When pulling a rope wound around an axle, how much force does the axle receive?

Question

Flywheel strength training uses a rope attached and wound around an axle that has a weighted disc attached. Pulling on the rope causes the axle to spin, and once the rope is fully unwound, the momentum of the axle and disc will pull the rope back in - the user is expected to continue pulling on the rope to bring it back to a stop.

Some of the pull force on the rope goes into rotating the axle/disc, but the axle still receives a horizontal pull force. How much force would the axle have to withstand if someone was pulling on the rope with X amount of force?

For a more concrete scenario: if someone can lift, say, 400lbs when at maximum effort, and they applied that same force to pulling on the rope (Ft above), how much force would the axle have to withstand (Fa above) if the moment of inertia of the disc is, say, 0.1 kg⋅m²?

Answer 1

Because the system is not translating, you can apply Newton's second law with zero acceleration. This means that the vector sum of all forces applied needs to be equal and opposite to the reaction force. Neglecting gravity, $F_t = -F_a$. That said the system will certainly accelerate rotationally and you can express this using the equation. $\Sigma M = I\alpha$, where $\Sigma M$ is the sum of moments about the center of mass, I is the moment of inertia, and $\alpha$ is the angular acceleration.