The "Laboratory Frame" Part I

The laboratory frame of reference is not truly inertial

A "laboratory" frame of reference is one that is attached to the surface of the Earth, and so it rotates along with the Earth. Normally we regard a laboratory frame as truly inertial, in which Newton's laws of motion can be applied in their simplest form. We generally do only that, as any errors that occur as a result are almost always small enough to be completely negligible.

Fig 1.  A silux macro that forces an object (o1) to behave as a point on the Earth's surface
at the equator would behave, as seen in a non-rotating frame attached to the center of the Earth.
(Note the centimetre-gram-microsecond system of units).

Back on 4 October 2014 I posted Figure 1 above, showing how an object in a laboratory frame at the Earth's equator is seen to move as measured in a truly inertial frame set at the center of the Earth. (The Earth's orbital motion is disregarded). From a macro like this, or just from the basic physics involved, results can be derived for the upwards displacement and velocity for a truly inertial "weightless" mass initially placed on the equator (Fig 2). I used values of 0.00007292115 radians/sec for the Earth's rotational velocity, and 6,378,137 meters for its equatorial radius.

An observer in a laboratory frame who stays attached to the Earth would see the weightless mass rise upwards, and drift to the West, slowly at first, then ever more rapidly.

Fig 2.  Spreadsheet of data for a weightless mass "floating" up from a point
on the Earth's equator.

Energy extraction?

From Figure 2, if a weightless mass rises upwards at the equator for say 5 seconds, it could then impact at a velocity of 0.16958 m/s against any structure fastened to Earth, after having risen through a distance of 0.423946 m. If the mass was say 10kg, that impact would deliver 
½ × 10 × 0.16958² = 0.14379 joules of energy. Could we really do that?

There are two problems:— achieving a weightless mass, and returning it back to the Earth's surface. The first could be easily solved e.g. with a constant-force spring from the mass to the earthed structure, but the second problem is much harder, if we hope to gain energy overall. We must distinguish carefully between a mass that is made weightless in a laboratory frame, which could be returned to the Earth's surface with no energy penalty, and one that is weightless in a truly inertial frame, which is required here. The latter would inevitably give an energy penalty equal to the centrifugal force caused by Earth's rotation (i.e. 0.033916 Newtons/kilogram at the equator) multiplied by the distance required to return it.

In the above case we would have a penalty of 
0.033916 N/kg × 10 kg × 0.423946 m = 0.14379 N-m = 0.14379 joules.

So the energy gained equals the energy lost, and this particular idea is ruled out.

I think that anyone who looks seriously into the idea of extracting energy from the rotating Earth will soon become as familiar with that figure of 0.033916 N/kg for equatorial centrifugal force as they probably already are with 9.80665 N/kg for (standard) gravitational force! (Assuming they site their thought experiments and models at the equator, as I always do).

[Postscript] — Calculation

The centrifugal force caused by Earth's rotation on a 1kg mass at the equator can be calculated thus:—

Centrifugal force = mω²r = 1 × 0.00007292115² × 6378137 = 0.033916 N