Question

A tunnel is dug along a chord of the earth at a perpendicular distance $\frac{R}{3}$ from the earth's centre. Assume that wall of the tunnel is frictionless. Find the force exerted by the wall on mass $m$ at a distance $x$ from the centre of the tunnel.

• A. $\frac{mg}{3}$
• B. $\frac{mgx}{R}$
• C. $\frac{mg\sqrt{\frac{R^2}{9}+x^2}}{R}$
• D. $\frac{mgx}{\sqrt{\frac{R^2}{1+x^2}}}$

Answer 1

The correct option is A $\frac{mg}{3}$


$F=mg\prime =mg(1-\frac{(R-\sqrt{\frac{R^2}{9}+x^2})}{R})$

$\Rightarrow F=\frac{mg}{R}\sqrt{\frac{R^2}{9}+x^2}$

$\Rightarrow F_{wall}=\frac{mg}{R}\sqrt{\frac{R^2}{9}+x^2}\frac{\frac{R}{3}}{\sqrt{\frac{R^2}{9}+x^2}}\Rightarrow \frac{mg}{3}$