Rejected Physics Problems

[Borodog]

I wrote this problem for an upcoming test and rejected it as too hard. However, I figured someone around here might find it interesting.

http://i27.photobucket.com/albums/c153/Borodog/image006-1.png

What is x in terms of the length of the rope L, d, and delta-h?

PS. What is the tension in the rope?

[purnell]

I find it interesting, but it's been almost 20 years since I solved this kind of thing. Will it help me to review a statics textbook, or is there more to this than I'm seeing?

[almostbusto]

i am getting a degree in math this semester and i have no idea how to solve that problem. but i have never taken a physics course.

anyway, i just wanted to say you should post the solution in white. or PM it to me. i'd like to see how its done

if i had to guess... at deltaH=0 X=.5D at deltaH=H X=D from that i would just guess that X would move uniformly from .5D to D as H was lowered from h to 0. obviously that is way wrong though because the solution is very simple and doesn't use all the givens. so it certainly wouldn't be a rejected problem if i was right.

[BluffTHIS!]

Boro,

I too have forgotten everything I studied in college physics and so don't have an idea. However wouldn't the elastic properties of any particular rope also come into play here?

[slickpoppa]

[ QUOTE ]
Boro,

I too have forgotten everything I studied in college physics and so don't have an idea. However wouldn't the elastic properties of any particular rope also come into play here?

[/ QUOTE ]

I think we are assuming that the rope is inelastic, and therefore would not be taut if the weight were not hanging on it.

Anyway, I don't see how this is really a physics question since the main question (solving for x) depends completely on trig and not any physical laws. Though I will say that it is not an easy question. I'm sad to say that I'm going to have to sleep on it.

[zadig]

d - 1
x= --------------- + delta h
sqrt(L^2 - d^2)

[evank15]

It's really pretty sad I can't do this.

Probably one of the reasons I got a C in classical mechanics last semester. Although that class was a lot more about Langrangian mechanics than Newtonian mechanics.

I hate classical mechanics. I can do EM, (some)QM, Thermo, Astro/Cosmo etc etc, but give me a [censored] rope on a wall and I freeze up.

[Nielsio]

That's easy. x = 5.5cm.

[MaxWeiss]

Here's what I have so far. I can't figure how to relate H to delta-h or the change in length to the change in angles, which I could then use to relate to L-sub-one, which I (correctly???) believe is half of L. Am I even on the right track??? God it's been years since I took any physics or math. My head hurts... but it's a good hurt.


http://img470.imageshack.us/img470/1346/worksofarbc4.jpg

[thylacine]

It does not seem too hard for college(?) level. Maybe they thought the problem had just one small physics insight and then it was all trig.

[madnak]

So we're expressing a relationship between where the weight will fall measured horizontally from the left edge (x) and the height difference (delta-h) between the two endpoints of a rope with length L across the area with a horizontal distance d?

And I assume figuring it out has to do with knowing how the weight affects the rope, because we need to know something about that. It's intuitively obvious that if the right endpoint is lower than the left endpoint, the weight will settle closer to the right side than to the left. But I don't think I can solve the problem without having some clue regarding exactly how far to the right the weight will settle relative to the difference in the endpoints. It seems clear that without more information about how the system operates (physics), the information I have (L, d, and delta-h) is insufficient.

I hate coming to that conclusion, but I guess it's better than being totally stumped. Am I correct in assuming that my limited experience with physics (some concepts, no math) will prevent me from solving this problem?

[Borodog]

zadig,

Your solution cannot be right; it is dimensionally incorrect (d-1).

[Borodog]

[ QUOTE ]
[ QUOTE ]
Boro,

I too have forgotten everything I studied in college physics and so don't have an idea. However wouldn't the elastic properties of any particular rope also come into play here?

[/ QUOTE ]

I think we are assuming that the rope is inelastic, and therefore would not be taut if the weight were not hanging on it.

Anyway, I don't see how this is really a physics question since the main question (solving for x) depends completely on trig and not any physical laws.

[/ QUOTE ]

Ah, untrue. You are making an assumption that is not an assumption if you understand the relevant physical concept. Perhaps it is intuitive for you (which is good), but this is most definitely a physics problem.

[ QUOTE ]
Though I will say that it is not an easy question. I'm sad to say that I'm going to have to sleep on it.

[/ QUOTE ]

Definitely not an easy question, especially for the level of the class.

[Borodog]

[ QUOTE ]
I find it interesting, but it's been almost 20 years since I solved this kind of thing. Will it help me to review a statics textbook, or is there more to this than I'm seeing?

[/ QUOTE ]

It would definitely help to review a statics textbook. For all I know it might have this very problem in it. It seems like it should be an archetypal problem, although to my knowledge, I have not seen it elsewhere.

[Borodog]

[ QUOTE ]
Boro,

I too have forgotten everything I studied in college physics and so don't have an idea. However wouldn't the elastic properties of any particular rope also come into play here?

