Free falling object and tension on rope

[Bucephalus01]

Hi

I'm wondering, if I had a mass of 10kg and I dropped it 10 metres, how would I work out how much force is exerted on a rope?

Thanks
David.

 

[torquil]

I'm assuming you mean a rope that will suddenly stop the mass after a 10m fall?

You can work out how much energy will be dissipated by the rope when it stops the mass by considering the kinetic energy of the mass at the time of stopping. But the detailed force as a function of time will depend strongly on e.g. how flexible the rope is.

A small force if the rope behaves as a rubber band, and a very large force if it behavious like a steel wire.

In the "idealised" limit where the rope is completely stiff, i.e. cannot stretch at all, the force will be infinite for an infinitely short time.

 

[Bucephalus01]

But the detailed force as a function of time will depend strongly on e.g. how flexible the rope is.

SO I think you're saying that the rope would have some kind of coefficient of elasticity or something. So where would I find the kind of formula that I'm looking for to find this out?
Let's just say I know the coefficient or, I can work out how much the rope stretches before it breaks.

David.

 

[torquil]

SO I think you're saying that the rope would have some kind of coefficient of elasticity or something. So where would I find the kind of formula that I'm looking for to find this out?
Let's just say I know the coefficient or, I can work out how much the rope stretches before it breaks.

Classifying the rope by a single coefficient is probably quite a simplification. For example, the same rope might be able to pull a force F for a time T without measureable damage, but a larger force G > F for a shorter time S < T. Of course, you probably want to stay well within the accepted range for you force, and this range is hopefully determined by the manufacturer with a large safety margin.

If you know the coefficient of elastiticy for you rope for small streches, you will be able to model the situation by treating the rope as a spring with a given spring constant using Hookes law for a spring. But this is probably not a very good model for the rope, perhaps apart from very small amount of stretch if you are lucky. I'm sure a much better model involving several coefficients can be constructed if you have this data available for you rope.

When it starts to break you are getting into some pretty complicated phenomena, which would be very difficult to model. The manufacturer should perhaps provide experimental data on how much on average the rope can be stretched for different amounts of time, and the resulting damage to the rope in each case.

But it is interesting: Consider a rope that can slowly lift a mass M, but doesn't stretch very much. Then consider a more flexible rope which is only able to slowly lift a mass M/2. There might be situations in which the more rugged rope will break, but the other rope won't. Because the "impact force", subjected to the more flexible rope are much smaller, e.g. in your proposed situation.

 

[torquil]

I think your best bet is to determine the exact experimental setup that the manufacturer has used to determine the strength of the rope, and see if you can related this to your situation. When it comes to ropes that are used in serious applications, I'm sure they provide data for some sort of "static strength" and some sort of strength in situations where the force is sudden.

 

[Bucephalus01]

Yeah you're probably right. I didn't realize it was this complicated to be honest.
Thanks for your replies.
David.

 

[AlephZero]

If you ignore the mass of the rope and assume it stretches a small amount compared with the distance the mass falls, and assume the rope remains within its elastic limit, you can answer this by finding the work done to stretch the rope when it stops the mass.

Kinetic energy of the mass = Wh where W is the weight and h is the distance it falls
Work done by the rope = Fs/2 where F is the maximum force and s is the amount it stretches

so F = 2Wh/s.

For a long heavy rope the situation is more complicated, because there will be axial traveling waves of force (or stress) moving along the rope at a finite speed and being reflected from both ends, and it's not "obvious" where or when the maximum force will occur along the length of the rope.

 

[Bucephalus01]

Thanks, that's an awesome response.
Cheers.
David.