Consider the ring-shaped body of Fig. 12.35. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its plane.
(a) Calculate the gravitational potential energy U of this system. Take the potential energy to be zero when the two objects are far apart.
(b) Show that your answer to part (a) reduces to the expected result when x is much larger than the radius a of the ring.
(c) Use F, = -dU/dx to find the magnitude and direction of the force on the particle (see Section 7.4).
(d) Show that your answer to part (c) reduces to the expected result when x is much larger than a.
(e) What are the values of U and F, when x = O? Explain why these results make sense.