Statics of Ropes and Cables

First, on an unrelated note, I would like to share an anwser I think I found to be correct on a question I posted here a while back. I asked about how a smooth collar that was Not Pinned could prevent a moment. The problem I was faceing was the fact that in order to have "play" so that the collar is loose, there had to be some physical gap to allow for sliding. But if there is a gap, then when a moment is applied to the collar, it will move due to this gap a minute amount, which will result in some horrible complications. I think my resasoning in my gripe was correct, and if you actually did know the amount of gap/play or slop (whatever you want to call it) between the materials, you could calculate a moment and force that is prevented more accurately. (The amount of error in reality would depend on the geometry of the collar and the amount of play between it and the journal.) But the little devil here is the fact that there can be literally NO space between the collar and the journal, and yet it can still have the ability to slide, provided that the frictional force is zero! What a tricky little devilish detail! I guess thats why they call it an IDEAL CASE!

Anywho, My question is on distributed loads on cables. Why is a distributed load on a cable with respect to dx, but a distributed weight of a cable with resepct to arch length ds? (Oh yeah, and ive got alot more questions on this subject to come!)