So, I pulled the following diagram from a table in an old edition of the Machinery's Handbook available through Google Books:
We can see in this diagram three distinct points that are shown along a vertical line and that are labeled with an upper case "A," "B" and "C" which represent respectively the "center of gravity," the "center of gyration" and the "center of oscillation."
We can also see in the diagram three different lengths labeled with the lower case "a," "b" and "l" and which represent respectively the distance from the top end of the line to the center of gravity, the distance from the top end of the line to the center of gyration and the distance from the top end of the line to the center of oscillation.
In the latter two cases, of course, the top end of the line would represent the center of rotation and also the pivot/suspension point for a pendulum, so the distance "b" would be the "radius of gyration" and the distance "l" would be the "radius of oscillation."
Since the center of gravity of the depicted line is shown at the midpoint of the line, we can assume, then, that there is a uniform mass distribution along the line.
Now, points and lines are mathematical concepts and by definition they don't have any width, area or volume, but they are still quite useful as tools to help us calculate real world conditions. Obviously, in the real world we have to deal with mass taking up actual volume, but using point masses and massive lines can still give us pretty good approximations for real world things like thin rigid massive rods or compact masses. If, of course, we needed greater accuracy we could do more complex calculations and actually take into account greater detail about the real world volumes and mass distributions.
Now, most all of us should be familiar with what the center of gravity is. Near the surface of the earth where the gravitational field lines are approximately parallel, the center of gravity of a system of mass is essentially the center of mass of that system. As shown in the following diagram, if that system of mass is rigid, then we can support the total weight of the mass at its center of gravity and without causing or affecting any rotation of the system. In other words, the center of gravity is the balance point.
In the following diagram, if all the mass of the thin rigid rod on the left were concentrated in the sphere or cylinder on the right, then the two systems would be rotationally equivalent when turned around the shown centers of rotation. Both would have the same rotational inertia - the same moment of inertia - and if they were both rotating with the same angular speed, they would have the same kinetic energies.
In this diagram, if all the mass of the long rigid rod on the left was concentrated in the sphere or cylinder on the right and both systems were used as pendulums, then they both would oscillate at the same frequency and so have the same period of oscillation when allowed to swing through a small angle.
They do not have the same center of gravity, however, nor do they have the same moment of inertia or radius of gyration. They are not rotationally equivalent.
In a simple pendulum where we use a point mass to represent all the mass of the system, the center of gravity, the center of gyration and the center of oscillation are all at that same point. Obviously, though, these things will most likely be at different points for a compound pendulum, with the location of the points then being dependent upon the actual distribution of mass in the system.
We can calculate the frequency and period of oscillation for either the simple pendulum or compound pendulum, though, if we use the "radius of oscillation (l)" in the following formulas - where the radius of oscillation of a compound pendulum, then, is the length of its equivalent simple pendulum.
Now, if we look at the following formula from the Machinery's Handbook that a member of this forum used while apparently believing it gave the radius of gyration for a two point mass system, we can see that there are no terms related to the actual geometry of the two masses themselves, so it indeed appears to be a two point mass approximation of something.
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