Question

A tunnel is dug along a diameter of the Earth. If $M_e$ and $R_e$ are the mass and radius respectively of 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. $\frac{G{M}_{e}m}{R_e^3}r$

Answer 1

Answer: (A)

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

Mass of inner solid sphere, $M$

$=\frac{M_e}{\frac{4}{3}\pi R_e^3}\times \frac{4}{3}\pi r^2=\frac{M_e}{R_e^3}{r}^3$

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 $m=mg=\frac{G{M}_{e}mr}{R_e^3}$