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A Calculus Challenge

DrD

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Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, #45

(c) DrD, 2018

It has been quite a while since I last posted anything here, but an interesting problem has come to mind that I wanted to share with you. If you really know calculus, this should be straight forward; if you don't know calculus, don't even try!

MEForumChallenge.pdf

 

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THE  CHALLENGE IS NOW ENDED. I WILL NOT RESPOND TO FURTHER ANSWERS. i EXPECT TO POST A SOLUTION AND A FEW COMMENTS IN THE NEXT FEW DAYS.

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As I check this on 26 December, there have been over 200 people look at this problem. Thus far, not a single one has ventured even the first bit of an attempt. What do you folks do when confronted by a problem that is not in the textbook?

DrD

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Today, 29 December, there have been over 300 people look at this problem, and yet not a single comment! What is it with all of you?

What are you thinking? Is the problem unrealistic, so that it would not be useful to know how to do it? I can cite cases where this geometry really appears.

Is the problem too hard? You are supposed to be engineers, able to confront new situations and find ways to deal with them. Why is this any different?

Is the problem too easy, and therefore beneath your dignity? If so, please humor an old man, and indulge him by answering the questions.

Whatever the situation, somebody please say something!!

DrD

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Hi Professor,

As far as the title goes it is to my understanding this is a pure calculus problem, which implies the solution should be strictly by integrals.

Being honest, the way I'd solve it we'll be by decomposing of areas and the use of area moments of inertia for common shapes.

I can't speak for everybody here but in my case knowing that the problem could be solved it by easy means different to calculus discourages me a little, however I think it's fine to practice some integral calculus.

Having said that, I will submit a solution (using integral calculus) after new year's eve festivities.

Happy New Year!

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My word! An answer ... I'm simply stunned!

If you are sure it can be solved without integration, go ahead. This will serve as a check on the results obtained by integration.

If you use formulas from a book, please be sure to give their derivation as well.

DrD

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Dear Jarek,

Your final result for the enclosed area is correct (if we ignore some notational inconsistency, r in place of R). Congratulations!!

That said, I  cannot follow your development. Would you explain your reasoning with some words, perhaps? In particular, you show the sum of two integrals equal to a third integral. How is that equality established? I cannot follow it.

Also, you write in the integrand in one place  xtgalpha; what is this? Is "tg" the tangent function? If so, it would help if you would enclose the argument in parentheses.

Now, to all others, I hope you see that the problem is not impossible. Why are you so unwilling to try it yourself?

DrD

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Dear Jarek,

Your figures help a lot to explain your logic. Thank you.

I'd say you definitely have the answer for the first part. Now, can you extend this to obtain the other answers ?

 

DrD

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Well, as of now (4 Jan 2019), over 1000 people have looked at this problem, one person has worked one part of the problem statement, and another says he know how to do this without integration (and thus far, without results). You folks are real balls of fire!!

DrD

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14 hours ago, DrD said:

Well, as of now (4 Jan 2019), over 1000 people have looked at this problem, one person has worked one part of the problem statement, and another says he know how to do this without integration (and thus far, without results). You folks are real balls of fire!!

DrD

Dear professor please find solution to your challenge, I have received y email . Typing formulas get a little bit of time

Shaft Moments.pdf

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Dear Amin,

There are a few problems with your answer.

1) If you are going to cite Roark, you should surely say which edition. There are many, and they are not all the same. I have two of them myself, but quite likely not the one you use.

2) You don't give your final result for the area. You simply cite Roark, but there is nothing here that I can check. Roark has been known to be wrong.

3)  You give an answer for something you call "static  moment," but you do not locate the centroid.

4) You say, "The same for moment of inertia." The same what? You have given any answer before (unless you meant your area result), so this is no answer at all.

5) You seem to think actually answering the question is beneath you because you know how to look in a book. What do you do if the book is not at hand?

6) You said, "When I had to deal with these kind of problems, I used to go to handbooks." Do you think handbooks have all the answers? Do you no longer do engineering?

Overall, I'd say this is a fairly poor effort by someone who really is not very interested.

