UKC

Why do so many arrested falls end up with the leader in an inverted position?

User error.

Because you can hit things on the way down with your feet which flip you.  That's the only way I've ever managed it.  Even dynoing on boulder problems and above a peg I stayed the right way up. 

Or you could just be doing a particular climb that invites a particularly uncontrollable fall like L'Helicoptere L'Hélicoptère (f7A). 

In the last few years I've climbed a lot at crags where the vast majority of people are redpointing sport routes, and I must have seen hundred of falls. And I think I've only ever seen 1 inverted fall, on a trad climb. Are they really that common?

Actually, I did invert once a long time ago when I snapped an undercut and pitched backwards, so that's one way. Rope and leg I would guess is the biggest danger.

This is mainly because the attachment point - the harness tie-in - is close to the climber's centre of gravity. If the harness is fitted low, the climber's centre of gravity can actually be above the attachment point.

Good point but that would only flip you after the rope comes tight.  That wasn't what happened with me at least, and I find being flipped as a fall is being stopped a bit hard to imagine intuitively, compared to rotating as I fall off and continuing to rotate in flight.

In my experience most falls do not end up with an inversion in fact a very small percentage.  Again in my experience inversion is usually caused by getting the rope behind ones leg. i.e leader error/ bad luck.

If you want to heighten the COG then tie off some coils! Or wear a chest harness.

I think the lesson is more about wearing a helmet than any issue with harness design.

The friction of the rope through the karabiners is enough to start the backward rotation. In a short fall, that rotation is very minor, but with a longer the "peel" there is likely to be a noticeable backward rotation. Possibly the higher aerodynamic drag of the upper part of the body, compared with the lower,  tends to resist the rotation.

For a stable non-inverted hanging position at the end of a fall you want to have the centre of gravity (COG) below the attachment point. Raising the COG would make this harder to achieve. Tying off coils and putting them round your shoulder would raise the centre of gravity slightly and make matters worse. Wearing a light chest harness would probably raise the attachment point more than the COG and thus make matters a little better.  

ps to last. I think the biggest factor (based on my own, limited, empirical experience of falling off) is which points of contact fail first. If the feet go first, the body stays fairly upright, whereas if the hands go first  one tends to rotate backward somewhat. If the hands burst of a layback, while the feet are still thrusting against the rock, the upper body goes rapidly backwards - which give one a fine view of the crag rushing away!

Do you have a lot of experience with top heavy climbers flipping upside down or are you just speculating? 

If a climber's centre of gravity is above their rope loop, what do they do on a hanging belay?  Why don't they flip upside down when arresting a fall?

I don't think air resistance is a significant factor with only 10-30m of rope out.

I have no experience of heavy climbers flipping upside down - I have never seen that and I didn't say that! My only speculation was re air resistance (hence I said "possibly" at the beginning of that sentence). The only time I ended up rotating beyond the horizontal to a slightly head down position was when one of my heels clipped a ledge halfway down. On that fall (which was about 70 feet) I had already rotated to an almost horizontal position by the time I clipped the ledge. That fall was a case where my hands slipped first.

I said the climber's centre of gravity is close to the attachment point - not above it - unless the harness is set too low.

Actually, I have always been surprised just how well most waist harnesses perform in falls - this is probably why chest harnesses are hardly used nowadays. (The only time I have used chest harnesses was for my children when they were very young.) 

Going back to the OP's original question, if the climber does rotate beyond the horizontal the arrest tends to pull the attachment point somewhat towards the feet, putting the climber's centre of gravity on the wrong side of the attachment point so that the climber tends to remain "inverted". It is obviously very important that waist harnesses are sufficiently tight that they can't slip over the hips. Luckily, the proportion of falls that end up with the climber inverted is quite small in my experience (which is also Wayne S's experience, above).