In the figure above, there is dumb-bell of mass \(m=2.5 \text{ kg}\) which is rolling without slipping on a rough horizontal surface with a velocity \(v\). The horizontal merges into an incline which subtends an angle \(A\) with the horizontal in clock-wise direction. Find the minimum value of \(A\) for which the dumb-bell gets separated from the horizontal surface for any velocity \(v\).

If \(A\) can be expressed as \(A=\arcsin { \left( \dfrac { \sqrt { P } }{ Q } \right) } \), where \(P\) is not a perfect square integer, find \(P+Q\).

Consider the dumb-bell as 2 light spheres connected to a heavy axle.

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