Why do gear forces need to be balanced at the pitch circle radius?

[bonodut]

Imagine we have two simple gears with different radii that are connected, with each gear on its own axis. If we fix the smaller gear so that it cannot rotate, then apply a force to the edge of the larger gear (perpendicular to its radius), the force applied to the smaller gear by the larger gear at the interface between them should be equal in magnitude to the force we are applying to the larger gear.

Since all the forces involved are perpendicular to the gears they are acting on, it should be true that if a force is being applied to one part of a gear, to keep it from moving we must apply an equal force to the opposing side of the gear.

In the scenario above where we want to keep the smaller gear fixed, it seems that if what I've said so far is correct, in order to prevent the two gears from rotating we would only need to apply a force equal in magnitude to the one acting on the larger gear.

Obviously this is not correct, but I can't pin down exactly where the argument fails.

 

[FactChecker]

Why do you think that is wrong?
Maybe I don't fully understand the problem. Although you don't say it, I will assume that the only force you are applying to the smaller gear is a tangent force to its edge.
If you are applying forces only on the contact edge of the gears and tangent to the gears, then what you say seems ok to me. The radius of the gears (and the fact that they are gears) would not matter.

However, if you are fixing the smaller gear at its axis and are talking about the torque there, then that is different.

 

[Stephen Tashi]

Obviously this is not correct, but I can't pin down exactly where the argument fails.

The situation(s) aren't described well. It isn't clear what a "side" of a gear means or where you intend to apply forces.

I suggest you think of the situation(s) in terms of a free body diagram. Pick one of the gears as the free body and explain what forces are acting on it.

 

[Dale]

When you have these kinds of questions, where your intuition clashes with a sketchy analysis, my first approach is to do a rigorous analysis. In this case, that would just be setting up the free body diagrams and solving the equations.

 

[Baluncore]

Obviously this is not correct, but I can't pin down exactly where the argument fails.

Why is it obviously incorrect? If the two shafts are fixed and the two meshed gears are free to rotate on the shafts then the perpendicular forces applied at the pitch circle radius will be balanced. That is because each gear is a virtual symmetrical balanced beam, with fulcrums at the centre of the shafts.
If you consider the forces on the shafts that support the gears, the shafts will provide a reaction force of twice the applied force. In this application the gear diameter, or gear ratio, is not important, the gears are called “transfer gears”.

You only need to consider the torque on the shaft when gears with different radii are fixed to a shaft, or when meshed gears are fixed to separate shafts.