# Elasticity problem

1. Mar 25, 2004

### Dr.Brain

A Uniform rod is fixed from top and becomes stretched under its own weight.Then which of the following will be true on its elongation:

a) Diameter at the top is smaller than at the top
b) Diameter at the bottom is larger than that of top
c) Diameter is uniform throughout
d) Dimater becomes smaller than previous one uniformly

I know the answer but i want your views and the way you think...I tried to check it out with the forumla for Young's Modulus and Poisson's Ratio...But I think my description for the right answer wasnt right...so contribute plz...

2. Mar 26, 2004

### HallsofIvy

Staff Emeritus
What? You know the answer but you won't tell us? Is this a test?

As the rod stretches vertically, it radius has to become smaller (conservation of mass, basically). Since it is stretching under its own weight, the bottom, subjected to relatively little weight, will only stretch a small amount compared with the top. The top has smaller diameter than the bottom.

3. Mar 28, 2004

### turin

How is this conservation of mass? It sounds more like conservation of mass-density, but that doesn't sound like a law that I've heard. There's got to be another explanation.

4. Mar 28, 2004

If it's stretching and the radius doesn't change, then there's more volume. If the density is constant, then that implies more mass. Stretching doesn't create mass, so the radius must decrease in order to suitably decrease the volume and maintain the mass.

5. Mar 28, 2004

### turin

OK, but that's what I don't get. Why assume this? (Other than the obvious, "because we observe the affect.")

6. Mar 28, 2004

Because of the word "uniform" in the description of the rod. A uniform rod.

7. Mar 28, 2004

### turin

I think that is a weak justification. The question can easily be interpretted as the rod starting out uniform. When it stretches, it either becomes non-uniform, or non-rod. Though, I guess that this does seem to be the best we can do with this problem. I really hate these first year physics questions.

8. Mar 28, 2004

It's not how I would personally phrase the problem, that's for sure.

9. Mar 29, 2004

### Dr.Brain

Even I do...but none of you have been able to interpret the question or even try to get near to the answer...

"DIAMETER AT TOP IS SMALLER THAN THAT OF BOTTOM"

Can anyone explain me ..why?...

U have made fun of "Conservation Of Mass" .. You are applying This law within body itself??? Wow! ..You people really innovative

The explanation to this problem has something to do with Young's Modulus or Poisson's Ratio as far as I am concerned. and remember it is extending under it own weight"....I am trying to use that fact....

10. Mar 29, 2004

### Staff: Mentor

Poisson's Ratio is positive

The tension in the rod is greater near the top (since it must support the weight of the section below), so the amount of axial stress (and thus strain) is greater towards the top. The lateral strain is related to the axial strain by Poisson's Ratio. Since most materials have a positive Poisson's Ratio, they shrink (get thinner) under axial stress. So it makes sense that the rod will be thinner at the top.

Now if you are asking why most materials have a positive Poisson's ratio: I haven't a clue.

11. Mar 29, 2004

### turin

Now we're getting somewhere! Excellent suggestion Doc Al! I will now proceed to investigate the meaning and canonical (first year?) applications of this "Poisson's Ratio."

12. Mar 29, 2004

What's wrong with HallsofIvy's explanation again?

13. Mar 29, 2004

### turin

I have yet to look into this Poisson's Ratio, but I got the impression that it is not just a statement of conservation of mass.
..........

14. Mar 30, 2004

### Dr.Brain

Poisson's Ratio varies between [-1] to [+0.5]

any say?

15. Mar 30, 2004

### Dr.Brain

Doc Al

That was a good reply. But I am not even 10% satisified with that
cuz'

Poisson's Ratio= negative of fractional change in diameter/fractional change in longitudinal length

16. Mar 30, 2004

### Staff: Mentor

This is true. So?

17. Mar 30, 2004

### Staff: Mentor

This is also true. What is your point?

18. Mar 30, 2004

### turin

OK, here's what I found:

1. Poisson's Ratio is not discussed in my old physics book (I don't think this is used any more, so it might not be relevant): Serway's "Physics: For Scientists and Engineers" 4th ed. Since this is the case, my investigation consisted of a terse internet search, resulting in information extracted and compiled from unqualified/unofficial sources. The justification for this is that I found 100% consistency out of the 10 pages I viewed that gave information of the subject.

2. Poisson's Ratio is as DocAl has described: it can certainly be used to relate the axial strain to the radial strain.

3. For a material with isotropic elastic properties, Poisson's Ratio must be positive. Since the rod is uniform, I think it is reasonable to apply this qualification to directional uniformity of the elastic properties. In other words, I think it is reasonable to assume that Poisson's Ratio is positive according to the wording of the problem.

4. Therefore, I conclude that the rod will be narrower at the top.

19. Mar 30, 2004

I hate to say it, but that's not convincing at all, either...

We're still skirting around the "why" of the issue.