Question

if hypothetically, radius of earth increases by a factor of 3, then by what factor its density should change ro keeo value of g same​

Answer 1

Given : radius of earth increases by a factor of 3

To Find : by what factor its density should change to keep value of g same​

Solution:

g = GM/r²

M = Mass of Earth

r = radius of earth

G = Universal Gravitational constant

Density = Mass /Volume

Density = ρ

Volume = (4/3)πr³

=> ρ = M/ (4/3)πr³

=> M = ρ  (4/3)πr³

g = GM/r²

=>

g = Gρ  (4/3)πr³/r²

=> g =  Gρ  (4/3)πr³/r²

=> g =  Gρ (4/3)πr

=> g =   (4/3)πGρr

g is constant

 (4/3)πG are constant

Hence ρr  should be constant

=> if radius of earth increases by a factor of 3

then ρ has to decrease by a factor of 3

Density should be 1/3rd

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