If the vertical distance between the fingers is constant, then the angle which the rod must turn through increases as they get closer together. The new position $y'$ also satisfies $xL=y'R$ so $y'=\fracML^2$ at the centre, so the angular acceleration $\alpha=\tau/J$ decreases as the pivot reaches the centre. Mathematically, suppose we start with $xR$ so the right finger slides toward the centre, increasing $R$ and decreasing $L$. The sliding finger then "overshoots" the static friction balance-point ($\mu_s L=\mu_s R$) and continues until the kinetic friction force on this finger becomes slightly greater than the static friction force on the stationary finger ($\mu_sL=\mu_kR$). This alternation could happen because the coefficient of kinetic friction $\mu_k$ (on the sliding finger) is slightly less than the coefficient of static friction $\mu_s$ (on the stationary finger). In practice the fingers slide alternately, first one then the other, until they meet at the CM. In ideal conditions they will then both slide towards the CM at the same time. When your fingers reach the same distance from the CM they bear the same weight. This brings the CM closer to that finger, which increases the weight that it bears and the friction force on it, and reduces the forces on the other finger. So if you push your fingers together just enough to make one slide, the ruler will slide over the one bearing less weight - ie the finger further from the CM. The finger which bears the greater weight $N$ (ie that nearer the CM) has the greater (static) friction force $F=\mu_s N$ with the ruler, where $\mu_s$ is the coefficient of static friction. If they are distances $x$ and $y$ from the CM, then by balancing moments we find that they bear weights $L$ and $R$ such that $L+R=W$ and $xL=yR$. The one which is closer to the centre of mass (CM) bears the greater share. The two fingers do not share the weight $W$ of the ruler equally. The explanation is the interplay between the friction forces and the distribution of the weight of the ruler as the fingers move along it.
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