Question

If the radius of the earth were to be increased by a factor of 3, by what factor would its density have to be changed to keep the 'g' same.

Answer 1

The formula for calculating the density of the earth is given as

ρ = M /V = [gR² / G] / [(4/3)πR³] . [where ρ=density]

as g = GM/R² and V = (4/3)πR³

so,

ρ = (3/4).(g / πRG)

now, if the radius is increased, the new density will be given as

ρ' = (3/4).(g / πR'G)

now as

R' = 3R

so,

ρ' = (3/4).(g / π(3R)G)

so,

ρ' = (1/3).(3/4).(g / πRG)

thus,

ρ' = (1/3)ρ

so, the new density would decrease to one-third its initial value.