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ensonik


Aug 30, 2010, 6:52 PM
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Pendulum fall speed
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I am currently reading through the ANAM 2009.

On page 32, the analysis of an accident that happened during a pendulum fall is pretty thorough, but mentions the following:

"..., if you fall from, say, ten feet to one side of and level with your pivot point, you will have the same speed at the lowest point of your swing as you falling straight down ten feet ..."

To say the least, this is surprising to a physics layman like me.

To describe this and be sure I really understand the full impact of a pendulum (yes, I over protect them even though I wasn't aware of this small yet non inconsequential detail), I drew up something a small child could have done better but nonetheless illustrates it well:



In other words, on both fall types, when I hit point B, I will be going at the same speed? (distances between A and B being equal of course).

(Yes, I know, I know: It's in the book ...., but you never know ... errors can slip in. So I want to confirm this on the internets where it's well known that errors are weeded out and only the real information bubbles through to the top)


bill413


Aug 30, 2010, 7:13 PM
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ensonik wrote:
I am currently reading through the ANAM 2009.

On page 32, the analysis of an accident that happened during a pendulum fall is pretty thorough, but mentions the following:

"..., if you fall from, say, ten feet to one side of and level with your pivot point, you will have the same speed at the lowest point of your swing as you falling straight down ten feet ..."

To say the least, this is surprising to a physics layman like me.

To describe this and be sure I really understand the full impact of a pendulum (yes, I over protect them even though I wasn't aware of this small yet non inconsequential detail), I drew up something a small child could have done better but nonetheless illustrates it well:



In other words, on both fall types, when I hit point B, I will be going at the same speed? (distances between A and B being equal of course).

(Yes, I know, I know: It's in the book ...., but you never know ... errors can slip in. So I want to confirm this on the internets where it's well known that errors are weeded out and only the real information bubbles through to the top)

As I read that statement, your diagram is in error. "Fall from...level with your pivot point...." So, in your pendulum diaram, point A should be at the same height as your anchor (and as the A in the straight down fall).


nailzz


Aug 30, 2010, 7:33 PM
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What Bill said ... plus:

The distance from what you have marked as A->B in your second diagram (the pendulum) would be greater than 10ft.

The distance between point A and your anchor would be 10ft.

The circumference of the circle would be 2(Pi)r. Your radius is 10ft and you are falling 1/4 of the circle, so your fall distance would be 2(Pi)10ft/4, or
15.7ft.

More than 50% longer.

I think what the book is saying, though, is that in either case you are moving downward for 10ft. Gravity is constant. It doesn't matter if you fall those 10ft straight downward or at an angle. At the bottom you are going to be travelling at the same speed (not counting things like friction and such).

At least that's what I'm taking away from the quoted text.


patto


Aug 30, 2010, 8:19 PM
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ensonik wrote:
To say the least, this is surprising to a physics layman like me.

Its conservation of energy.

If you fall 10meters and weigh 100kg then your body will have E=mhg worth of kinetic energy (~speed energy). E= 100 * 10 * g(~9.8) = 9800J of energy.

In practice a climbing rope stretches so a pendulum fall will be slightly slower as its not a perfect pendulum.


majid_sabet


Aug 30, 2010, 8:30 PM
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sounds like a case in yosemite . is it ?


TarHeelEMT


Aug 30, 2010, 9:51 PM
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Your chart is wrong. It's 10 vertical feet.

The motion of the pendulum just redirects the force that you gather in those ten vertical feet of falling into horizontal motion. At the base of the pendulum, you're traveling sideways with the amount of energy one gathers when falling ten vertical feet - just as if you hadn't been redirected by the pendulum and fell straight down.


The pendulum just changes the direction of the force.


cantbuymefriends


Aug 31, 2010, 1:45 AM
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I guesstimate that the speed at the bottom of the the pendulum will be roughly the same as in a straight-down fall before the rope starts to stretch.

As mentioned above, the velocity will be in the horizontal direction and not vertical though.


socalclimber


Aug 31, 2010, 3:44 AM
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Let us not forget that a pendulum low down can take your straight into the deck. I saw it happen once. The person was lucky because it was just dirt. Now if there had been talus, it would have been a whole different story.

Also bare in mind that if the swing ends in a corner, the end result can be very bad. You will likely be receiving a "trunk" blow. In other words, slamming the side of your body against the wall. Lot's of vital organs there that can get punctured or ruptured.


Good question.


ensonik


Aug 31, 2010, 3:49 AM
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majid_sabet wrote:
sounds like a case in yosemite . is it ?

It is.


Partner devkrev


Aug 31, 2010, 3:53 AM
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ensonik wrote:
I am currently reading through the ANAM 2009.

