Two Sided Pulley!

There is a light, inelastic thread is stretched round one half of the circumference of a fixed cylinder as shown above.

As a result of friction, the thread does not slip on the cylinder when the magnitudes of the forces acting on the ends satisfy the inequality \[\frac{1}{2}F_A\le F_B\le 2F_A\]

Determine the coefficient of friction \(\mu\) between the thread and the cylinder.

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