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Discuss the variation of g with

(a) altitude (b) depth

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Solution

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( a ) The acceleration due to gravity on the earth is given by

g=GMR2i.e.GM=R2g . . . . . . .( 1 )

The acceleration due to gravity at height 'h' from the surface of the earth is given by

gh=GM(R+h)2i.e.GM=(R+h)2gh . . . . . . ( 2 )

From ( 1 ) and ( 2 ) we have,

ghg=R2(R+h)2

ghg=(1−2hR)

( b ) The acceleration due to gravity on the surface of the earth is given by.

g=GMR2 . . . .( 1 )

let 'Q' be the density of the material of the earth.

Now, mass = volume × density

M=43πR3×ρ

Substituting in equation ( 1 ) we get

g=GR2×43πR3×ρ=43πGRρ

g=43πGRρ . . .. .. ( 2 )

gd=43πG(R−d)ρ . . . .. .( 3 )

Dividing equation( 3 ) by ( 2 ) we get

gdg=R−dR=(1−dR)

gd=g(1−dR)

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