What is the change of value of g with the depth of earth explain with derivation.

Asked by yashwantjain12 | 7th Dec, 2015, 11:25: AM

Expert Answer:

Considering the Earth to be a homogenous sphere of radius R and mass M.
Let ρ be the mean density of the Earth.
     g be acceleration due to gravity.
begin mathsize 14px style straight g equals GM over straight R squared end style      ... (1)
Consider the body at the depth 'd' from the surface of the Earth where the value of acceleration due to gravity is g'.
Here, the force of gravity acting on the body is only due to inner solid sphere of radius (R-d) and mass be M'.
begin mathsize 14px style straight g apostrophe equals fraction numerator GM apostrophe over denominator left parenthesis straight R minus straight d right parenthesis squared end fraction end style        ... (2)
Dividing (2) by (1), we get,
begin mathsize 14px style fraction numerator straight g apostrophe over denominator straight g end fraction equals fraction numerator left square bracket GM apostrophe divided by left parenthesis straight R minus straight d right parenthesis squared right square bracket over denominator left square bracket GM divided by straight R squared right square bracket end fraction fraction numerator straight g apostrophe over denominator straight g end fraction equals fraction numerator straight M apostrophe straight R 2 over denominator straight M left parenthesis straight R minus straight d right parenthesis squared end fraction space space space space... space left parenthesis 3 right parenthesis Now comma space the space density space of space the space Earth comma space straight rho equals straight M over straight V straight rho equals fraction numerator straight M over denominator open parentheses begin display style 4 over 3 πR cubed end style close parentheses end fraction straight M equals straight rho cross times open parentheses 4 over 3 πR cubed close parentheses space space space space space... space left parenthesis 4 right parenthesis Density space of space the space Earth space below space apostrophe straight d apostrophe space metre space from space the space surface space of space the space Earth comma space straight rho equals fraction numerator straight M apostrophe over denominator straight V apostrophe end fraction straight rho equals fraction numerator straight M apostrophe over denominator open parentheses begin display style 4 over 3 straight pi left parenthesis straight R minus straight d right parenthesis cubed end style close parentheses end fraction straight M apostrophe equals straight rho cross times open parentheses 4 over 3 straight pi left parenthesis straight R minus straight d right parenthesis cubed close parentheses space space space space space... space left parenthesis 5 right parenthesis Substituting space eqn space left parenthesis 4 right parenthesis space and space left parenthesis 5 right parenthesis space in space left parenthesis 3 right parenthesis comma space we space get comma fraction numerator straight g apostrophe over denominator straight g end fraction equals fraction numerator straight rho cross times open parentheses 4 over 3 straight pi left parenthesis straight R minus straight d right parenthesis cubed close parentheses space cross times straight R squared over denominator straight rho cross times open parentheses 4 over 3 πR cubed close parentheses cross times left parenthesis straight R minus straight d right parenthesis squared end fraction fraction numerator straight g apostrophe over denominator straight g end fraction equals 1 minus straight d over straight R straight g apostrophe equals straight g open parentheses 1 minus straight d over straight R close parentheses end style
If we go down in a mine of depth 'd' , the value of 'g' decreases due to decreased mass of attracting Earth.
At the centre of the Earth, the value of g becomes be zero.

Answered by Faiza Lambe | 7th Dec, 2015, 11:43: PM