On page 32, the analysis of an accident that happened during a pendulum fall is pretty thorough, but mentions the following:

"..., if you fall from, say, ten feet to one side of and level with your pivot point, you will have the same speed at the lowest point of your swing as you falling straight down ten feet ..."

To say the least, this is surprising to a physics layman like me.

To describe this and be sure I really understand the full impact of a pendulum (yes, I over protect them even though I wasn't aware of this small yet non inconsequential detail), I drew up something a small child could have done better but nonetheless illustrates it well:

[image]http://imgur.com/memJH.png[/image]

In other words, on both fall types, when I hit point B, I will be going at the same speed? (distances between A and B being equal of course).

(Yes, I know, I know: It's in the book ...., but you never know ... errors can slip in. So I want to confirm this on the internets where it's well known that errors are weeded out and only the real information bubbles through to the top)

Watch the opening scene in hard grit.

Actually watch the whole movie, its good.


Toast_in_the_Machine


Aug 31, 2010, 4:39 AM
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So this got me thinking about if there was an easy formula for pendulum. Especially looking at it with rope and a drop to calculate the velocity into a vertical wall. It was easy to get to the speed of a full pendulum. (Best web example is here:http://www.worsleyschool.net/...sement/pendulum.html)

But my question was more: Knowing the height up and the distance over (and of course the rope length), was there an easy way to calculate (1) the force / tension when the rope "caught" and you began to swing and (2) what would the velocity be when you hit the wall. So I started to put pen to paper, but before I did, I did a quick look on the web and found this gem:

What is the proper equation for maximum tension force in a rope for a falling load

With the reference to the person's question being, you guessed it RC.com:
answers.yahoo.com wrote:
Fall Factor source:
The Standard Equation for Impact Force. Goldstone, R. (PDF, 2009)
ref:
http://www.rockclimbing.com/cgi-bin/forum/gforum.cgi?do=post_attachment;postatt_id=746
http://www.rockclimbing.com/...hment;postatt_id=746.

I thought that was funny.

(Now back to my pen and paper....)


majid_sabet


Aug 31, 2010, 6:48 AM
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ensonik wrote:
majid_sabet wrote:
sounds like a case in yosemite . is it ?

It is.

Man, my memories is good but there was a similar case in late 90's on El Cap where a guy took a 15 feet pengi and crushed his side to the wall except he had tons of gear on his shoulder harness ended up with two broken ribs. Rescuers rapped 900 feet and jugged with the injured climber up to the top. pengi falls are extremely dangerous cause you are dealing with down and side forces simultaneously and you can not predict where you may end up.


gunkiemike


Aug 31, 2010, 8:49 AM
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patto wrote:

In practice a climbing rope stretches so a pendulum fall will be slightly slower as its not a perfect pendulum.

That's the key. The other statements (ANAM, nailzz, tarheel) that speed would be equal because you've fallen the same vertical interval are only true in the theoretical case of a rigid, frictionless pendulum arm.

So now the question becomes: for a typical rope modeled as a Hooke's Law spring, can we (er...rg?) model the final speed of a pendulum fall? The rope stretches (absorbing energy) as it does work against the climber. The tension in the rope increases proportionally (sinusoidally I suspect) as the fall angle increases from vertical. But the calculations are way beyond my pay grade.


fresh


Aug 31, 2010, 10:40 AM
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ptlong2


Aug 31, 2010, 11:49 AM
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Ha ha ha!

But the Hooke's Law version is an easy enough problem to solve. Stretch does subtract some of the energy but it also adds to it by increasing the fall distance. The answer is that the difference between the a stretchy rope and a truely static rope is negligable, in theory.

Real life, with real ropes and slack added, there's going to be more of a difference but I can't say how much. Either way a big pendo fall into a corner is a bad idea, RG will concur.


darkgift06


Aug 31, 2010, 12:04 PM
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all the calcs & no one added the increase of forces based on the centrifugal force generated by the fall.


gunkiemike


Aug 31, 2010, 12:50 PM
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"Centrifugal" force is simply the climber's perception of the tension in the rope.


Rudmin


Aug 31, 2010, 12:59 PM
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gunkiemike wrote:
"Centrifugal" force is simply the climber's perception of the tension in the rope.

Centrifugal force would actually be the climber's perception of a field (similar to gravity) pulling hair cloths, everything away from the axis of the pendulum. It is an artifact of adopting a rotational frame of reference.

The tension in the rope is a centripetal force, and is an actual force no matter your frame of reference.


patto


Aug 31, 2010, 2:16 PM
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I can't believe this is still being discussed.

For a perfect pendulum with a perfectly static rope the speed is the same. The tension on the rope simplifies to the extraordinarily simply F=3mg. In other words 3 times the person's weight.

