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Suppose we have a horizontal rigid bar held in place by two reinforcement beams (one above the bar and one below the bar), both of which are attached to a wall, such that they "clamp" the bar from one side. This system is like a cantilever support, where one end of the bar is held firmly and the other end is free. See the diagram below.

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Now let's say the bar is of uniform density and of mass $m$, so there is a gravitational force $mg$ acting at the center. To prevent the bar from falling down, we need some force upward acting on the bar. The only possible other forces are provided by the support beams on the left. Let's say they exert normal forces $N_{1}$ (up) and $N_{2}$ (down).

Now if both support beams touch the bar at exactly the same $x$-coordinate (where we say $x$ is a coordinate of a horizontal axis), the bars supply a force $N = N_{1}-N_{2}$ upward at one $x$-coordinate of the bar.

To prevent the center of mass of the bar from moving, the net force must be zero, so we must have $N - mg = 0$. However, if we consider the torque about the center of mass of the bar, we find the net torque is $\tau = -r_{N}\cdot N + 0\cdot mg \ne 0$. The net torque is nonzero, so the bar must rotate. However, it shouldn't rotate, by design of the problem.

Is there a mistake made in my reasoning leading to this contradiction? Is the assumption that the two support beams touch the bar at exactly the same $x$-coordinate invalid?

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If you tried to hold the bar simply by pinching it at one point, you are right: you would not be able to exert the torque that is needed to keep it horizontal. In practice a clamp of this kind produces a contact force all along the length of the part of the clamp that is touching the bar. In this way it can provide the torque as well as the net force.

For a simple case you could consider where there are just the forces you mention, plus one more: a downwards reaction force on the bar where it touches the wall. This will be present if we suppose the bar is embedded in the wall or something like that.

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