[/ QUOTE ]

As slickpoppa figured, we are neglecting any elasticity in the rope (if we didn't, the rope would be curved, rather than straight). Although I believe the equilibrium position of the mass does not change in the fully elastic version (I haven't worked it out though; that's solely based on physical intuition).

[Borodog]

college kid,

I'm afraid you are not only on the wrong side of the tracks, it's the wrong set of tracks entirely. /images/graemlins/wink.gif

It's fairly clear from the picture that the two lengths up rope from the walls to the pulley are unequal, which agrees with our physical intuition. If the height difference is zero, the mass should hang in the middle. If one end is raised, the pulley should move toward the other side, making the 2 lengths of rope unequal in length.

Also, writing things like "g = 9.8" will get you tasered in my class. /images/graemlins/grin.gif

[gumpzilla]

I confess I'm having a bit of trouble seeing what you're going for here, mostly because I don't know what the approximations that you're asking us to make are. To be a nit, if the "rope" were completely rigid, I wouldn't expect throwing a weight on there to bend it at all, and then the problem kinda goes kaput. (EDIT: This is sort of an interesting issue in physics education. Making the right abstractions to talk about this kind of thing is non-trivial.)

But, I'm guessing what you're going for is that the tension in the rope must be constant at equilibrium, and to avoid having a discontinuity at the location of the mass, this must mean the angles formed on each side are identical. This does have the nice benefit that in the case where \delta_h = 0, you find that the equilibrium position is in the middle, as it should be in the symmetric case. It will also mean that it skews towards the lower end, which fits the intuitive picture presented earlier nicely.

[Borodog]

[ QUOTE ]
It does not seem too hard for college(?) level. Maybe they thought the problem had just one small physics insight and then it was all trig.

[/ QUOTE ]

thylacine,

These students are non-scientists/engineers; this is the first physics class many have ever seen. They are, in general, very weak on algebraic manipulations and trigonometry. This problem is far too hard for most of them.

[Borodog]

[ QUOTE ]
I confess I'm having a bit of trouble seeing what you're going for here, mostly because I don't know what the approximations that you're asking us to make are.

[/ QUOTE ]

That the rope has negligable mass compared to the hanging mass, and negligable elasticity. I.e., not that it is rigid, but simply that it's length is constant, and when you put a weight on it, the lengths of rope are straight and not curved. Certainly not an uncommon, or even bad, approximation. Sorry if this was unclear; I thought it was implied by the diagram.

[Borodog]

[ QUOTE ]
But, I'm guessing what you're going for is that the tension in the rope must be constant at equilibrium, and to avoid having a discontinuity at the location of the mass,

[/ QUOTE ]

Yes; this is the critical bit of physics that makes the problem soluble.

[ QUOTE ]
this must mean the angles formed on each side are identical.

[/ QUOTE ]

Which is non-obvious without applying Newton's 2nd Law, which definitely makes this a physics problem. /images/graemlins/wink.gif

[ QUOTE ]
This does have the nice benefit that in the case where \delta_h = 0, you find that the equilibrium position is in the middle, as it should be in the symmetric case. It will also mean that it skews towards the lower end, which fits the intuitive picture presented earlier nicely.

[/ QUOTE ]

Yep.

[Borodog]

almostbusto,

Here is the proper approach, in white:

<font color="white">
As has already been said, at equilibrium the tension in the rope is constant throughout (pretty apparent given the 2nd question, "What is the tension in the rope?"). If you apply Newton's 2nd Law (Fnet = ma) to the mass hanging from the rope, particularly the horizontal component, you will discover that the angles the two lengths of rope make with the horizontal must be identical. From there, it's all trig and algebra.
</font>

I'd still like to see a correct full solution posted by someone else rather than just giving it.

[MaxWeiss]

In my head, I was making it out to a triangle without the wall being present on the right side, which is why I thought they will be equal lengths (in the triangle). I assumed for some reason that the wall on the right was just an arbitrary stopping point, but you are saying the rope is attached there, right??? Also, why is writing g=9.8 bad?? Should I just have g as a variable, or am I supposed to use 9.8, or am I supposed to do something else entirely???

[Borodog]

[ QUOTE ]
In my head, I was making it out to a triangle without the wall being present on the right side, which is why I thought they will be equal lengths (in the triangle). I assumed for some reason that the wall on the right was just an arbitrary stopping point, but you are saying the rope is attached there, right???

[/ QUOTE ]

Yes, the rope is attached to both walls, at different heights.

[ QUOTE ]
Also, why is writing g=9.8 bad?? Should I just have g as a variable, or am I supposed to use 9.8, or am I supposed to do something else entirely???

[/ QUOTE ]

g = 9.8 what? Furlongs? Eggs? Fortnights per carat-acre?

Also, numbers suck. Just work the problem algebraically, as God intended.

[MaxWeiss]

Borodog can tell me if I'm wrong, but I think all the variables you are talking about are arbitrary (but of course dependent on each other according to the proper laws). That's at least what I've been assuming when thinking about it. I don't see how we can solve for X in terms of other variables if they aren't all arbitrary, other than the laws about tension and similar triangles and all that.

