From a uniform sphere of radius , a spherical cavity of radius 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 . Find the gravitational field intensity at point which is at a distance of from the center of the sphere.
If this value can be expressed as , where and are coprime positive integers, then evaluate .
Details and Assumptions:
- Point is at a distance of from the geometrical center of the original, larger sphere, not from the center of mass of the newly formed body.
- Point 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.
- denotes the universal gravitational constant:
- Neglect Earth's gravitational field and deformities within the sphere.