Explain why the value of g is zero at the centre of the earth .

Explanation

The value of g changes while going deep inside the earth . It goes on decreasing as we go from the earth' s surface towards the earth' s centre . We shall treat the earth as a sphere of uniform density . If we consider a particle of mass m at point P at a distance ( R - d ) from the earth' s centre , where R is the radius of the earth and d is the depth below the earth' s surface , the gravitational force on the particle due to the earth is

`F=(GmM')/((R-d)^(2))`, where M' is the mass of the sphere of radius `(R-d)`.
`M'=(4)/(3)pi(R-d)^(3)xx(M)/((4)/(3)piR^(3))=(M(R-d)^(3))/(R^(3)).` This is because the outer spherical shell is not effective . In this case, the acceleration due gravity is
`g=(F)/(m)=(G)/((R-d)^(2)), (M(R-d)^(3))/(R^(3))=(GM(R-d))/(R^(3))`,
where M is the mass of the earth. Thus, g decreases as d increases. It is less than that at the earth's surface `((GM)/(R^(2)))`. At the earth's centre, `d= R therefore g=0`.