"While we're on the subject, if you take account of friction between the rope and the top piece, then tension in the belayer's side of the rope is about 60% of the tension in the climber's side. The sum of these two forces is the force on the top piece. So, the top piece feels about 1.6 times the climber's weight, in a static situation. This ignores any other friction in the system, such as between the rope and intermediate protection or between the rope and the rock."

Can somebody explain the number 1.6 to me in more detail? Is this number based on an estimation of the friction between ropes and carabiners--or is it mathematically more precise than that?

My understanding is that it is an empirical result from experiments in which the tension in the belayer's side of the rope was found to be around 0.6 (or 0.67) times the tension in the climber's side. Since the total downward force on the anchor is the sum of these two tensions, the total force on the anchor is 1.6 (or 1.67) times the tension in the belayer's side of the rope (or, equivalently, the impact force on the climber).

Jay

This is the limiting case where the climber and belayer's ends of the rope are both parallel. As the angle deviates from this, the contribution of the belayer's side to the tension in the top piece will reduce. Counteracting this, however, is the fact that reduced contact area between rope and carabiner leads to reduced friction, increasing the tension in the belayer's side. So... it's complicated (read: more or less impossible) to calculate from first principles, and can only be estimated based on experiments.

The angle issue is minor enough that I think we can concentrate on a model in which both ropes are vertical. If you're going to get picky about rope angles, then you shouldn't consider the "parallel" case to be "limiting." The positioning of the belayer relative to the top anchor can result in a negative angle between the ropes at the top anchor (ie, the ropes cross) as easily as a positive angle.

Jay

Guess I wasn't clear enough. A bit more detail:

In the below force diagram, F is the force on the climber, and f is the frictional force through the carabiner, so that tension on the belayer side is F - f. Then as the angle changes, the vertical (Fv) and horizontal (Fh) components of the forces on the top carabiner are as shown. The total force on the top piece (Ft) is then calculated by good old Pythagoras.



Plugging in some numbers (setting F arbitrarily to 1, and f to 0.33 so that F - f = 0.67), we get the following for Ft vs. angle:

.

That's what I was getting at, anyway - that maximum force will be seen where belayer and climber are pulling in the same direction.

My main point was just that we should ignore the affect of the angle, at least at this stage of answering the OP's question, and just assume that both ropes are vertical, or at whatever angle the experiments were conducted that came up with the 1/3 "friction factor."

My much more picayune point was that if you are going to concern yourself with the effect of the angle on friction, then you shouldn't ignore cases where the angle is less than zero, and hence the friction is greater (I think) than where both ropes are plumb.

And, BTW, wouldn't f itself be a function of theta, rather than a constant independent of theta?

Jay

I'm just splitting hairs, really.

So... it's complicated (read: more or less impossible) to calculate from first principles..... the only way to get good estimates of forces is by experiments with various configurations.

This isn't true if by first principles you mean starting with the same assumptions you did. It is relatively straightforward to calculate Ft while including f as a function of theta.

As theta varies up to a pretty large angle the difference in force is small enough to be considered noise, as you've already admitted. And of course Jim Titt pointed out there are much larger real effects that are being ignored by this "first principles" approach.

But that doesn't mean that real forces cannot be estimated with models.

The difficulty lies in predicting with any accuracy how f varies...

But that's my point, really. You could put together a model entirely from first principles, but it would be really quite difficult and most likely a fair way off from reality. Easier to collect a bunch of experimental data to feed into a semi-empirical model.

But it isn't difficult.

If that were really your original point it would have been better had you simply stated that in the first place. It would have led to a different and possibly less trivial discussion.

It is when you start to consider factors such as rope age, stiffness and fuzziness, carabiner model, wetness, etc.