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Let us think about a rod whose length is L and end points are A and B. The road is parallel to the ground therefore the weight of the rod is perpendicular to the line between A and B.

The weight of the rod is F and two forces with the magnitude of F are exerted on both ends and the direction of these forces are opposite to the direction of weight.

The vertical force balance: 2F-F = m.a, therefore there is an acceleration towards the upper side.

However the torque balance makes me confuse. Which point do I need to use? If I use the mid-point the total torque is zero. However if I use one of the end points the magnitude of total torque is FL/2. Then what does it mean? Does it have an angular acceleration or not?

I thought about that maybe it means there is no angular acceleration around the mid-point axis and there is an acceleration around the end point axis. However if there is no initial angular velocity and no angular acceleration around the mid-point axis the relative positions of the points on the rod seem constant to me. Therefore it sounds ridiculous to me to say there is an acceleration around end points if there is no acceleration around mid-point.

Could someone please explain me what I am missing?