Gravitation Test - 72

At points inside the earth the statements of gravitational field need to be modified. If one could drill a hole to the center of the earth and measure the force of gravity on a body at various distances from the center, the force would be found to decrease as the center is approached, rather than increasing as $$l/r^2$$. Qualitatively, it is easy to see why this should be so: as the body enters the interior of the earth (or other spherical body), some of the earth's mass is on the side of the body opposite from the centre of the earth and pulls the body in the opposite direction. Exactly at the center of the earth, the gravitational force on the body is zero.
The magnitude of the gravitational constant $$G$$ can be found experimentally by measuring the force of gravitational attraction between two bodies of known masses $$m_1$$ and $$m_2$$ at the known separation for bodies of moderate size the force is extremely small, but it can be measured with an instrument called a torsion balance, invented by the Rev. John michell and first used for this purpose by Sir Henry Cavendish in $$1798$$. The same type of instrument was also used by Coulomb for studying forces of electrical and magnetic attraction and repulsion.

The earth does not have a uniform density; it is most dense at its centre and least dense at its surface. An approximation of its density is  $$\rho (r) = (A - Br)$$, where $$A = 12,700 kg/m^3, B = 1.50 \times 10^{-3} kh/m^4$$ and  $$r$$  is the distance from the centre of earth. Use $$ R = 6.4 \times 10^6$$ m for the radius of earth approximated as a  sphere, Imagine dividing the earth into concentric, elementary spherical shells. Each shell has radius $$r$$,  thickness $$dr$$, volume  $$dV$$ and mass $$dm = \rho (r) dV$$. By integrating  $$dm$$  from zero to $$R$$  the mass of earth can be found. Knowing the fact that a uniform spherical shell gives no contribution to acceleration due to gravity inside it, we can also find  $$g$$  as a function of $$r$$.