Question

A tunnel is dug along a chord of the earth at a perpendicular distance 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 distance of the particle from the centre of the Earth.



$$\begin{gathered} \mathrm{Now},d^2=x^2+\left(\frac{R^2}{4}\right)=\frac{4x^2+R^2}{4} \\ \Rightarrow d=\left(\frac{1}{2}\right)\sqrt{4x^2+R^2} \end{gathered}$$
Let M be the mass of the Earth and M' be the mass of the sphere of radius d.
Then we have:
$$\begin{gathered} M=\left(\frac{4}{3}\right)\pi R^3\mathrm{\rho } \\ {M}^1=\left(\frac{4}{3}\right)\mathrm{\pi }{d}^3\mathrm{\rho } \\ \therefore \frac{M^{\mathit{1}}}{M}=\frac{d^3}{R^3} \end{gathered}$$

Gravitational force on the particle of mass m is given by
$$\begin{gathered} F=\frac{G{M}^{1}m}{d^2} \\ \Rightarrow F=\frac{\mathrm{G}{d}^{3}Mm}{R^3{d}^2} \\ \Rightarrow F=\frac{GMm}{R^3}d \end{gathered}$$
∴ Normal force exerted by the wall, F_N = F cos θ$=\frac{\mathrm{GM}md}{{\mathrm{R}}^3}\times \frac{\mathrm{R}}{2d}=\frac{\mathrm{GM}md}{2{\mathrm{R}}^2}$