JEE MAIN - Physics (2005 - No. 28)
Average density of the earth
is a complex function of $$g$$
does not depend on $$g$$
is inversely proportional to $$g$$
is directly proportional to $$g$$
Explanation
Mass of earth = Volume $$ \times $$ Density of earth($$\rho$$)
$$\therefore$$ M = $${{4 \over 3}\pi {R^3}} $$$$ \times $$ $$\rho $$
We know, $$g = {{GM} \over {{R^2}}}$$
$$ \Rightarrow g = {{G \times \rho \times {4 \over 3}\pi {R^3}} \over {{R^2}}}$$
$$ \Rightarrow $$ $$g = {4 \over 3}\rho \pi GR$$
$$\therefore$$ $$g \propto \rho $$ or $$\rho \propto g$$
$$\therefore$$ M = $${{4 \over 3}\pi {R^3}} $$$$ \times $$ $$\rho $$
We know, $$g = {{GM} \over {{R^2}}}$$
$$ \Rightarrow g = {{G \times \rho \times {4 \over 3}\pi {R^3}} \over {{R^2}}}$$
$$ \Rightarrow $$ $$g = {4 \over 3}\rho \pi GR$$
$$\therefore$$ $$g \propto \rho $$ or $$\rho \propto g$$
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