Question

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 $==0, y>0$. (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 sections and infinite length, with axis along the $<$ axis. (i) The mass 13 uniformly distributed in a dumbbell whose axis is oriented alone the $z$-axis (g) The mass is in the form of a uniform wire wound in the geometry of en infinite helical solenoid, with axis along the $z$ axis.

          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 $==0, y>0$.
(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 sections and infinite length, with axis along the $<$ axis.
(i) The mass 13 uniformly distributed in a dumbbell whose axis is oriented alone the $z$-axis
(g) The mass is in the form of a uniform wire wound in the geometry of en infinite helical solenoid, with axis along the $z$ axis.

Added by William J.

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