Right angle force on perfectly stretched string

[xtimmyx]

Right angle force on "perfectly stretched string"

I have a theory which although physically impossible I would like to know some sort of answer to.

Image a string stretched between two trees. The string is perfectly stretched, totally horizontal, and has no elasticity, but can bend as a normal string.

My theory is that when a vertical force is applied to the middle of the string, no matter the size, the horizontal force upon the trees from the string is infinite. Or impossible to calculate.

I know that there is no such thing as a totally inelastic string and so on, but what would physics be without "what if´s"? :)

I've attached a picture of the problem with some units to it if someone would like to calculate it.

 

[Zaphys]

From an usual classical statics approach it is a non-sense, as you said, so short answer can be given to you. If you want to theorize about this you´ll need to redefine laws of mechanics.

If use of "conventional" machanics at your example, this is what happens:

If string is stable at that configuration (horizontal) then all of its points are, including the one were the force is aplied and so we focus on it.

The conventional 2-D static stability condition for this point is

$$\sum\vec{F}=\vec{0} \Leftrightarrow \sum F_x=0 \sum F_y=0$$

And by definition we have

$$\sum F_y=-1 N$$ as seen in the picture, because you assume that the string doesn't bend (well in fact you don't but is implicit since you say it's "perectly stretched" and "has no elasticity") and so no more forces act upwards in such a way they would annul the exsisting force.

Therefore, because of Newton's 2nd law the string have to be moving, here you have your non-sense.

 

[xtimmyx]

From an usual classical statics approach it is a non-sense, as you said, so short answer can be given to you. If you want to theorize about this you´ll need to redefine laws of mechanics.

If use of "conventional" machanics at your example, this is what happens:

If string is stable at that configuration (horizontal) then all of its points are, including the one were the force is aplied and so we focus on it.

The conventional 2-D static stability condition for this point is

$$\sum\vec{F}=\vec{0} \Leftrightarrow \sum F_x=0 \sum F_y=0$$

And by definition we have

$$\sum F_y=-1 N$$ as seen in the picture, because you assume that the string doesn't bend (well in fact you don't but is implicit since you say it's "perectly stretched" and "has no elasticity") and so no more forces act upwards in such a way they would annul the exsisting force.

Therefore, because of Newton's 2nd law the string have to be moving, here you have your non-sense.

Thank you, however I might have used the wrong words to describe it when talking about elasticity. What I mean is that the string can bend just like a normal one. But that the string's length is constant.

For example, If you have a hammock hanging between two trees, as far as I know, the less stretched the hammock is (more "loose" or what you would call it"), less force from it is acted on upon the trees, right?

Therefore my assumption is that as the hammock/string get's more and more stretch (by increasing the distance between the trees or shortening the hammock/string), the force on the trees would increase and ultimately reach "infinity" or not calculable.

 

[xtimmyx]

A colleague of mine who is a climber made some interesting points about this, and also found the formula for calculating load on the anchor points. And according to that formula, the tension on the string goes towards infinity as the angel approaches 0º. Se link below for the formula (under equalization):

http://en.wikipedia.org/wiki/Anchor_(climbing)#Equalization

 

[Zaphys]

Yes sure it does, but remember that infinity is not a quantity. What is meant by the mathematical behavior of that expression is that, considering "conventional" mechanics laws the tension of the string is arbitrarily high as you set the angle arbitrarily small. But at angle=0º force is not defined since 1/0 is not a real number.

I already considered the string with constant length for my conclusion above :)

Salutations

 

[Born2bwire]

Yes sure it does, but remember that infinity is not a quantity. What is meant by the mathematical behavior of that expression is that, considering "conventional" mechanics laws the tension of the string is arbitrarily high as you set the angle arbitrarily small. But at angle=0º force is not defined since 1/0 is not a real number.

I already considered the string with constant length for my conclusion above :)

Salutations

I think Zaphys may have a point here, the question becomes whether or not the anchor formula that was provided is still relevant in this case. It seems hard for me to imagine that is still true. For example, what about a chain? A chain would have zero elasticity and in the limit of small link size would be equivalent to your string. Yet, I can't imagine that a stretched chain would break or bring down its supports if you were to hang a small or even appreciable amount of weight in the middle.