Discuss the variation of g with
(a) altitude (b) depth
( 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)