What would happen if the Earth was
 hollow, and all else remained equal ?
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What spin rate would be required
It seems highly likely that it would then accelerate away from the Sun. As it is unlikely for the Earth to become hollow, we could build a light weight machine to mimic the Earth or Saturn as closely as possible, to test the idea.

It is quite possible that a levitating, rotating, magnetic body will do within the Earth's magnetic field, what the rotating magnetic planets do within the Sun's magnetic field. That is, it will surround itself with an area of high voltage electricity and light.

If this machine is lightweight, and hollow, and the electric field created surrounds it completely, then it may form its own distorted magnetic or gravitational field. The machine would then attempt to accelerate towards the "center of gravity" of the field, which would be above it.

As the field would be continuously created, the machine would never reach the "center of gravity", but would continue to accelerate. The rate of acceleration would depend on the  mass of the machine and the strength of the field. The field strength is almost certainly related to the rate of rotation and the magnetic characteristics of the rotating parts.

As "Flying Saucers" sometimes are reported as having rotating parts on the outer perimeter, it would seem likely that the planets are creating their own field/s by the rotation of their outer perimeter and/or rings. The Earth's equatorial rotational velocity is about 0.5 kilometers / second, the band of electrical activity is at an altitude of about 100 kilometers, with a thickness of about 10 kilometers.

Saturn and its rings spin at a much faster rate, an average of about 13 kilometers/second. This appears to create a light source starting at about 650,000 kilometers from the surface, and about 100,000 kilometers thick.

What spin rate would be required