Question

A tunnel is dug along a diameter of the earth. Find the force on a particle of mass m placed in the tunnel at a distance x from the centre.

Answer 1

Mass of the Earth, $M=\left(\frac{4}{3}\right)\pi R^3\mathrm{\rho }$ ...(i)
Consider an imaginary sphere of radius x with centre O as shown in the figure below:



$$\begin{gathered} \mathrm{Mass}\mathrm{of}\mathrm{the}\mathrm{imaginary}\mathrm{sphere},M\text{'}=\left(\frac{4}{3}\right)\mathrm{\pi }{\mathrm{x}}^3\mathrm{\rho }...\left(ii\right) \\ \mathrm{From}\left(i\right)\mathrm{and}\left(ii\right),\mathrm{we}\mathrm{have}: \\ \frac{M\mathit{\text{'}}}{M}=\frac{x^3}{R^3} \end{gathered}$$


∴ Gravitational force on the particle of mass m is given by
F$=\frac{GMm}{x^2}$
$\Rightarrow F=\frac{GM{x}^{3}m}{R^3{x}^2}=\frac{GMm}{R^3}x$