Question

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

[Assume , walls of the tunnel are frictionless]

• A. $\frac{G{M}_{e}m}{R_e^3}r$
• B. $\frac{G{M}_{e}m}{R_e^2}$
• C. $\frac{G{M}_{e}m{R}_e^2}{r^5}$
• D. $\frac{2G{M}_{e}m}{R_e^3}r$

Answer 1

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

Given,
Mass of the Earth$=M_e$,
Radius of the Earth$=R_e$

Gravitational field inside a solid sphere of mass $M$ and radius $R$ at a distance $a$ from its centre is given by, $$E=\frac{GMa}{R^3}$$Since, the particle of mass $m$ is inside the Earth at distance $r$, so the gravitational field on the particle due the Earth will be,

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

So, the force on the particle of mass $m$ will be,

$F=m{E}_e=\frac{Gm{M}_{e}r}{R_e^3}$

Hence, option (a) is the correct answer.

Key Concept: The gravitational field inside a solid sphere of radius $R$ and mass $M$ at distance $r$ is given by $E=\frac{GMr}{R^3}$