Question

A tunnel is dug along a diameter of earth. The force on a particle of mass $m$ and distance $x$ from the centre in this tunnel will be :

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

Answer 1

The correct option is D $\frac{G{M}_{e}mx}{R^3}$

Field at a distance $x$ from the center will be equal to field at a distance $R-x$ from the earth's surface.

So
$g_x=g(1-\frac{(R-x)}{R})=\frac{gx}{R}$

or

$F=\frac{mgx}{R}=\frac{mx}{R}\frac{G{M}_e}{R^2}=\frac{G{M}_e}{R^3}mx$