A rotating shaft carries three masses, $\backslash ($ A$,$ B $\backslash ),$ and $\backslash ($ C $\backslash )$ rigidly attached to it; the centers of mass are at $\backslash (40\backslash$mathrm$\{$~mm$\},50\backslash$mathrm$\{$~mm$\}\backslash ),$ and $55$ mm respectively from the axis of rotation; $\backslash ($ A $\backslash )$ and $\backslash ($ C $\backslash )$ are $\backslash (10\backslash$mathrm$\{$~kg$\},6\backslash$mathrm$\{$~kg$\}\backslash ).$ The rotating unbalance on the shaft itself due to inconsistencies of shaft material density and casting irregularities can be considered as an extra $\backslash ($ D $\backslash )$ mass of $5$ kg attached to the shaft at $45$ mm from the axis of rotation. The axial distances between planes carrying masses $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ are $500$ mm $,$ and that between $\backslash ($ B $\backslash )$ and $\backslash ($ C $\backslash )$ is $600$ mm ; the eccentricities of $\backslash ($ A $\backslash )$ and $\backslash ($ C $\backslash )$ are at $\backslash (90^\{\backslash$circ$\}\backslash )$ to one another.

By sketching a couple polygon and force polygon, find for complete balance:

$($a$)$ The angles between $\backslash ($ A$,$ B $\backslash )$ and $\backslash ($ A$,$ D $\backslash ).$

$(10$ points$)$

$($b$)$ The axial distance between the planes of revolution of $\backslash ($ C $\backslash )$ and $\backslash ($ D $\backslash ).$

$(10$ points$)$

$($c$)$ The mass $\backslash ($ B $\backslash ).$

$(05$ points$)$