Bucephalus01 said:
SO I think you're saying that the rope would have some kind of coefficient of elasticity or something. So where would I find the kind of formula that I'm looking for to find this out?
Let's just say I know the coefficient or, I can work out how much the rope stretches before it breaks.
Classifying the rope by a single coefficient is probably quite a simplification. For example, the same rope might be able to pull a force F for a time T without measureable damage, but a larger force G > F for a shorter time S < T. Of course, you probably want to stay well within the accepted range for you force, and this range is hopefully determined by the manufacturer with a large safety margin.
If you know the coefficient of elastiticy for you rope for small streches, you will be able to model the situation by treating the rope as a spring with a given spring constant using Hookes law for a spring. But this is probably not a very good model for the rope, perhaps apart from very small amount of stretch if you are lucky. I'm sure a much better model involving several coefficients can be constructed if you have this data available for you rope.
When it starts to break you are getting into some pretty complicated phenomena, which would be very difficult to model. The manufacturer should perhaps provide experimental data on how much on average the rope can be stretched for different amounts of time, and the resulting damage to the rope in each case.
But it is interesting: Consider a rope that can slowly lift a mass M, but doesn't stretch very much. Then consider a more flexible rope which is only able to slowly lift a mass M/2. There might be situations in which the more rugged rope will break, but the other rope won't. Because the "impact force", subjected to the more flexible rope are much smaller, e.g. in your proposed situation.