The Mass of the Earth. The gravitational force of attraction, F_g, between two b

Question

The Mass of the Earth. The gravitational force of attraction, F_g, between two bodies of mass M and m, the centres of which are a distance r apart is given by the expression F_g = GMm/r^2 where G is the universal gravitational constant and has a value of 6.668 times 10^-11 m^3/kg-sec^2. In particular, if the mass of the Earth is M, the gravitational force acting upon an object of mass m on the Earth's surface is given by F_g = mg where g is the acceleration due to gravity. a) Use the relations above to calculate the mass of the Earth, assuming that g = 9.780 m/sec^2 and r = r_e. (b) Use your result for the mass of the Earth to estimate the mean density of the Earth. (c) How does this mean density for the whole Earth compare with those for the minerals in the rock samples from the upper mantle (olivine, pyroxene, garnet)? Densities of Mantle Materials (Mg/m^3 = g/cm^3)

Explanation / Answer

a) Fs= GM m/ r2

also Fs=mg

therefore

GM m/ r2 =mg

M= g r2/ G

r= 6371 kms

putting the values and making units of values same , we get

M = 5.95 X 1024Kg

b) Density = Mass/ Volume

volume of sphere = 4 pi r3/ 3

therefore

density = 5.95 X 1024Kg/ (4 pi (6371)3/ 3)

= 5.51 g/cm3

c) The mean density of the whole earth is more than the individual densities of the minerals.

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