That’s only true if you release the object and forces from the plane no longer act on it. Remember, the rope has a weight on it (probably a pretty substantial one), so the force vector is a combination of the centripetal acceleration from the plane, plus the force of gravity acting on the weight. So it falls down, and out of the plane of the circle. But the rope is still pulling on the weight, and the weight now wants to get pulled back in to the circle. But the plane keeps pulling it around, and if the acceleration in the circle is constant there will be a resultant total force on the bucket that is essentially zero. In reality I imagine the bucket probably swings around a bit in a small circle because of inertia. Instead of a lariot, think of water flowing down a drain.
Drag. The airplane can move through the air at, say, 90 mph, while the bucket can accomplish maybe 20 mph. So, since it is tied to the airplane’s number of revolutions it must accomplish same in the same amount of time, and must then seek a shorter travel path. Unless the rope adjusts for the difference by twisting (which I suspect would occur).
So, I can see it possibly overcoming some of the inertial forces that would drive it outside the circle of the airplane’s rotation, but it’s still hard to see it coming to near stationary positioning.
And of course, as soon as mail was retrieved or added, the bucket’s travel path would change, quite possibly dramatically.
Have any of you guys actually witnessed this?
Remember, the rope is curved. So the force on the bucket pulling toward the plane is lagging far behind the actual position of the plane. As I understand it, if you can get the rope/antenna long enough then it is essentially motionless in the center of the circle.