Once the chain is swung and wraps around the desired structure could the magnetism be strong enough to support a person swinging from it?
It's possible with a sufficiently strong magnet.
For work I have found myself operating close to a table made of a permanent magnet which was used as a base on which an heavy object was supposed to magnetically levitate and move.
Before being allowed to get close to such table, I had to follow a safety training and learn all the safety procedures involved, which boil down to:
- always cover the magnetic table with thick plastic covers before doing any work on it
- do not wear anything which can be attracted by the magnet on your hands, arms and neck
- if you happen to drop anything while working on it, do not try to stop its fall
The magnet was so strong that if an allen key or a wrench would have found some fingers between itself and the plate, it would have easily crushed their bones to get closer. Even dropping a simple washer on it would have resulted in excruciating efforts to get it back.
Use a magnetorheological fluid to bind a ball and socket chain
You obviously will want some kind of electro magnet because permanent magnets will bind to themselves prematurely and constantly get all tangled up, but you also can't afford to use up a lot of voltage because a chunky high voltage wire will weigh your rope down too much to be able to throw it any reasonable distance.
So the optimal solution will be one that uses an electromagnet in a way that receives the best possible mechanical advantage so that it requires less input energy than what is required to hold your weight. This is where the magnetorheological fluid comes in. Magnets are on thier own are generally pretty easy to slide apart; so, just using a simple electromagnet would require one with much more binding force than your actual weight.
Magnetorheological fluids however are fluids that harden into a plastic like substance when exposed to a magnetic field. However, they come with an interesting property in which this effect in increased when you also try to compress it; so, if you have this fluid, run a weak magnetic field through it, and then compress it, it hardens into a tough solid material able to hold up more weight than the input energy would suggest.
So, what you do is you make the last few feet of your "rope" out of a ball-and-socket chain "lubricated" with MF surrounded by an electromagnetic sleeve. When you throw your rope and the end wraps around something, you press a button, and the MF hardens cementing the segments of your ball-and-socket chain together. Then when you got to put your weight on it, it compresses the MF inside of the chain joint collars which further hardens the bind preventing it from unwrapping until you turn the power off.
I think if you had VERY strong magnet(s) in the chain then this would work fairly well.
Have the last link in your chain be magnetised, and then a few links back from that have another magnet - with the poles arranged so that when these 2 links touch, the magnetic attraction is aligned and then they will hold together very well indeed.
Ideally, you would want the rest of the chain to be constucted from a non magnetic material, and one that had sufficient tensile strength to carry the load you want to swing from it. In addition, you'd want the rest of the chain to not be a brittle substance, otherwise you risk it cracking or even breaking over time due to repeated stress causing fatigue
While you can definitely get magnets strong enough to hold your weight, as well as the additional force from swinging or climbing, such magnets are likely not practical to keep around.
However, if you can get your chain to wrap around the anchor point a couple times, you could probably rely on the loops to hold the weight and the magnet can just keep the chain from unlooping.