The Physics Behind Why a Rope Breaks When Falling from a Cliff

[Dacourt]

Hi
In a school interview the physics teacher me why a rope attached to the top of a cliff, which would support a climber's weight while dangling gently, would break if the same climber fell from a height while attached to the rope.
Im guessing it's due to increased momentum but the rope equations and ratings I've seen don't really reflect that. It's also the kind of thing it's hard to Google. I've even asked a physicist (medical). It's probably quite simple but has been on my mind for 32 years! Any input appreciated.

 

[Vanadium 50]

Rope equations?

 

[Doc Al]

why a rope attached to the top of a cliff, which would support a climber's weight while dangling gently, would break if the same climber fell from a height while attached to the rope.

For the rope to bring the falling climber to rest in a short distance (the rope cannot stretch that much) it must exert a much greater force on the climber than just his weight. If that force exceeds the breaking strength of the rope -- oops! Climbing ropes are rated for being able to withstand such falls.

 

[mjc123]

I think "would break" is a little dramatic; I rather suspect climbing ropes are designed to cope with a fall from height, up to a point, not just a dangling climber.
However, to answer the question. What force is exerted on the rope by a dangling climber of mass m? Suppose he falls from height and is falling at velocity 10 m/s when the rope arrests him. What is the momentum transferred to the rope? If his velocity drops from 10 to 0 over, say, 0.1 s, what is the (average) force exerted on the rope during this time? See where this is going?

 

[A.T.]

https://en.wikipedia.org/wiki/Dynamic_rope

 

[Dacourt]

So

I think "would break" is a little dramatic; I rather suspect climbing ropes are designed to cope with a fall from height, up to a point, not just a dangling climber.
However, to answer the question. What force is exerted on the rope by a dangling climber of mass m? Suppose he falls from height and is falling at velocity 10 m/s when the rope arrests him. What is the momentum transferred to the rope? If his velocity drops from 10 to 0 over, say, 0.1 s, what is the (average) force exerted on the rope during this time? See where this is going?

So it's F=ma where a is deceleration and the force on the rope is greater than just dangling?

 

[A.T.]

So it's F=ma where a is deceleration ...

... plus 1g, if F is the total rope force. Otherwise F is the difference to just hanging still.

 

[Dacourt]

... plus 1g, if F is the total rope force. Otherwise F is the difference to just hanging still.

That makes sense. Thank you!

 

[sophiecentaur]

the rope cannot stretch that much

Not as much as a bungy! but simple climbing ropes are made of nylon (or similar) which is very stretchy, for that purpose - to spread the impulse force over a relatively long time. A polyester rope would snap a falling climber in half as it has a much higher modulus.
This link suggests that the maximum G that a human can survive with a fall is 15g. It isn't clear what evidence that figure is based on but it could be something to work with and suggests that's the extreme value that your rope should be subjecting you to. Assume the climber falls with a rope that's just within its strength limit.
If you equate the kinetic energy gained from a fall of h (mgh) to the energy stored in the (ideal hooke's law) stretched rope (kx²/2) where k is the modulus and x is the amount of stretch or Fx/2 where F is the maximum force. Acceleration is ma and acceleration is 15G so
x/h=2g/a, giving a/h (fractional extension) will be about 1/7. That seems a lot to me but the basic calculation seems to check out. The only thing I left out is the energy absorbed by (friction within) the rope. That would reduce the 1/7 figure by quite a bit, perhaps.
A heavier duty rope would not stretch as much - and kill the climber with excess G and a lighter rope would snap and the climber would die.
In practice, I believe they use harnesses with friction coupling and a higher modulus rope. The coupling provides the friction.

 

[Doc Al]

Not as much as a bungy! but simple climbing ropes are made of nylon (or similar) which is very stretchy, for that purpose - to spread the impulse force over a relatively long time. A polyester rope would snap a falling climber in half as it has a much higher modulus.

Good point!

 

[thetexan]

I suspect that any rope that has little or no stretchyness would be more likely to break since the distance over which the rope must arrest the fall is much less, meaning more force per distance of stoppage.

My father who was an aeronautical engineer who worked on the Lockheed Electra during its crisis in the late '50's had this same problem. The wings and engine mounting assemblies were so rigid that the natural wing flutter and engine vibrations tore the airplane apart. There was no give.

tex

 

[rumborak]

One thing that probably needs to be pointed out is, rope has a certain stretch *per unit length*. That actually means, by the very nature of the activity, the higher from where you fall, the more rope you fall on, which in turn means it gives more. So, long falls don't necessarily mean more injury than short ones, provided the rope catches you.

That said, dynamic rope has a certain number of "factor-2 falls" ( https://en.m.wikipedia.org/wiki/Fall_factor ) after which the manufacturer suggests retiring it.

 

[sophiecentaur]

One thing that probably needs to be pointed out is, rope has a certain stretch *per unit length*. That actually means, by the very nature of the activity, the higher from where you fall, the more rope you fall on, which in turn means it gives more. So, long falls don't necessarily mean more injury than short ones, provided the rope catches you.

That said, dynamic rope has a certain number of "factor-2 falls" ( https://en.m.wikipedia.org/wiki/Fall_factor ) after which the manufacturer suggests retiring it.

The length of rope can be a lot greater than the fall distance it it passes over the top 'eye' and back down again (all the rope can stretch). This is all good news to the guy on the way down.
Of course, the simple mathematical statement about the peak G being independent of the length of the fall, is right as far as it goes. However, the longer the fall, the longer the time that the victim is actually exposed to high forces (Impulse = Force times time) so I'd rather fall 2m than 20m, however suitable the rope happens to be.