A tunnel is dug along a diameter of the earth of mass  $M_e$ and radius $R_e$ . The force on a particle of mass m placed in the tunnel at a distance r from the center is :

If we imagine a sphere of radius r form the center of earth, according to shell theorem the mass which is inside this sphere will only have exert gravitational force on the particle.

So, $\mathrm{F}=\mathrm{G}\frac{{\mathrm{M}}_{\mathrm{e}}}{\frac{4}{3}{\mathrm{\pi R}}_{\mathrm{e}}^3}\frac{4}{3}{\mathrm{\pi r}}^3\left(\frac{\mathrm{m}}{{\mathrm{r}}^2}\right)=\frac{{\mathrm{GM}}_{\mathrm{e}}\mathrm{m}}{{\mathrm{R}}_{\mathrm{e}}^3}\mathrm{r}.$