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Q.
If the radius of the earth be increased by a factor of $5$ , by what factor its density be changed to keep the value of $g$ the same?
Gravitation
Solution:
$g =\frac{ GM }{ R ^{2}} $
$\Rightarrow g =\frac{ G \rho \times \frac{4}{3} \pi R ^{2}}{ R ^{2}} $
$\Rightarrow g =\frac{4}{3} \pi GR \rho ...$(i)
$g '=\frac{4}{3} \pi G (5 R ) \rho' ...$(ii)
given that $g=g'$, so $\rho'=\frac{\rho}{5}$