$3$ Five discs $()/(($ A$,$ B$,$ C$,$ D $/)),$ and $()/(($ E $/))$ are to be keyed to a rotating shaft at the positions shown in the fig. below. Initially, the disc $()/(($ B $/))$ is in exact balance, but $()/(($ A$,$ C$,$ D $/)),$ and $()/(($ E $/))$ have the out of balance to the extent $()/((1,3,2/)),$ and $2$ kg respectively at $250$ mm $.$ The relative angular positions of the eccentricities of $()/(($ A$,$ C $/)),$ and $()/(($ D $/))$ are as shown, and the system is to be put into complete dynamic balance by adjustment of the lateral and angular position of $()/(($ E $/))$ and by drilling a hole with centre at $500$ mm radius in the disc $()/(($ B $/).)$ Determine:

$($i$)$ the correct lateral and angular position of $()/(($ E $/).)$

$($ii$)$ the mass of the metal to be removed from $()/(($ B $/))$ and the angular position of the centre of the hole relative to $()/(($ A $/).)$

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