For a perfect pendulum with a dynamic rope the speed is less. The tension on the rope will induce stretch the rope and absorb energy.

For an imperfect pendulum with a dynamic rope it is all too complicated to work out, too many variables.


jt512


Aug 31, 2010, 3:12 PM
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patto wrote:
I can't believe this is still being discussed.

For a perfect pendulum with a perfectly static rope the speed is the same. The tension on the rope simplifies to the extraordinarily simply F=3mg. In other words 3 times the person's weight.

For a perfect pendulum with a dynamic rope the speed is less.

Maybe you're satisfied with the answer "less." It's obviously still being discussed because others want a little more specific answer.

In reply to:
For an imperfect pendulum with a dynamic rope it is all too complicated to work out, too many variables.

Too complicated for whom? Are you seriously suggesting that the problem is so complicated that it can't be solved?

Jay


(This post was edited by jt512 on Aug 31, 2010, 3:14 PM)


ptlong2


Aug 31, 2010, 5:20 PM
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patto wrote:
For a perfect pendulum with a dynamic rope the speed is less. The tension on the rope will induce stretch the rope and absorb energy.

Yes, less. But the stretch also lengthens the fall which adds energy to the system, nearly enough to make it a wash. For a Hookian rope the difference is maybe a percent or two, depending on the rope modulus you select. If you add damping to the rope it reduces the speed a bit more, but not much.


In reply to:
For an imperfect pendulum with a dynamic rope it is all too complicated to work out, too many variables.

Got a computer?



(This post was edited by ptlong2 on Aug 31, 2010, 5:29 PM)


jt512


Aug 31, 2010, 6:25 PM
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ptlong2 wrote:
patto wrote:
For a perfect pendulum with a dynamic rope the speed is less. The tension on the rope will induce stretch the rope and absorb energy.

Yes, less. But the stretch also lengthens the fall which adds energy to the system, nearly enough to make it a wash. For a Hookian rope the difference is maybe a percent or two, depending on the rope modulus you select.

If that's true then can you solve for the peak tension as follows?

The potential energy (PE) of the fall is converted to kinetic energy (KE) and strain energy (SE) in the rope. That is,

(1) PE = KE + SE .

(2) PE = mgh = mg(L + s) ,

where L is the length of the unstretched rope and s is the maximum rope stretch in the fall.

(3) KE = (1/2)mv ,

(4) SE = (k/2L)s (from rgold's paper),

where k is the rope modulus.

Substitute (2), (3), and (4) into (1), to give

(5) mg(L + s) = (1/2)mv + (k/2L)s .

From rgold's paper

(6) s = TL/k ,

where T is the maximum rope tension.

Now, substitute (6) into (5), compute v assuming an ideal pendulum, and solve for T.

Jay


(This post was edited by jt512 on Aug 31, 2010, 6:26 PM)


ptlong2


Aug 31, 2010, 6:38 PM
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jt512 wrote:
compute v assuming an ideal pendulum, and solve for T.

If you're assuming the speed of an ideal pendulum then you don't need the energy balance equation. T=(k/L)s is also equal to the centrifugal force plus the weight (assuming no vertical component of the velocity at the bottom of the swing).

It's easy enough to take that force balance and combine it with the energy balance to get a quadratic in s. Then you have s and v, as well as T.


Rudmin


Sep 1, 2010, 8:25 AM
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s and T are quite linearly related. You solve one and you have the other. The problem is that the centrifugal component of T is a function of v. That is m*v^2/(L+s). Simple enough to solve recursively, but it puts a crimp in your solution.

To run it through some simple numbers. Let's say 10 metres of rope with a 100 kg climber and a stiffness of 24 kN per metre. (that's assuming that only the 10 metres of rope is being used).

What I get is a velocity of about 50.2 km/hour and a rope tension of about 3kN. In essence, the rope stretch does next to nothing. If we take a quarter of the stiffness, as if you had almost a full rope length out, we get a velocity of 49.8 km/hour. So looks like no amount of climbing rope will help you in a pendulum.


(This post was edited by Rudmin on Sep 1, 2010, 8:26 AM)


ptlong2


Sep 1, 2010, 9:49 AM
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What I posted above is wrong. That is, you can't expect a force balance at the bottom of the pendulum. By coincidence it gives an answer that is very close to that of a numerical approach, at least for the first swing.

Rudmin, I'm not sure exactly what you did but when I run your numbers the machine gives slightly lower results. Same basic answer though. Interestingly the tension peak occurs after the bottom of the swing.

Now in a real pendulum fall you have a real rope, a belayer and some initial slack. I think it would still hurt.


(This post was edited by ptlong2 on Sep 1, 2010, 9:50 AM)

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