$A,B,C$ and D are four masses in a complete dynamic balance. They are carried by a rotating shaft at radii $120,130,210$ and $180$ mm respectively. The planes in which the masses revolve are spaced $450$ mm between masses A and B$,300$ between B and C and $150$ between C and D$.$ The mass of B$,$ C and D are $10kg,5kg,$ and $4$ kg respectively. Find the followings:

$($a$)$Draw two space diagrams: $($i$)$ the positions of the planes along the shaft and $($ii$)$ the angular positions of the masses $($angles measured from positive x $-$axis$).$

$($b$)$ Construct a table and tabulate all necessary data.

$($c$)$ Choose appropriate scales for the couple and force diagrams and write them clearly.

$($d$)$ Draw the couple diagram and relative angular settings of the three unbalance masses.

$($e$)$ Draw the force diagram.

$($f$)$ Determine the magnitude and angular position of the balance mass.