Classical Mechanics - Special Interest Problem

There is a famous science fiction series by Larry Niven called the Ringworld series. The stories are based upon the inhabitants who live on a remarkable marvel of engineering. A giant ring (with a radius roughly that of earth's orbit about our sun) rotates about a normal G-2 star like our sun. The rotation provides "gravity" for the inhabitants who live on the inside of the ring. The ring is about 100 miles wide and the walls are high enough to keep in an atmosphere. A simple calculation of area should convince you that the living space dwarfs that of earth. The inhabitants have "devolved" from their original highly advanced predecessors and ... well, you'll just have to read the books.

After Niven wrote the first book, it received world-wide acclaim. But there was a problem. Although physicists generally liked the book as much as anyone else, Niven was beseige with letters of physics protest. As any advanced physics student could have told him, Ringworld is not stable in the plane of the ring. That is, if the ring is nudged slightly from the exact center, it will fall into the star. Because of this, Niven later wrote one book in the series based upon fixing this very problem. The proof of this instability can be found in Example 5.2 of your text, the 4^th edition of Classical Dynamics  by Marion and Thorton. The integration is difficult and an approximation is used.

There is a similar problem that is not quite so difficult. In lecture, we solved for the gravitational force between a uniform rod of mass M, length L and a point mass m located a perpendicular distance x from the center of the rod. The result for the force perpendicular to the rod was

F_x = ^GMm / _x(L2 _{+ x}2₎

Assume a Squareworld of length 2L and mass M on each side. The star (m) is at the origin and Squareworld is displaced a distance z from the origin. (Do not be concerned about the absurd gravitational peculiarities of a rotating Squareworld.) Find the expression for F_z exerted on the star by Squareworld. Show that at the equilibrium point (z = 0), equilibrium is unstable along the z direction. For the two vertical segments in the diagram below, you will be able to use the equation above directly. However, the top and bottom segments in the diagram are not centered on the star and you will to need to start from the original integral expression to find the parallel  component of the force that will be along the z direction.