If the system is frictionless, the rope is weightless, and the weights are not supported, then the lighter weight will rise and the heavier weight(s) will fall. Therefore if the rope is pulled very slowly, the lighter weight will rise first.
If the rope really is weightless and the pulleys really are fictionless (and inertialess) then it doesn't matter how hard or fast you pull, the lighter weight will rise first. This is at odds with your intuition simply because you have no (or insufficient) experience with weightless and frictionless environments. This is one reason why space is so bloody dangerous, in addition to the dangers posed by, say, diving, where similarly to space, your equipment has to work perfectly or you die.
In the real world, pulling fast enough will make the closer weight rise first.
>If the rope really is weightless and the pulleys really are fictionless (and inertialess) then it doesn't matter how hard or fast you pull, the lighter weight will rise first.
I'm considering a thought experiments that make me believe that this is not the whole story.
Imagine a two weight system set up similar to the original diagram in which the weights are the same weight, and the gravity is very little. Yanking on the rope I imagine them rising at the same speed. Now, we take a very small flake off of one weight and repeat the experiment. It seems clear that both weights will still rise from the start, just the lighter weight will rise at a faster speed.
I think this should extend to three weights of any positive mass -- if you yank the rope fast enough (and it might be very fast) all three should rise from the start.
If you take the rope from 0T to >30T instantly, then you will have all the weights lift up. As long as you keep the rope tense, they will all rise with different accelerations until they hit the top.
If you apply tension over time, even very quickly, then the lighter weight will rise first.
In your thought experiment you make the weight difference so small that otherwise insignificant factors (e.g. friction) take over. Go the opposite direction and make the weight difference enormous.
Assuming no friction and given a fast tug of the rope I still don't think the near equal weight would hold still or sink while the other near equal weight flew up quickly.
With the large weight difference, I think the required speed of pulling the rope to make them both rise just becomes impractically fast.
Imagine the system in space - all the weights would move up, so they all have an upward force applied to them from the rope tugging. It's just a matter of pulling so fast that that upward force overcomes gravity.
Your conclusion is incorrect given your assumptions. You are assuming no floor, in which case all the weights will rise. Only the man will fall. I think the problem was intended to include a floor that simply isn't drawn.
Do you really want me to write the full six page analysis? No, I didn't think so. Unfortunately, concise replies such as the one I gave and which require the reader to think a little often result in replies such as yours where it is difficult to discern whether you are being deliberately clever, deliberately trollish, or genuinely confused.
Let me expand.
I only assumed the weights were unsupported, not the man. I did that to assist the reader in understanding the analysis. I did not assume the man was unsupported - I had hoped my initial description of what happens implied that. Possibly it didn't.
I would expect that the problem is intended to include the floor - that's not my point. Having made the analysis for the unsupported weights, the evolution of the situation when there is a floor becomes obvious.
I'm not trolling, and only half joking. Your first paragraph where you talked about the heavier weight falling seemed like a joke about the drawing not including a floor. Which would be fine, except someone making that silly assumption should go all the way with it and assume the man is unsupported too.
Now I understand that you were helping people understand the analysis, and I'm sorry my reply sounded trollish.
If the rope really is weightless and the pulleys really are fictionless (and inertialess) then it doesn't matter how hard or fast you pull, the lighter weight will rise first. This is at odds with your intuition simply because you have no (or insufficient) experience with weightless and frictionless environments. This is one reason why space is so bloody dangerous, in addition to the dangers posed by, say, diving, where similarly to space, your equipment has to work perfectly or you die.
In the real world, pulling fast enough will make the closer weight rise first.
The truth lies somewhere in between.