I'd also like more insight. I got the same problem incorrect. How do you know that the centre of mass changed enough for the system to move out of translational equilibrium?
My impression of this concept after reading the answer is that since the fulcrum is not physically attached to the rod (I assume), a change in the centre of the mass can cause the system to move out of equilibrium. As the system rotates (clockwise in this example), the centre of mass shifts to the right and so now the net force in the downward direction exceeds the net force in the upward direction (normal force from the fulcrum).
Anyone else have other ways of conceptualizing this problem?
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Edit: I think I have a solution. Hopefully someone can correct me. I think question 12 - 14 for this passage all relate to different scenarios, not necessarily the scenario presented in the passage.
For instance, if you want a system to be in equilibrium such as is the case of two masses on a bar balanced on a fulcrum, then both net translational and net rotational forces must = 0. If one of them changes, then you can no longer be in equilibrium.
Now, if we move away from this scenario and talk about another unrelated scenario, then you can have no net force, but still have a net torque. Likewise, you could have no net torque and a net force. A net torque = 0 tells us that we not angular acceleration. A net force = 0 means we have no translational acceleration. However, in scenarios unrelated the one described in this specific passage (two masses on a bar balanced on a fulcrum), there is no reason why we can't have a net force and no net torque (think of an object in free-fall). It has a net force in the downward direction due to gravity, but no net torque because it does not rotate. Likewise, if we think of the Price is Right wheel, that has a net torque (i.e., a net angular acceleration) due to the force we apply to it, but it is in translational equilibrium since there is no net force (it doesn't fall since it is physically attached).
In summary (and someone please correct me if I am wrong), I think that this passage is a very specific scenario where both net rotational and translational forces must = 0 for the system to be in equilibrium as the centre of mass can change. In the scenarios I described above, I don't think we can change the centre of mass.
Hopefully someone can double check this.