Determining angle alpha with an elastic rope and a mass hanging from the rope.

[blueboy499]

Homework Statement

I am confused on how to begin finding the angles (theta) for answering part a?

 

[gneill]

Homework Statement

I am confused on how to begin finding the angles (theta) for answering part a?

Are you given the Young's modulus for the nylon rope?

 

[blueboy499]

In the book, it is given as 5x10⁹ N/M². But this problem is not out of the book, so it's a bit of a stretch.

 

[gneill]

In the book, it is given as 5x10⁹ N/M². But this problem is not out of the book, so it's a bit of a stretch.

"Stretch". Heh. Good one

You'll need some way to determine the tension in the rope, so presumably Young's modulus would help. How does the Young's modulus for a material relate to Hooke's Law?

 

[blueboy499]

The change in the length of the material = ((tensile stress)/(young's modulus))(final length of the material).

But the stress = force/area and since I'm not given the force or the angles or enough other geometric values, how can I solve for this?

 

[blueboy499]

If you use Newton's second law, you'll get two scalar equations, and you'll have two unknowns*...


*Assuming the system is in static equilibrium.

So then what do you do with those 2 unknowns?

 

[jhae2.718]

So then what do you do with those 2 unknowns?

Never mind; the $x$ equation is useless; it just states that the tensions are the same. At first glance it looks like the problem is statically indeterminate.

 

[gneill]

The change in the length of the material = ((tensile stress)/(young's modulus))(final length of the material).

But the stress = force/area and since I'm not given the force or the angles or enough other geometric values, how can I solve for this?

You've got the radius of the rope's cross section and the initial length of the segments. You've got the Young's modulus. You should be able to determine how much the rope segments stretch w.r.t. the angle, and thus the tension for any angle.

 

[blueboy499]

"w.r.t."?

 

[gneill]

"w.r.t."?

w.r.t. : Abbreviation; "with respect to".

 

[blueboy499]

The way I still see it, I still don't have: the force for the F/A = tensile stress, the final length of the rope, and the change in length of the rope. Am I missing something here, or is there some other method we should be going about this?

 

[gneill]

The way I still see it, I still don't have: the force for the F/A = tensile stress, the final length of the rope, and the change in length of the rope. Am I missing something here, or is there some other method we should be going about this?

Young's modulus should give you an effective "spring constant" for the rope. Take a look at the Wikipedia article on Young's Modulus, at the section "Force exerted by a stretched or compressed material".

 

[blueboy499]

Young's modulus should give you an effective "spring constant" for the rope. Take a look at the Wikipedia article on Young's Modulus, at the section "Force exerted by a stretched or compressed material".

Each of those stated equations require knowing the change in length to calculate the force.

 

[gneill]

Each of those stated equations require knowing the change in length to calculate the force.

You're looking at the problem backwards You want to have an expression for the force in terms of the angle, and then determine the angle at which the block is balanced by the resulting tensions.

Which component of the rope tensions should you be concerned about?

 

[blueboy499]

Thank you for all of your help, but I only just now discovered that the professor changed the problem to include the vertical distance the rope sags after loading and reaching equilibrium. I should be able to figure it out from here. Thanks again! :)

 

[gneill]

Thank you for all of your help, but I only just now discovered that the professor changed the problem to include the vertical distance the rope sags after loading and reaching equilibrium. I should be able to figure it out from here. Thanks again! :)

Ah, well that makes it an entirely different problem then! No need to consider the stretching of the rope in that case. Good Luck!