Gravitational field intensity of a carved out spherical body

From a uniform sphere of radius \(2R\), a spherical cavity of radius \(R\) is cut in such a way that the sphere and the spherical cavity share a common tangent, as shown in the diagram. The mass of the new body is \(M\). Find the gravitational field intensity at point \(A\) which is at a distance of \(6R\) from the center of the sphere.

If this value can be expressed as \(\dfrac{a}{b} \cdot \dfrac{GM}{R^2}\), where \(a\) and \(b\) are coprime positive integers, then evaluate \(a+b\).

Details and Assumptions:

  • Point \(A\) is at a distance of \(6R\) from the geometrical center of the original, larger sphere, not from the center of mass of the newly formed body.
  • Point \(A\) lies such that it is collinear with the centers of the sphere and the spherical cavity and nearer to the common tangent shared by them.
  • \(G\) denotes the universal gravitational constant: \(G = 6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}.\)
  • Neglect Earth's gravitational field and deformities within the sphere.

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