Question

A tunnel is dug along a diameter of the earth. If $M_e$ and $R_e$ are the mass and radius, respectively, of the earth, then the force on a particle of mass $m$ placed in the tunnel at a distance $r$ from the centre is :

• A. $\frac{G{M}_{e}m}{R_e^3}r$
• B. $\frac{G{M}_{e}m}{R_e^{3}r}$
• C. $\frac{G{M}_{e}m{R}_e^3}{r}$
• D. $\frac{G{M}_{e}m}{R_e^2}r$

Answer 1

The correct option is A $\frac{G{M}_{e}m}{R_e^3}r$

Let us first calculate the mass of inner solid sphere of radius r.

Mass of inner sphere $M^{\prime }=\frac{M_e}{\frac{4}{3}\pi {R_e}^3}\times \frac{4}{3}\pi r^3=\frac{M_e}{{R_e}^3}{r}^3$

Now,$g=\frac{G{M}_e{r}^3}{{R_e}^3}\times \frac{1}{r^2}$

or $g=\frac{G{M}_{e}r}{{R_e}^3}$

Force on particle of mass m$=mg=\frac{G{M}_{e}mr}{{R_e}^3}$

Note that those layers of the sphere which have radii larger than r do not have any contribution to the gravitational force.