Part 9 - Estimation of magnitude of the unbalanced centrifugal forces driving tectonic movement

Based on the above observations, let us assume as a working hypothesis that the Earth can be modelled as a rotating body where the centre of mass is offset from the principal axis of rotation. For the purposes of this paper the author will consider the two principal approaches to determine the circumferential forces associated with an unbalanced rotating body

Model 1. Rigid body dynamics

As discussed in Section 5, the Pacific plate has all the appearances of being in compression while the almost diametrically opposed African plate appears to be subject to tensile forces

The simplest model is to consider the Earth as an eccentrically rotating solid body such as an unbalanced flywheel. Although this model (shown in Figs 8 & 9 and enumerated in Appendix 1) accounts for the compressive and tensile stresses developed in the outer rim, it does not describe the unbalanced centripetal forces which the author believes to be linked to tectonic forces resulting in plate movement.

Fig 8 below. Force/Vector diagram showing the centripetal force P and the circumferential or tensile force F in the outer rim of a solid rotating body. Rotating Machines are designed to ensure that the developed circumferential stress F, is less than the design hoop stress.

Fig. 8

Fig 9 below. Force/Vector diagram showing the differential circumferential stress induced in the outer rim of a solid rotating body, whose centre of mass is not co-incident with the principal axis of rotation. Under these conditions the tensile forces are increased on the 'heavier' side and are decreased on the 'lighter' side. Thus the hoop stress in the rim will be in tension on the 'heavier side' and in compression on the 'lighter side'.

Fig. 9

Model 2. Outer rim is able to slide relative to main body

In order to determine the magnitude and direction of the forces postulated as being responsible for tectonic movement the model used in one in which the thin crust is able to slide relative to the solid body at the crust /mantle interface. By way of illustration Fig. 10 shows that if an unbalanced disc with an outer annular ring containing fluid is rotated about its principal axis, the liquid will move to the ‘lighter’ side. Fig. 11 shows an analogous situation with the sliding continental plates.

Fig. 10 & Fig. 11

If we consider the crust as being able to move relative to the mantle, albeit it over a long geological time scale, then a simple force diagram (Fig 12) can be constructed by making the following assumptions: the crust is a thin shell that is able to slide relative to the mantle, the forces due to eccentricity are superimposed on the stress caused by the general rotation and gravity, and the stress that is of interest for the purposes of tectonic movement is the differential stress due to this eccentricity.

Fig.12

By approaching the problem in terms of a thin shell moving relative to the mantle, it is possible to consider what increments of the tensile force are responsible for putting the Pacific basin under compression (note crumpled profile) and the African Plate under  tension. (note the Rift Valley). The calculations to derive the expression of the circumferential stress at the surface of the earth, which are based on the consideration of the eccentrically induced loads on the thin crust are detailed in Appendix 11. The term radius of eccentricity’ was introduced to denote the distance between the centre of mass and the major axis of rotation.

From Appendix 2 the following relationship was derived:

Maths

Eq.2 shows that the magnitude of the circumferential forces or in this case the derived circumferential stress is dependent on the distance between the geometric centre and the centre of mass, i.e. the ‘radius of eccentricity’. In a limiting case, if the ‘radius of eccentricity’ is zero, the rotating body will be balanced and the net force will be zero. Fig 13 shows this relationship in between F (circumferential stress) and the E (radius of eccentricity).

Fig. 13 below. Relationship between the radius of eccentricity and the circumferential stress

Fig. 13

Having derived an equation which relates the circumferential stress with the rotational velocity and the centre of eccentricity it would seem appropriate to consider the possibility of tectonic activity on the planet Venus. Due to the low peripheral velocity of Venus (1 Revolution in 243 days = 6.5 kmhr -1 ) as compared with 531.5 kmhr -1 on Earth, the centrifugal forces available as compared to the similar sized planet Earth will be in the ratio of (42.5) 2 / (531.5)2 = 1806.25/282492.25 = 0.006: 1. This would give a stress value of 3.9x10-3 Nmm-2 . (0.059 psig). The unbalanced centripetal forces thus needed for tectonic activity are negligibly small.

In order to better understand the magnitude of the calculated circumferential stress in the continental crust, it is helpful to relate the model to more familiar applications. (These are shown pictorially in Cartoons 2, 3 &4) The stress value of 7.29x10-2 Nmm-2 if applied to a 1 tonne braked motor vehicle with a rear surface area of 1000 mm x 1300 mm=1.3x106 mm2 will yield a push force 94770 N. In imperial units this equates to a push of 21305 lbf (pound force) or 9.5 tonf (ton force). Rounded up and put more simply, this equates to the vehicle being pushed by 118 people each of whom weighs 180 pounds (81.8 kg) (see Cartoon 1). As the incline between the height of the Andes (taken as 5 km) and distance between the Peru- Chile trench and the Cordillera –Real (taken as c.1000 km) is c. 1:250, the vehicle can be considered to be on a level surface for purposes of scaling. However a 3 tonne hoist will easily pull the vehicle up a 1:3 incline onto a pick-up truck It is also worth noting that an upward acting net force of 2.37 x 10-2 N/mm2  (3.5 psig) on a 60 metre long wing span of an aircraft is sufficient to keep a large 350 tonne aircraft flying.

A puff of wind with dynamic pressure as low as 0.135 x 10-2 N/mm2 (0.2 psig) acting on the large surface area of a ship’s sail will cause a boat to move across water. Thus the unbalanced centrifugal forces created by placing the centre of mass of the Earth 1 km off centre are large and cannot be ignored. The calculated circumferential forces if applied to the cross sectional area of the South American plate are more than sufficient to push it over the Nazca plate.

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