S 3.1 In the sci-fi classic Ringworld, Larry Niven suggested that an advanced civilization would be...

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S 3.1 In the sci-fi classic Ringworld, Larry Niven suggested that an advanced civilization would be capable of constructing a variation of a Dyson sphere where all the mass contained in a solar system (with exception of the star) was used to construct a circular ring that would encircle the star at a habitable orbital radius. It was then pointed out that this ring world would be unstable, and would eventually fall into the star. In this problem, we will show this. R M (a) Consider a thin ring with radius R and mass m spread uniformly on it (see above figure). A mass M is placed at a distance from the center of the ring. Show that the gravitational force on M is given by GmM T Fr = (R cos - x) T Jo (R + x - 2Rx cos 0)/2 -do, (1) (b) Unlike the spherical case, this integral cannot be expressed in terms of elemen- tary functions (elliptic functions are needed instead). To proceed further, take u = x/R < 1, and expand the integrand of Fx to a Taylor series about u = 0 and drop terms that are proportional to un for n > 1 (or, in physics jargon, keeping terms linear in u). With this approximation, show that the force on M is Fx = 1 GmM 2 R3 (2) and is pointed away from the center of the ring. (c) Argue using physics-that the ring world in Larry Niven's book is unstable. S 3.1 In the sci-fi classic Ringworld, Larry Niven suggested that an advanced civilization would be capable of constructing a variation of a Dyson sphere where all the mass contained in a solar system (with exception of the star) was used to construct a circular ring that would encircle the star at a habitable orbital radius. It was then pointed out that this ring world would be unstable, and would eventually fall into the star. In this problem, we will show this. R M (a) Consider a thin ring with radius R and mass m spread uniformly on it (see above figure). A mass M is placed at a distance from the center of the ring. Show that the gravitational force on M is given by GmM T Fr = (R cos - x) T Jo (R + x - 2Rx cos 0)/2 -do, (1) (b) Unlike the spherical case, this integral cannot be expressed in terms of elemen- tary functions (elliptic functions are needed instead). To proceed further, take u = x/R < 1, and expand the integrand of Fx to a Taylor series about u = 0 and drop terms that are proportional to un for n > 1 (or, in physics jargon, keeping terms linear in u). With this approximation, show that the force on M is Fx = 1 GmM 2 R3 (2) and is pointed away from the center of the ring. (c) Argue using physics-that the ring world in Larry Niven's book is unstable.