Consider a system shown in the figure, where $AB$ and $BC$ are two beams with

lengths l$\_(1),$ and $l_2,$ respectively, resting on two spheres of radius R$\_(1)$ and $R_2.$ The bigger sphere of radius $R_1$ and the smaller sphere of radius $R_2$ have a friction coefficient of ${}_1$ and ${}_2,$ respectively, between the sphere and the ground. These two spheres are

attached by an inextensible string of $C_1{C}_2$ at their center, as shown in the figure.

Two people of weight, $W_1$ and $W_2$ are always at a distance from each other, as shown. $($a$)$ For this equilibrium condition and maximum tension in the string $C_1{C}_2,$ find the value $$ and maximum tension in the string $($The values should be independent of $a)$

$($b$)$ After the maximum tension is reached, the string suddenly breaks, and the bigger sphere starts to roll and reaches the end of beam $AB$ with its centre and the end $A$ of beam $AB$ aligning together in a straight line. At this time, to balance, the man with weight $W_2$ moves to point B$.$ Analyse whether the man did the correct thing or not.