A particle moves without friction in a conservative field of force produced by various mass distributions. In each instance, the force generated by a volume element of the distribution is derived from a potential that is proportional to the mass of the volume element and is a function only of the scalar distance from the volume element. For the following fixed, homogeneous mass distributions, state the conserved quantities in the motion of the particle:
(a) The mass is uniformly distributed in the plane z = 0.
(b) The mass is uniformly distributed in the half-plane z = 0, y > 0.
(c) The mass is uniformly distributed in a circular cylinder of infinite length, with axis along the z axis.
(d) The mass is uniformly distributed in a circular cylinder of finite length, with axis along the z axis.
(e) The mass is uniformly distributed in a right cylinder of elliptical cross section and infinite length, with axis along the z axis.
(f) The mass is uniformly distributed in a dumbbell whose axis is oriented along the z axis.
(g) The mass is in the form of a uniform wire wound in the geometry of an infinite helical solenoid, with axis along the z axis.
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