# 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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