According to my Physics Textbook:
"The force between two finite rigid bodies is not necessarily along the line joining their centre of mass".
If the Gravitational Force is Central then why it will not act between two finite bodies along their Centre of mass?
Please elaborate that highlighted text with some simple example.
This is only true in a uniform field, and this is why: the center of mass is the average mass weighted position of an extended object. Meanwhile, the total gravitational force is the sum over all parts of the object, weighted by mass: the mass-weighted integrals for the average and the sum are the same. In reality, the center of gravity differs from the center of mass, since a variable gravitational field changes the later sum over parts.
Imagine that the system consists of two point masses A and B connected by a rigid rod, attracted to a third point mass C. The net force acting on A&B due to C is the sum of the force vectors of AC and BC. If A and B are equidistant from C, then indeed the resultant force vector (A&B)C goes through the center of mass of A&B. But if the rigid rod is tilted a bit so that A is closer to C than B is to C, the force vector AC is greater than the force vector BC, so the sum of the two vectors no longer passes through the center of mass of A&B.
