$10-$ Three rotating masses, $A=14kg,B=11kg,$ and $C=21kg$ are carried on a shaft with centers of mass $275mm,400mm$ and $150$ mm $,$ respectively from the shaft axis. The angular positions of B and C are $60$ and $135$ respectively from A measured in the same direction. The distance between the planes of rotation of A and B is $1\mathrm{.}35$ m and between those of A and C is $3\mathrm{.}6m,B$ and C being on the same side of $A.$ Two balance masses are to be fitted each with its centre of mass $225$ mm from the shaft axis in planes midway between those of A and B $,$ and of B and C $.$ determine the magnitude and angular position with respect to A of each balance mass. $($Uni of London$).$

Answer: $36\mathrm{.}7$ kg between A and $B$ at $187$ to $A$;$29\mathrm{.}3kg$ between B and C at $310$ to A $.$