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\includegraphics[max width=\textwidth, alt={}, center]{467d7747-6a07-40ea-bb47-41ea3117f233-7_442_1006_251_532} A uniform \(\operatorname { rod } A B\), of weight \(W \mathrm {~N}\) and length \(2 a \mathrm {~m}\), rests with the end \(A\) on a rough horizontal table. A small object of weight \(2 W \mathrm {~N}\) is attached to the rod at \(B\). The rod is maintained in equilibrium at an angle of \(30 ^ { \circ }\) to the horizontal by a force acting at \(B\) in a direction perpendicular to the rod in the same vertical plane as the rod (see diagram).

1. Find the least possible value of the coefficient of friction between the rod and the table.
2. Given that the magnitude of the contact force at \(A\) is \(\sqrt { 39 } \mathrm {~N}\), find the value of \(W\).