Is gravitation really a conservative force?

Question

A rotating gyroscope that is mounted in a gimbal maintains its rotational axis relative to space when it is moved – and not to the surface of the earth, as illustrated below (the figure shows only the gyroscope without the gimbal). Also see this training film for illustration.

When a gyroscope is on board of an airship, the orientation of the gyroscope’s rotational axis remains unchanged (relative to space) during flight, while the orientation of the airship constantly changes (relative to space).

The figure below shows the airship and the mounted gyroscope beneath it.

Additionally, there is a “swing” beneath the gyroscope. It is mounted, so it can move around the gyroscope and gets accelerated towards the center of the earth.

The figure below shows the gyroscope and the swing in detail. Furthermore, this figure illustrates the orientation of both the gyroscope and the swing when the airship is “north” of the earth.

The following figures show the orientation of the gyroscope and the swing when the airship is “east”…

“south”…

and “west” of the earth.

The orientation of the gyroscope’s rotational axis is the same in all locations. The swing points in a changing direction, because it gets accelerated in a changing direction when it moves around the earth.

As a consequence, the orientation of the gyroscope’s rotational axis remains unchanged while the swing moves around it. Technically speaking, it is a rotor-stator assembly, where the gyroscope represents the stator and the swing represents the rotor, respectively.

This assembly can do work to a limited extent as long as the torque acting onto the gyroscope is not big enough to push it out of its orientation. Because this work is done along a closed path, gravitation shows to be not a conservative force. The reason why gravitation can do work along a closed path is because the “field lines” of gravitational force are not parallel and a mass moving around the earth at a constant height gets accelerated in a changing direction (relative to space).

Do you have objections against this conclusion? If yes, what did I miss?

Answer 1

In this situation, the Earth is completely redundant. It has no effect on the gyroscope. The gyroscope will keep its orientation even if the zeppelin orbited a completely empty region of deep space. Gravity plays no role. Deep space satellites like Voyager 1 and 2 have on-board gyroscopes to keep themselves oriented. Despite begin millions of miles from anything, the gyroscopes work just fine.

As for the work done by the gyroscope, energy can be extracted by, for example, fixing a turbine to it and anchoring the turbine to the swing. The rotation of the zeppelin and the turbine will put a torque on the gyroscope, since the turbine's back voltage will resist turning. By Newton's Third Law, the gyroscope will put a torque on the turbine, generating electrical power. This will extract energy from the gyroscope's rotational energy and the zeppelin's kinetic energy. Eventually, the gyroscope will slow down enough that it no longer has the angular momentum to resist turning with the swing. Gyroscopic power extraction stops here.

Answer 2

For clarity here is the gyroscope from wikipedia

A gyroscope in operation with freedom in all three axes. The rotor will maintain its spin axis direction regardless of the orientation of the outer frame.

As we observe all orientations do not affect the spin direction in absolute space, the airship could be looping anyway .

The whole construct has a center of mass and a mass which will be affected by a gravitational field , this will not change the spin orientation of the gyroscope.