# Measuring Earth rotation rate about its axis with gyroscopes

With gyroscopes having two basic properties, precession and rigidity in space, shouldn't we see an apparent movement (due to rigidity in space) of a gyroscope's rotor (gimbals) in relation to the earth's rotation?

I am fully aware of the inner workings of an aircraft's mechanical standby attitude indicator's auto erection system (for those that have it). This question would be pertaining to a precision mechanical gyroscope with an electrically driven rotor on your kitchen table with no such correction mechanisms in place.

I understand most gyroscopes have a measurable drift rate but remember that the earth rotates 360 degrees per 24 hours (15 degrees per hour). Shouldn't this apparent movement be noticeable to the observer?

Note: Rigidity is Space is independent of "Gravity". (You can find video on YouTube of a gyro maintaining its orientation in a zero G environment.)

• Please give a reference for the assertion of "no such movement." Note that there are many fake videos on YouTube, so those can't be trusted to make scientifically valid statements. On the other hand, Foucault pendulums do make the proper rotations based on rotation of the Earth and latitudinal location. – Bill N Jun 7 '17 at 22:30
• Your claim that "we notice no such movement" should be backed up with references to suitable experiments or it should be removed. – Emilio Pisanty Jun 7 '17 at 22:33
• Some satellites are gyro-stabilised, so they do "move", once per orbit, with respect to the earth. – hdhondt Jun 8 '17 at 0:44
• I have removed this claim due to the lack of "references to suitable experiments" and replaced it with a more suitable question. – Leeroy Jenkins Jun 8 '17 at 5:08
• Note that there exist cameras -- that you can buy for less than £2,000 -- whose image stabilisation is already either close to or up against the limits imposed by the Earth's rotation. It is likely that the next generation of these cameras will need to compensate for the precession of their gyros due to the Earth's rotation. So, yeah, the Earth is not, it turns out, flat. – tfb Jun 15 '17 at 12:06

# Mechanical gyroscopes

For an observer at the surface of the Earth, the Coriolis force would be responsible of moving the gyroscope axis of rotation. It is well-known that this force is weak on small system, by which I mean it is easily overwhelmed by non-inertial forces such as friction, viscosity, etc. This is why Foucault pendulums have to be sizeable (the smallest I am aware of has a string about 5 meters long). Thus I reckon you would need a gyroscope way bigger than the mass-produced ones. I am not aware of any purpose-built gyroscope to test Coriolis forces. But why would anybody bother when it is so much easier to build a Foucault pendulum?

# Laser gyroscopes

Mechanical gyroscopes have been completely replaced by laser ones in all inertial system of planes, rockets, etc. They are based on the optical brother of the Coriolis force, called the Sagnac effect. The principle is rather simple and can be demonstrated on the following simple sketch (kudos to Krishnavedala on Wikipedia: I just converted the image to PNG)

So a light source (nowadays a laser) sends two beams across the optical loop (the beam is reflected by the mirrors, represented by the grey rectangles): one in one direction and the other one in the other direction. Then the two beams meet at the detectors, which just records a shift $\Delta$ of interference fringes between them. This difference is equal to

$$\Delta = \frac{4A}{\lambda c}\omega\sin\phi$$

where $A$ is the area enclosed by the loop, $\lambda$ is the wavelength of the light, $c$ is the speed of light in vacuum, $\omega$ is the angular velocity of Earth, and $\phi$ is the latitude.

So again, the effects is bigger for a bigger detector, and commercially available laser gyroscope have a size of the order of 10 cm, and that gives a fringe shift several orders of magnitude too small to be measurable.

But the experience has been done with a very large ring, whose perimeter was about 2 km (or 1.2 miles), by Michelson, Gale and Pearson in 1925 (the Michelson of the other famous experiment). The measured shift was 230 parts in 1000 for an accuracy of 5 parts in 1000, as compared to a predicted shift of 237 parts in 1000, with the above formula. So clearly Michelson et al did measure the rotation of Earth. The Wikipedia article has more details, including the references to the three original papers of Michelson et al.

