I read the ratings of different gear, such as cams and such, but I did not really understand what the ratings mean in terms of comparison with an actual fall. I did the analysis below. I am assuming a worst case 10 m fall. This would be climbing climbing to a height of 10 m, and then falling back 5 m back to the site of the anchor, then 5 m back down to the starting point. Basically falling the entire length of the rope. As expected, this calculation holds whether the fall is 20 m or 100 m, in the end, it depends only on the mass of the climber and the elongation of the rope.

There is a little error in this in that the climber would hit the ground before the rope would grab hold of the climber and start stretching and bringing things to a stop. But lets assume there is no floor, and the climber falls past the start; and then the rope starts holding and stretching.

Mass of Climber: M=80 kg
Acceleration due to gravity: g=9.8 m/s^2
Distance of fall: d= 10 m
Length of rope: x= d= 10 m
Dynamic elongation of rope: E= 30%
Duration of free fall: t= SQRT(2*d/g)= 1.4 sec
Velocity at the end of free fall: v= g*t= 14 m/s
Length of rope stretch: xs= x*E= 3 m
Duration of arrested fall: ta= xs/v= 0.2 sec
Acceleration back to rest: a= v/ta= 65 m/s^2= 6.7 g

Force of arrested fall: f=M*a= 5.2 kN
General equation: f(E,M)= 2*M*g/E, where E is dynamic elongation in %, M is mass in kg, and g is 9.807 m/s^2

Is this correct?

fernandoe@yahoo.com wrote:There is a little error in this in that the climber would hit the ground before the rope would grab hold of the climber and start stretching and bringing things to a stop.

Yeah, bummer; experiment fails because the ground interferes. Lots of ways around this, but it will be hard to convince your 80 kg guinea pig to repeat your test after the first failure.

Kidding aside: I never tried to calculate the deceleration, nor the forces of the arrested fall. When it comes to falling while rock climbing, my priority is not hitting the ground. And as your setup clearly illustrates, if the first piece of protection is at 5 m, the second piece should be much lower than 10 m to have a chance at that.

My little study here, says that, regardless of where you place the protection, 5 m, 10, 20 m, the braking force of the rope, and the forces on the protection pieces, does not change. It is dependent on the mass of the climber, and the stretch of the rope. If you fall a smaller distance; then there will be less rope in play, less stretch, and the deceleration to zero will still be about 6.7 g.

I did this because I am looking into Metolius mini cams, to save weight. I see that the strength of these cams is 10 kN vs. the 14 kN on the larger Camalot C4's I use. But, what does this mean exactly...

fernandoe@yahoo.com wrote:My little study here, says that, regardless of where you place the protection, 5 m, 10, 20 m, the braking force of the rope, and the forces on the protection pieces, does not change.

Only when ignoring the fall factor. When the climber falls right at the moment that he's placing the first protection above the belay on a multi-pitch route, the height of the fall until the rope gets taut is twice the length of the rope between belay and climber. That's a fall factor of 2, and indeed, that is regardless of how far the climber is actually above the belay, and the forces involved are the same for 5, 10 or 20 m. However, when there is already one piece of protection in at, say 5 m, and he falls while he's at 10 m, the free fall is 10 m, a fall factor of 1. And falling from 5 m above the last piece of protection but 60 m above the belay means a fall factor of 5 / 60 or 0.083.

The lower the fall factor, the lower the forces on the protection, climber and belay. It's not a linear relationship, but for all practical purposes I know that it's important to get the fall factor down, and on a multi-pitch climb that means placing protection shortly above the belay, and sometimes I place something just one or two meters higher, and perhaps another piece a few meters above that.

If you want to learn the physics, try this free MIT course: Physics of Rock Climbing. I'm interested in the practical side, but not enough in the theory to spend my time on it.

This assumes a factor 1 fall, that one piece of protection is placed. This is what i was trying to get at, whether using a lightweight piece with 10 kN was as safe option vs. using a heavier piece with 12 kN. I looked at the case for a factor 2 fall. The numbers are double. It still holds out, that if you are using a climbing rope that properly stretches, the actual fall distance is not important. What is important is the ration of the fall distance to the amount of rope in use. Thank you for the reference to the book. I will buy it. Actually I am a physicist by training; but optics is my specialty.

FYI, the strength of Black Diamond X4s (their version of Aliens or Metolious Master Cams) start at 9kn for the largest size and go down to 5kn for the smallest size. The strength of the largest Black Diamond Stoppers, which I have always considered to be bomb proof, is 12 kn. If you are worried about marginal placements, use thin double ropes and load limiting runners (e.g. Screamers).

fernandoe@yahoo.com wrote:... I looked at the case for a factor 2 fall. The numbers are double. It still holds out, that if you are using a climbing rope that properly stretches, the actual fall distance is not important. What is important is the ration of the fall distance to the amount of rope in use.

Yep, that's exactly it. The ratio is what's called the fall factor, and, although perhaps counterintuitive, the actual fall distance doesn't matter when it comes to the forces involved.

fernandoe@yahoo.com wrote:Thank you for the reference to the book. I will buy it. Actually I am a physicist by training; but optics is my specialty.

You're welcome. It's not a book though, but a course - and you can download the course materials for free!

I assumed Frenando was familiar with the fall factor as it is discussed in most comprehensive climbing how to books. If find it quite impressive that he was able to derive it mathematically.

Stronger is better, of course, but what matters more is reducing the fall factor as quickly as possible. If I can get a piece in from the belay stance, I will. That doesn't reduce it much, but it helps, and it might keep me off my belayer as well. And though there is not consensus on this, some start off by clipping the strongest anchor piece or one of the bolts.

I've never taken or had to catch an FF2, but there are plenty of stories out there about injuries, anchor failures, and dropped climbers. Placing gear well, and placing it early, is key.

Bob Sihler wrote: If I can get a piece in from the belay stance, I will. That doesn't reduce it much, but it helps, and it might keep me off my belayer as well.

Mark Twight called this 'protecting the belay' and it is considered a standard safe practice to place a piece of gear as soon as possible after leaving the belay. Mark has a story of his partner taking a factor 2 fall in the mountains onto his less than stonker belay. He accredits the fact that he was rooting through his pack for something to eat and accidently allowed the rope to slip through his belay device, effectively reducing the force, to preventing the belay from being ripped.