DrD

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Dear Wasulia,

Congratulations, you got the correct answer, almost. Two comments come to mind:

(1) You made an error in factoring out R^2 in the last line;

(2) You should learn that, when doing calculus, angles are always expressed in radians, so your factors 360, 180, etc are really errors. Learn to think in radians; it will help you a lot.

You took the long way around, but you almost got here. Good effort.

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Dear Jarek,

You are correct in saying x-bar is zero. I'm amazed that no one else has seen that yet!

Without saying yes or no to your result for y-bar, I have two comments for you:

(1) I urge you to provide a sketch showing how you formed the integrals you offer as a route to the answer.

(2) You already have an expression for the area, so I would urge you to substitute it and seek to simplify the result.

 

DrD

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Dear Jarek,

I cannot quite agree with your result. I think you may have an algebraic error in the calculation of y2-bar. I'd look there first. Interesting approach, I must say.

DrD

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Dear Professor,
Thank you for help and advices. I will look it again. In previous version I found several mistakes so this time will be the same :)
Best regards,
Jarek

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Dear Jarek,

Just as a matter of curiosity, how do you write the equations in your work? They are clearly type written, but I'm curious what software you use for this. Is it MS Word, some other word processor, or what? It looks good, but I'm interested to know more.

DrD

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Dear Professor,
I am using MS Word 2007, however not the new build-in equation processor but Mathtpye (http://www.dessci.com/en/products/mathtype/). It is easy to install. When you save file as the pdf version there is no problem. When you use this new processor the equations look quite messy.

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Dear Jarek,

Thanks for the comment on Mathtype. I have heard of it, but know little about it. I use LaTeX for all of my work, but I like the way your work looks very much.

DrD

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Thanks for the information, Jarek. I do not like MS Word, and would never use it given any choice in the matter. But, I know many others like it, so this is useful.

DrD

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On 1/6/2019 at 3:25 AM, DrD said:

Dear Amin,

There are a few problems with your answer.

1) If you are going to cite Roark, you should surely say which edition. There are many, and they are not all the same. I have two of them myself, but quite likely not the one you use.

2) You don't give your final result for the area. You simply cite Roark, but there is nothing here that I can check. Roark has been known to be wrong.

3)  You give an answer for something you call "static  moment," but you do not locate the centroid.

4) You say, "The same for moment of inertia." The same what? You have given any answer before (unless you meant your area result), so this is no answer at all.

5) You seem to think actually answering the question is beneath you because you know how to look in a book. What do you do if the book is not at hand?

6) You said, "When I had to deal with these kind of problems, I used to go to handbooks." Do you think handbooks have all the answers? Do you no longer do engineering?

Overall, I'd say this is a fairly poor effort by someone who really is not very interested.

DrD

Dear Professor 

Today at Friday, at weekend, I set aside time for myself and check the site. My book is 6th edition, 1989, I get it from book fair, with government subsidies. For a while I was kind of "privileged" to access online math calculus. I have attached a new version of representation. I hope it will be better. Thank you for your comments. Hand books are not bad in general. They save time time. not for getting answers easily, but because cuts discussions short. When they ask " how do you get formula?" , the answer will be "here is the formula in the book."

Shaft Moments2.pdf

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Dear Amin,

Well, I have both the 2nd and 5th editions of Roark, so I am unable to tell whether the book is correct (or if you read the book correctly). That's the problem with using a reference source (although Roark is widely recognized).

I'm amused that you said, "When they ask "how do you gget formula?" the answer will be "here is the formula in the book." Does that really convince you? Are you more convinced by something in a book, or be a carefully worked out derivation that you can follow for yourself? I'll take the derivation every time!!

It is certainly true that handbooks have their uses. The problem is, they also have their limitations. If there are assumptions made in the derivation, looking the formula up in a book may not tell you about those assumptions.

I'm sorry that you evidently find the area moment of inertia expressions just too tedious to be worth your time. For those who want to find out if they really can use calculus, this remains a very useful exercise.

DrD

 

Dear HKS,

You promised results after the New Year. Just which New Year did you have in mind?

DrD

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