A tunnel is dug along the diameter of the earth. If a particle of mass m is situated in the tunnel at a distance x from the centre of earth then gravitational force acting on it, will be

The correct option is GMₑmx/Rₑ³.

To find the gravitational force acting on a particle of mass m situated inside a tunnel at a distance x from the center of the Earth, we analyze the gravitational force inside a solid sphere.

According to the shell theorem, for a particle located at a distance x from the center of a uniform solid sphere of radius R_e (where x < R_e), the net gravitational force exerted by the outer shell of thickness (R_e - x) is zero. Thus, the particle only experiences a gravitational force due to the mass of the sphere of radius x containing it.

Assuming the Earth to be a uniform solid sphere of mass M_e and radius R_e, the uniform density (ρ) of the Earth is:
$\rho =\frac{M_e}{\frac{4}{3}\pi R_e^3}$

The mass of the sphere of radius x (denoted as M_x) is given by:
$M_x=\rho \times \left(\frac{4}{3}\pi x^3\right)$

Substituting the density (ρ) into the equation for M_x:
$M_x=\left(\frac{M_e}{\frac{4}{3}\pi R_e^3}\right)\times \left(\frac{4}{3}\pi x^3\right)=M_e\frac{x^3}{R_e^3}$

Using Newton's law of gravitation, the force (F) acting on the particle of mass m at a distance x from the center is:
$F=\frac{G{M}_{x}m}{x^2}$

Substituting the expression for M_x into the force formula:
$F=\frac{Gm}{x^2}\times \left(M_e\frac{x^3}{R_e^3}\right)$

Simplifying the equation yields:
$F=\frac{G{M}_{e}mx}{R_e^3}$

Consequently, the gravitational force acting on the particle is directly proportional to its distance x from the center of the Earth and is equal to GMₑmx/Rₑ³.