I wasn't sure if I should post this thread in the earth science section (or mathematics) but I guess the topic is general; apologies if otherwise.

I wonder if the center of gravity of the earth, or any object, is in the same position as its gravitation "center of attraction"?

The center of gravity is an imaginary point where all the object's mass can be positioned and from a mechanical perspective the object will "behave" as if it were acted on at this central point (perhaps internal rotations modify this?). The center of gravity is a weighted average of mass * distance integrated linearly over the object in question.

However gravity follows an inverse square law

and this integration is no longer linear. If the integral is performed then the gravitational center of attraction would shift closer to the external attracted object than the center of mass.

If this reasoning is valid, then have we an accurate estimate for the weight of the world? For example we can measure the force of attraction between two objects in a laboratory and given that we know their (inertial) mass the we can estimate the constant G. However when we "weigh" such objects in the earth's gravitational field, can we extrapolate it's weight based on the measured (weight) force and our understanding of its distance to the earth's center?

The center of mass for a symmetrical earth may be at it's center but if the gravitational center is displaced then the weight of the earth would be estimated with a bias (systematic error).

Further, we know the earth is not homogeneous; its density varies with depth. This must further compound our ability to estimate the weight of the earth if we base this just on the weight of objects on its surface relative to inter weight gravitational attraction measurements.

If the attracted object is sufficiently far away compared to the earth's radius the I guess the gravitational center will be adequately positioned at it's center of mass. The moon, for example, has an orbit consistent with such relative dimensions. However can we ascribe its orbit to an estimation of the earth's total mass?

Since the moon acts on the sea and dissipates energy in the form of tides, would this energy expenditure modify its orbit and still add bias to the earth's estimate. Also, the earth is also in orbit with the moon (conjoined pairs) and the mass of the moon influences its motion. Does this mean we still need to know the weight of the moon to know the weight of the world?