Question

a sphere of diameter r is cut from a sphere of radius r such that the center of mass of the remaining mass be at maximum distance from original center then the distance is

Answer 1

Let   x = distance between COM from the original center of the sphere, after the small sphere is cut off from it.

Let the density of the sphere be = d
Mass of big sphere (full) = 4/3 * π r³ * d
Mass of the small sphere cut off from it = 4/3 * π (r/2)³ * d
Mass of the remaining object = 7/6 π r³ * d

Position of the COM of the complete sphere = 0
 =>  - x * 7/6 π r³ d  + r/2 * 4/3 π (r/2)³ * d = 0
=>   x = r / 14