A$,$ B$,$ C$,$ and D are four masses carried by a rotating shaft at radii $100mm,150mm,150$ mm $,$ and $200$ mm $,$ respectively. The planes in which the masses rotate are spaced $500$ mm apart, and the magnitude of the masses $B,C,$ and D are $9kg,5kg,$ and $4$ kg $,$ respectively. By analytical method, find the required mass A and the relative angular settings of the four masses so that the shaft shall be in complete balance which leads to longer machine life.