Why is Larry Niven's Ringworld Unstable? In his 1970 science fiction novel Ringworld, author Larry Niven describes the eponymous Ringworld, a gigantic structure shaped as a ring with a radius of around 1 AU, rotating around a star in the center of the ring. This system is described as physically stable; however many readers have complained that it is actually unstable and the structure will drift away in time.
My first thought reading this is that the Ringworld's center of mass is identical to the star's center of mass, so the system should be stable. Why is it unstable?
Some additional details of the structure:


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*Radius: ~1 AU

*Mass: ~1 Solar Mass

*Year duration: ~220 hours


The Ringworld is also made of material strong enough to withstand the stresses affecting it in such a system.
 A: http://www.alcyone.com/max/writing/essays/why-niven-rings-are-unstable.html
The short version: a Dyson sphere is stable because even if the sphere gets off-center, there is no net attraction or repulsion (the extra mass of the parts further away help offset any attraction from parts closer to the star).
The ring doesn't have this advantage because, being confined to a plane (mostly), it doesn't have sufficient mass further away to counteract the runaway attraction that would happen if the ring moved off-center.
A: It is only for a spherically symmetric shape that you can treat an extended body as if it were a point mass at the CoM. 
The Ringworld is 


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*stable against axial displacements after which it will gently bob back and forth around the star.

*unstable against transverse ones because the gravitational attraction of the near-side is greater than that of the far-side.

A: Both are unstable (ring and sphere) for the same reason: the potential everywhere inside either one is zero. 
This is true for gravitational as well as electrical and magnetic forces, which are all inverse square law / central force situations, and it is that pattern which causes the result.
The proof requires calculus, but is considered an elementary derivation easily done by first-year students in physics or math.
