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7.Gravitation
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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 ?
A
$1/25$
B
$1/5$
C
$1/\sqrt{5}$
D
$5$
Solution
The acceleration due to gravity on surface of earth is
$g=\frac{G M}{R^{2}}$
$g=\frac{G \rho V}{R^{2}}$
where, $G$ is the gravitational constant, $M$ is mass of earth, $R$ is radius of earth, $\rho$ is the density and $V$ is the volume of earth.
$\therefore g=\frac{G \rho \frac{4}{3} \pi R^{2}}{R^{2}}$
$\therefore g=\frac{4}{3} G \rho \pi R$
$\Rightarrow g \propto \rho R$
since, all other quantities are constant.
Hence, if the radius of the earth be increased by a factor of $5,$ density will decrease by $\frac{1}{5}$ to keep the value of $g$ the same.
Standard 11
Physics
