Several planets (Jupiter, Saturn, and Uranus) are encircled by rings, perhaps composed of material that failed to form a satellite. In addition, many galaxies contain ring-like structures. Consider a homogeneous thin ring of mass M and outer radius R (Figure).

(a) What gravitational attraction does it exert on a particle of mass m located on the ring's central axis a distance x from the ring center?

(b) Suppose the particle falls from rest as a result of the attraction of the ring of matter. What is the speed with which it passes through the center of the ring?

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