The acceleration due to gravity g and the mean density of earth p are related by which relation.?

{G=gravitational constant, R= radius of earth{

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Solution

Radius of earth =R
Gravitational constant = G
Acceleration due to gravity=g
Mass of earth =M
Density of earth = p
We know by newtons law of gravity,
$F equals fraction numerator G M m over denominator R squared end fraction. B u t space F equals m g comma space m equals space m a s s space o f space o b j e c t. m g equals fraction numerator G M m over denominator R squared end fraction o r comma space space space space space g equals fraction numerator G M over denominator R squared end fraction. space space minus negative negative greater than E q 1 N o w space comma space c o n s i d e r space e a r t h space a s space s p h e r e. space V o l u m e space o f space e a r t h space equals 4 over 3 Ï�R cubed Density space of space earth space equals straight p equals fraction numerator Mass space of space earth over denominator Volume space of space earth end fraction straight p equals fraction numerator straight M over denominator begin display style 4 over 3 end style Ï�R cubed end fraction Rearranging comma space Mass space of space earth comma space straight M equals 4 over 3 Ï�R squared straight p space minus negative negative greater than Eq 2 Using space Eq 2 space in space Eq 1 straight g equals GM over straight R squared equals straight G over straight R squared cross times straight M equals straight G over straight R squared cross times 4 over 3 Ï�R cubed straight p Cancelling space straight R squared space and space rearranging straight g equals fraction numerator 4 Ï�GRp over denominator 3 end fraction Rearranging comma straight p equals fraction numerator 3 straight g over denominator 4 Ï�GR end fraction comma space space space where space straight p space is space density space of space earth.$

This is the answer given too. It is just substituting density/volume for Mass of earth , and some rearrangements.

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