Ryfujk

I am wondering how Newton's 3rd Law of Motion would apply in the case of a man walking the length of a rotating spacecraft.

Please reference the drawing below.

Say that there was a spacecraft sitting still in interstellar space, far away from any stars and planets. The spacecraft then starts to rotate via an electric motor until it reaches a rotating speed that will mimic Earth's gravity for a man inside the spacecraft. The man walks from one end of the spacecraft to the other end. While he is walking, will the spacecraft remain stationary or will the spacecraft move in the opposite direction, thus obeying Newton's 3rd Law of Motion, for every action there is an equal and opposite reaction.

Also, if the spacecraft is set in motion by the man's walking, will the spacecraft remain in motion and travel in that direction indefinitely?

EDIT

If the length of this spacecraft is extended out to 1 km long and the man is walking at 3km/hour, at this rate it will take him 20 minutes to walk from one end to the other. Now since he transferred a lot of kinetic energy to the spacecraft in those 20 minutes, the spacecraft should be moving at a good rate of speed (based on the fact that the spacecraft is constantly accelerating while he is walking).

When he reaches the other end and comes to a stop, will his body mass coming to a stop transfer enough kinetic energy back to the spacecraft to bring the spacecraft to a stop?

In the frame of reference where the spacecraft is initially at rest, the momentum of the spacecraft (including you) will initially be zero. By conservation of momentum, it will continue to be zero the entire time you are walking and after you have stopped.

While you are walking, you will have a certain amount of momentum in your forward direction. Because the total momentum of the spacecraft-plus-you is zero, the spacecraft will have momentum of the same magnitude in the opposite direction. This means it will be moving very slowly towards your rear. Because it is much more massive than you, it will be moving backwards proportionally more slowly. Somebody watching from the outside would not be able to notice the motion.

Once you got to the other end, as you stopped moving forwards, it would stop moving backwards, thus keeping the net momentum at zero. This would be true no matter how you moved around inside the spacecraft before stopping.

It is wrong to think that you will be continually transferring kinetic energy while you are walking. You transfer a small amount as you accelerate to your walking speed, and then you perform no additional (net) work on the spacecraft until you stop. The momentum has to balance.

The effort you put into walking is to make up for frictional losses in your joints and against the ground, and it all gets turned into heat.

The spacecraft + astronaut can be treated as a system where the center of mass is moving. When the astronaut stops, the space station stops its astronaut-induced motion, because the momentum in that system is conserved.

The Third Law is how momentum is transferred between the parts of that system, but conservation of momentum is going to dictate the velocity of those parts relative to one another.

I think your confusion might be based around your contention that

he transferred a lot of kinetic energy to the spacecraft in those 20 minutes

Instead of considering the entire walk, consider each step. The astronaut is pushing against the "floor," and as he moves forward the spacecraft moves in the opposite direction such that momentum (in the spacecraft-astronaut system) is conserved. If he then stops, the station will stop as well. If he puts his opposite leg forward to take another step it is the same (momentum-wise) as if he stopped before taking another step. Prove this to yourself by considering a treadmill: as you walk the speed of the treadmill stays constant - it does not accelerate. Considering each step as a discrete action makes it more clear that he is not continually imparting more and more velocity to the spacecraft (remember, in the frame of the spacecraft he isn't accelerating once he begins walking at 3km/hr).

Maybe a simpler way to consider this that doesn't require kinesiology is the case in which the astronaut floats in the center at one end and pushes off with his legs towards the other end (left to right in your figure). The same forces are at play; he has essentially just imparted some velocity to the spacecraft in the opposite direction of his motion. If there were a hole at the right end of the figure he would shoot out into the void and the spacecraft would continue with the (relatively small) increased velocity. If we replace the jumping with a controlled explosion and the astronaut with exhaust, this is how rocket engines work. If we patch the hole and he hits the wall, he and the spacecraft will lose the velocity induced by the original jump and the system is as it was originally.