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Assume that a tunnel is dug through earth from North pole to south pole and that the earth is a non-rotating, uniform sphere of density $\rho $. The gravitational force on a particle of mass $m$ dropped into the tunnel when it reaches a distance $r$ from the centre of earth is
$\left( {\frac{3}{{4\pi }}\,mG\rho } \right)r$
$\left( {\frac{{4\pi }}{3}\,mG\rho } \right)r$
$\left( {\frac{{4\pi }}{3}\,mG\rho } \right)r^2$
$\left( {\frac{{4\pi }}{3}\,{m^2}G\rho } \right)r$
Solution
Force on particle of mass $\mathrm{m}$ at a distance $\mathrm{r}$ from earth is due to a spherical part of earth of radius $r$.
$\mathrm{F}=\frac{\mathrm{Gm}^{\prime} \mathrm{m}}{\mathrm{r}^{2}}, \quad \mathrm{M}^{\prime}=\rho, \frac{4}{3} \pi \mathrm{r}^{3}$
$F=\mathrm{G}\left(\rho \frac{4}{3} \pi \frac{\mathrm{r}^{3}}{\mathrm{r}^{2}}\right) \mathrm{m}=\left(\frac{4 \pi}{3} \mathrm{mG} \rho\right) \mathrm{r}$
