Question

A tunnel is dug along a chord of the earth at a perpendicular distance $\frac{R}{2}$ From the earth's centre. The wall of the tunnel may be assumed to be frictionless. Find the force exerted by the wall on a particle of mass m when it is at a distance x from the centre of the tunnel.

Answer 1

Let $d$ be the distance from center of earth to mass 'm' then,
$d^2=x^2+\left(\frac{R^2}{4}\right)=\frac{4x^2+R^2}{4}$
$\Rightarrow d=\left(\frac{1}{2}\right)\sqrt{4x^2+R^2}$
M be mass of the earth $M^{\prime }$, the mass of the sphere of radius $d$
$M=\left(\frac{4}{3}\right)\pi R^3\rho$
$M^{\prime }=\left(\frac{4}{3}\right)\pi d^3\rho$
$\frac{M^{\prime }}{M}=\frac{d^3}{R^3}$

$\therefore$ Gravitational force on mass m due to the sphere of mass $M^{\prime }$ is,
$F=\frac{G{M}^{\prime }m}{d^2}$
$=\frac{G{d}^{3}Mm}{R^3{d}^2}=\frac{GMm}{R^3}d$
So, Normal force exerted by the wall
$=F\cos \theta =\frac{GMmd}{R^3}\times \frac{R}{2d}=\frac{GMmd}{2R^2}$