Question

If the radius of earth be increased by a factor of $5$, by what factor its density be changed to keep the value of $g$ the same ?

• A. $\frac{1}{25}$
• B. $\frac{1}{5}$
• C. $\frac{1}{\sqrt{5}}$
• D. $5$

Answer 1

The correct option is B $\frac{1}{5}$

Using formula for acceleration due to gravity,
$g=\frac{G{M}_e}{{R_e}^2}$
When the radius is increased to five times,
$g^{\prime }=\frac{G{M_e}^{\prime }}{{\left(5R_e\right)}^2}$
Here mass of the earth is,
$M_e=\frac{4}{3}\pi R_e^3\rho$
In order to maintain the same value for acceleration due to gravity:
$g=g^{\prime }$
$\Rightarrow \frac{G{M}_e}{{R_e}^2}=\frac{G{M_e}^{\prime }}{{\left(5R_e\right)}^2}$
or, $\frac{\frac{4}{3}\pi R_e^3\rho }{{R_e}^2}=\frac{\frac{4}{3}\pi (5R_e{)}^3{\rho }^{\prime }}{{\left(5R_e\right)}^2}$
$\Rightarrow \rho =5{\rho }^{\prime }$
$\therefore {\rho }^{\prime }=\frac{\rho }{5}$
Thus density has to be changed by a factor of $\frac{1}{5}$