[thylacine]

[ QUOTE ]
[ QUOTE ]
It does not seem too hard for college(?) level. Maybe they thought the problem had just one small physics insight and then it was all trig.

[/ QUOTE ]

thylacine,

These students are non-scientists/engineers; this is the first physics class many have ever seen. They are, in general, very weak on algebraic manipulations and trigonometry. This problem is far too hard for most of them.

[/ QUOTE ]

Student complaint: no-one taught us that triangles have pulleys! /images/graemlins/mad.gif

[gumpzilla]

[ QUOTE ]

Also, numbers suck. Just work the problem algebraically, as God intended.

[/ QUOTE ]

Haha, yes. This is another thing that interests me about physics education for non-physics types. They want to insert numbers as quickly as possible. I have no idea why - it always makes the problems harder, in the end, because you sort of lose information at each step. Kept symbolic, you can usually look at your answer and get useful information about what it predicts that will frequently let you digest whether it makes any sense or not.

I feel like understanding why people do this would be to understand a key truth in teaching useful physics thinking.

[thylacine]

[ QUOTE ]

Also, numbers suck. Just work the problem algebraically, as God intended.

[/ QUOTE ]

Is this what you say to motivate students at Liberty College? /images/graemlins/wink.gif

[danny]

x=h tan(theta)
d-x=(h-deltah) tan(theta)
Eliminating h and solving for x:
x=(d + deltah tan(theta))/2

We also have sin(theta) = d/L
So tan(theta) = d/sqrt(L^2 - d^2)

This gives the solution:

x = d (1 + deltah/sqrt(L^2 - d^2))/2

[Borodog]

Danny,

Assuming that deltah is positive when the left rope attachment point is higher than the right attachment point, your solution is correct.

http://i27.photobucket.com/albums/c153/Borodog/Mass-on-a-rope-problem.png

Normally I tell my students to always check the dimensionality of their solutions, and if possible, check to make sure it reduces to the proper expressions in important limiting cases. Note that the horizontal position reduces to d/2 when deltah = 0 and when L &gt;&gt; d, deltah. Also, when L &gt;&gt; d, the tension in the rope simply becomes mg/2.

[Borodog]

[ QUOTE ]
[ QUOTE ]

Also, numbers suck. Just work the problem algebraically, as God intended.

[/ QUOTE ]

Haha, yes. This is another thing that interests me about physics education for non-physics types. They want to insert numbers as quickly as possible. I have no idea why - it always makes the problems harder, in the end, because you sort of lose information at each step. Kept symbolic, you can usually look at your answer and get useful information about what it predicts that will frequently let you digest whether it makes any sense or not.

I feel like understanding why people do this would be to understand a key truth in teaching useful physics thinking.

[/ QUOTE ]

gumpzilla,

I feel like there are 2 reasons for this.

1) Students have been ill prepared by their mathematical training. In modern math classes in public schools, the main focus is on plugging in numbers first, and then algebraical manipulation to find the correct _number_ second. They are also greatly harmed by the ignorance of units that pervades mathematics education, which (having taught high school mathematics and physics for a year at a state magnet school _for_ science and mathematics), is near universal.

2) Even the best texbook I have found for this level of student (Serway and Faughn) explicitly _tells the students_ to plug in numbers first! /images/graemlins/shocked.gif /images/graemlins/confused.gif This is purely a concession to the students' inadequate math background, but in my opinion it only makes the problem worse. Demand that they do the work properly. If they can't, they probably shouldn't be in college anyway, at least not in any curriculum that requires physics. The rest will shape up and be better for it.

I am seriously considering writing my own textbook. To the point where I have a meeting Wednesday to talk with a publisher. I write out my lecture notes before classes to organize my thoughts, and since I would rather have the students pay attention than struggle to write down everything I say and write, I make the notes available online. Knowing that they will be read, I take pains to make them thorough and clear, with diagrams, worked out examples, etc. So I end up writing my own text anyway. I probably have about 100 pages written at the undergraduate algebra-based level already, split between mechanics and E&amp;M/modern.

Anyway, in my textbook, questions would be asked without reference to numbers. Only after a problem has been solved symbolically would a followup part ask about specific values (which they do need practice in; substituting in, checking units, etc.).

[recipro]

Just curious, and I don't feel like solving it, but wouldn't this problem be pretty simple using Lagrangian mechanics?

[Borodog]

[ QUOTE ]
Just curious, and I don't feel like solving it, but wouldn't this problem be pretty simple using Lagrangian mechanics?

[/ QUOTE ]

Certainly no simpler than the Newton's 2nd Law approach, since the system is in equilibrium.

Applying Lagrangian mechanics to this problem is like adjusting the volume on your TV with a torque wrench.

[recipro]

[ QUOTE ]
Certainly no simpler than the Newton's 2nd Law approach, since the system is in equilibrium.

Applying Lagrangian mechanics to this problem is like adjusting the volume on your TV with a torque wrench.

[/ QUOTE ]
Hmm, my bad. I was thinking the rope had weight, so it was a minimize potential energy problem. (It still is, of course, just a much simpler one.)