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. $\frac{GM_em}{R^3_e}r$
• B. $\frac{GM_em}{R^3_er}$
• C. $\frac{GM_emR^3_e}{r}$
• D. $\frac{GM_em}{R^3_e} r$

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^3_e} \times \frac{4}{3} \pi r^2 = \frac{M_e}{R^3_e} r^3$
Now, $g = \frac{GM_er^3}{R^3_e} \times \frac{1}{r^2}$ or $ g = \frac{GM_er}{R^3_e}$
Force on particle of mass $m = mg = \frac{GM_emr}{R^3_e}$