Several planets (Jupiter, Saturn, Uranus) are encircled by rings, perhaps compos

Question

Several planets (Jupiter, Saturn, 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 3.0 x 1022 kg and outer radius 4.9 x 108 m (the figure). (a) What gravitational attraction does it exert on a particle of mass 75 kg located on the ring's central axis a distance 3.3 x 108 m from the ring center? (b) Suppose that, starting at that point, 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?

Explanation / Answer

(a)

   according to the given problem we can seethat all points on the ring are at the same distance

   r = ?(x2 + R2)from the particle

   from the theory we have the equation for thegravitational potential energy as

   U = - G M m / r

       = - G M m /?(x2 + R2)

   the force will be along the x axis so wetake the derivative of U wrt x

   so the force of attraction willbe

   F = G M m x (x2 +R2)-3 / 2

(b)

   as the particle falls from rest to thecentre the magnitude of the loss in potential energy willbe

   U = G M m [(1 / R) - (1 /?(x2 + R2))]

       = (1 / 2) mv2

   v = ........

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