I had stopped that problem, gave up. :(
But now I could retry it. When you say
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when the cylinder rotates, the cylinder moves horizontally (since no slip with ground). So even though the particle seems stationary, it also has moved horizontally.
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it makes me think I misused the word "slipping" in English (not my native tongue) but I found on the Internet
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The wheel rolls without slipping only if there is no horizontal movement of the wheel at the contact point P (with respect to the surface/ground). Thus, the contact point P must also have zero horizontal movement (with respect to the surface/ground).
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So I do not understand how can the cylinder move horizontally if there's no slipping. I need some explanation.