body can be treated as a particle GM M e 2 r e GM M s 2 r s e Solve and e on you s on you F F y y Dividing these two equations gives 2 F F s on y e on y M M s e r e r s e 30 1 99 10 kg 5 98 10 kg 24 6 6 37 10 m 1 50 10 m 11 2 6 00 10 4 13 2 Model Assume the two lead balls are spherical masses Solve a F 1 on 2 F 2 on 1 Gm m 1 2 2 r 6 67 10 11 N m kg 10 kg 0 100 kg 2 2 6 7 10 N 9 0 10 m 2 b The ratio of the above gravitational force to the gravitational force on the 100 g ball is 9 6 67 10 N 2 0 100 kg 9 8 m s 6 81 10 9 Assess The answer in part b shows the smallness of the gravitational force between two lead balls separated by 10 cm compared to the gravitational force on the 100 g ball 13 3 Model Model the sun s the moon m and the earth e as spherical masses Solve F s on m m and F e on m GM M s 2 r s m GM M e 2 r e m m Dividing the two equations and using the astronomical data from Table 13 2 2 F F s on m e on m M r e m s r M s m e 30 1 99 10 kg 5 98 10 kg 24 8 3 84 10 m 1 50 10 m 11 2 2 18 Note that the sun moon distance is not noticeably different from the tabulated sun earth distance 13 4 Solve F sphere on particle and F earth on particle p GM M s 2 r s p GM M e 2 r e p Dividing the two equations F sphere on particle F earth on particle M M s e r e r s p 2 5900 kg 24 5 98 10 kg 6 6 37 10 m 0 50 m 2 1 60 10 7 13 5 Model Model the woman w and the man m as spherical masses or particles N m kg 50 kg 70 kg 6 67 10 11 2 2 Solve F F w on m m on w GM M w 2 r m w m 1 0 m 2 2 3 10 N 7 13 6 Model Model the earth e as a sphere Visualize The space shuttle or a 1 0 kg sphere s in the space shuttle is away from the center of the earth R e r s 6 37 10 m 0 30 10 m 6 67 10 m 6 6 6 Solve a F e on s GM M e s 2 r R s e 6 67 10 11 2 2 N m kg 5 98 10 kg 1 0 kg 2 6 67 10 m 6 24 9 0 N b Because the sphere and the shuttle are in free fall with the same acceleration around the earth there cannot be any relative motion between them That is why the sphere floats around inside the space shuttle 13 7 Model Model the sun s as a spherical mass Solve a g sun surface b g sun at earth GM 2 r s e s s GM 2 R s 6 67 10 6 67 10 11 2 2 N m kg 1 99 10 kg 2 6 96 10 m 8 30 274 m s 2 11 2 2 N m kg 1 99 10 kg 2 1 50 10 m 11 30 5 90 10 m s 3 2 13 8 Model Model the moon m and Jupiter J as spherical masses 6 67 10 11 22 2 2 N m kg 7 36 10 kg 2 1 74 10 m 6 1 62 m s 2 11 2 2 N m kg 1 90 10 kg 2 6 99 10 m 7 27 25 9 m s 2 Solve a g moon surface b g Jupiter surface GM 2 R J J m GM 2 R m 6 67 10 13 9 Model Model the earth e as a spherical mass Visualize The acceleration due to gravity at sea level is Table 13 2 9 83 m s see Table 13 1 and 2 R e 6 37 10 m see 6 Solve g observatory GM e h R e 2 GM e 2 2 R e 1 h R e g earth 1 2 h R e Here g earth 2 GM R e e value 9 83 m s Solving for h 2 9 83 0 0075 m s 2 is the acceleration due to gravity on a non rotating earth which is why we ve used the h 9 83 9 8225 1 R e 2 43 km 13 10 Model Model the earth e as a spherical mass Solve Let the acceleration due to gravity be 3g surface when the earth is shrunk to a radius of x Then g surface e and 3 g surface GM 2 R e GM 2 x e 3 GM GM 2 e e 2 R e x R e 3 x 0 577 R e The earth s radius would need to be 0 577 times its present value 13 11 Model Model Planet Z as a spherical mass 11 Solve GM 2 R Z b Let h be the height above the north pole Thus 8 0 m s 6 67 10 a Z surface g Z 2 2 2 N m kg 6 5 0 10 m 2 M Z M 3 0 10 kg 24 Z g above N pole GM R Z Z h 2 GM Z 2 2 R Z 1 h R Z 2 g Z surface h R Z 1 8 0 m s 2 6 1 10 0 10 m 5 0 10 m 6 2 0 89 m s 2 13 12 Model Model Mars M as a spherical mass Ignore air resistance Also consider Mars and the ball as an isolated system so mechanical energy is conserved Visualize Solve A height of 15 m is very small in comparison with the radius of earth or Mars We can use flat earth gravitational potential energy to find the speed with which the astronaut can throw the ball On earth with y i energy conservation is 0 m s and v f 0 m 1 2 mv mgy 2 f 1 2 mv mgy i 2 i f v i 2 g y e f 2 9 80 m s 15 m 17 1 m s 2 Energy is also conserved on Mars but the acceleration due …
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