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Suppose for example that we have a ladder in space and it has zero translational motion, but non-zero rotational motion and non-zero angular acceleration around its center of mass. The ladder is rotating faster and faster around its center of mass while having no translational motion.

There is no question that you have to apply a certain torque about its center of mass to produce this particular angular acceleration. A real life example is the specific torque needed for opening a door with a certain angular acceleration. This specific torque is produced by a specific combination of (a): Magnitude of external forces and (b): Direction/distribution of the external forces acting on the body.

Now let's take the exact same force distribution acting on the ladder but take the torque about an arbitrary fix axis in space outside the ladder. This will produce a torque about that axis with an amount that we can calculate. But what is the meaning of the torque about an axis outside the body? Is there practical real life examples where torque around an axis outside a body apply? Is it useful in certain theoretical calculations?

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Regarding the first paragraph, in order to produce rotation around the center of mass of the ladder without translation relative to another frame of reference in space, you would need to apply a force-couple (equal, but opposite, parallel forces) to locations that would cause some type of rotation. As an example, visualize two rocket thrusters of equal and opposite forces applied to each end of the ladder.

I don’t see how the “real life” example in the second paragraph applies in space (maybe its not intended to). The door has a fixed hinge and a single force is applied. The hinge prevents translational motion relative to the hinge. How would you visualize representing the hinge in space?

Regarding the 3rd paragraph, I’m having a hard time visualizing what you are saying. (drawings/pictures are so needed in discussions like this). However, it is common in statics to take moments (torques) about axes outside a rigid body as part of getting the sum of moments and forces to be zero.

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