With the advent of lasers, and modern instrumentations, the precision of laser rings has vastly increased, for much smaller rings. For example the C-I ring in Christchurch, New Zealand, has a perimeter of only 3.477 m but it has a sensitivity limit of 20 microarcs per seconds for a collection time of one hour. The Christchurch team (website) wrote a nice paper describing not only all their wonderful devices but also giving an in-depth historical, theoretical and experimental review of laser rings [a].

In that paper, I learned that some industrial laser ring using fibre optics can actually sense Earth rotation. Quoting Stedman et al:

Fibre-optic gyros have proved useful in a very wide variety of fields, such as oil prospecting. The Sagnac signal generated by the Earth’s rotation is sensitive to the orientation of the device, and if the orientation is defined by the ring laser being totally embedded in an oil-well drill bit, the alignment of exploratory bores can be monitored deep underground. Thanks to advanced technological tricks including dither, such small gyros monitor the projection of the Earth’s rotation to an accuracy of $0.01^\circ h^{−1}$ or $5\times10^{−8} rad s^{−1}$, a fraction $7 \times 10^{−4}$ of the Earth’s rotation rate $\Omega_E$ (Kim and Shaw 1986).

The given reference is [b].

[a] G E Stedman, Ring-laser tests of fundamental physics and geophysics, Reports on Progress in Physics 60 (1997), 615

[b] B. Y. Kim and H. J. Shaw, Fiber-optic gyroscopes: in harsh, confining environments this advanced gyroscope, a close cousin to the ring laser gyro, offers great advantages, IEEE Spectrum 23 (1986), 54-60

• So because commercially available laser ring gyro inertial navigation systems are too small to detect the earth's rotation, how are they used to navigate accurately on a ball spinning a 1000 mph at the equator? Well written but wrong... youtu.be/xDYsnrSqvrQ – Leeroy Jenkins Jun 18 '17 at 14:47
• There is no logical connexion between the two parts of your first phrase. More importantly, if you are really interested in physics, you could easily reproduce Michelson-Gale-Pearson experimental setup, using reasonable quality lasers for the Sagnac ring, and laser ranging for precision layout of the ring, resulting in a much higher precision with much less work, and a much smaller ring size. Instead you prefer armchair discussions. And see the update to my answer. – user154997 Jun 18 '17 at 17:15
• Lol... You must think very highly of yourself. Watch that video if you haven't already. It might clear things up a little for you,...that is if your interested in physics. – Leeroy Jenkins Jun 19 '17 at 2:34
• Note that the "quantum whistle" of superfluids is also sensitive to the orientation of the of the fluid wrt the earths rotation – R. Rankin Jun 19 '17 at 9:04
• Why do you think it got downvoted?... same reason my post that followed this, that you all can’t see now, was deleted. Do your own research. Stack exchange is good if you want the highly indoctrinated mainstream college answers. – Leeroy Jenkins Sep 5 '18 at 18:35

"In 1851 Léon Foucault ﬁrst described his famous pendulum experiment in the Comptes rendus [1]. As he explains in his report, the principle is easy to understand at the pole: by inertia the plane of oscillation remains constant with respect to an inertial reference frame, so that in a sidereal day (23 h 56 min) it rotates or veers through a full 360◦with respect to the Earth, in the opposite direction. Foucault checked that the rotation of the support has indeed negligible effect on the plane of rotation. He was inspired by the observation of a thin ﬂexible rod on the axis of a lathe, which keeps oscillating in a constant plane in spite of the rotation of its support. The problem is more complicated at lower latitude, but Foucault rightly convinced himself that the angular speed of the veering of the swing plane should be proportional to the sine of the latitude. This point was, however, puzzling, and in his paper Foucault draws the attention of “géomètres” (as mathematicians were called at that time) to solve the problem in a more rigorous way..."