Moment of inertia about the z-axis

[sberxa]

Homework Statement

Find I(z) (moment of inertia about the z-axis) bounded by the cone z=sqrt(3(X^2 + y^2)) and the sphere with radius a. Density is inversely proportional to the distance from the z-axis.

Homework Equations

double integration

The Attempt at a Solution

basicall, I set up my integral, solved it and got a negative number and was hoping somebody could let me know if my original integral is wrong.

it is:

$$\int\int\int(B^3)(sin\Phi)^3$$ d$$B$$d$$\Phi$$d$$\Theta$$

please ignore the exponent in the above intergral. I can't figure out this website! It's supposed to be on the same level as the rest of the integral to show you the order I am intergrating with respect to. Thanks!

I put the limits of B from 0 to a, of Phi from 0 to (pi/6) and the limits of Theta from 0 to 2(pi). Intergrate first with respect to B then, phi, then theta.

I'm basically trying to use spherical coordinates to solve this integral.

Thanks!

 

[Dr.D]

What is this cap gamma representing?

You know, it would help a lot if you defined your symbols. Then maybe we could help you a bit more. This is clear as mud!

 

[sberxa]

woops sorry.

I'm still getting used to writing these sorts of formulas on this website.

I'll try and edit it better.

 

[sberxa]

Is that a little better?

 

[Dr.D]

Is what a little better? You did not say you had edited your original post.

Also, what is B?

 

[sberxa]

I could find the symbol for 'row' on here. I simply used B as an arbitrary variable, just as row is. It represents the symbol row in the formula used to solve volume problems in three dimension using spherical integrals. My understanding (correct me if I'm wrong) represents the distance from the origin to the outer limits of the shapes described above.

Please let me know if further clarification is required.

Thank you for your help and patience.

Note: Original post was edited.

 

[Dr.D]

row is something you do in a boat.

Do you perhaps mean the Greek letter rho?

Now you say that B is an arbitrary variable. I take that to mean that it represents nothing in particular, so I have no idea how to interpret what you have written How is this supposed to relate to the problem at hand?

I strongly suggest that you go back to your book, look at the definition for a MMOI, and read the problem statement again, particularly the part about the variation of the mass density, and then try putting this together again. This is not a trivial problem, but you don't seem to be anywhere close to getting it at this point. Perhaps it would help if you drew yourself some pictures, some 3-D sketches of what the situation is, and what you need to express in this triple integral.

 

[sberxa]

Wow...
Who would have guessed that this Forum is not only a valuable tool for helping eager students expand their knowledge of Physics and Mathematics, but also of English spelling too... wow. have to say I'm impressed.

I'd also like to point out that now you are just being a smart aleck. You obviously know exactly what I mean when I say 'row' even though it may not be spelled correctly. I apologize my knowledge of the Greek alphabet is not nearly as expansive as your own. I believe I am doing pretty well communicating what I am trying to get across considering English is not my first language. I also can't believe that you refuse to make any attempt to help me just because I can't find the symbol for rho on this website.

But I would like to thank you. For what, you might ask? Well, for wasting my time. Here I was thinking you might actually offer me something valuable, some little hint that could push me in the right direction. If in fact my integral is wrong for this case, then simply say so and maybe, if you have any desire to be a good person, offer me a little bit of insight into how you believe this problem might be solved.

But obviously, you are simply a fool who searches threads and makes unhelpful comments like the ones you have made in this entire thread.

Might I give you a suggestion, with your 400 and some posts. Stop wasting your time on a website that is meant to help one to better understand one of the many concepts that make this world great. For you are obviously not helping.

Now what will I do... I know, I'll go onto the next post that pops up and tell them, 'go look at your textbook and try again.' Obviously I have done that already or I wouldn't be here. That is not what this website it supposed to be. It is meant to be used as a tool as well as an area to correct errors to help others.

Well, I hope you have a good day! :)
Thank you for your time!

 

[Dr.D]

Contrary to your assertion, I am not a fool who searches threads and makes unhelpful comments such as you think I have made here. Although you are evidently too thick to recognize it, I have been trying to help you learn to express yourself properly.

I told you that you need to define your symbols, and you still have not done that. At one point you said, "My understanding (correct me if I'm wrong) represents the distance from the origin to the outer limits of the shapes described above." That statement does not have a subject, and as such does not make sense. I don't know what you are saying. It is important to make sense!

The only way any of us learns the Greek alphabet is by working with it. But you need to understand it is still necessary to define what you want the symbol to mean. Rho does not have a universally defined meaning.

Now if your time has been wasted, it was you who chose to waste it. No one forced you to spend time looking at this web site. You could have been sitting there with pencil and paper and the book, thinking seriously about the problem and working your way through it, and quite probably learning something. Instead, if your time has been wasted it is because you were looking for something for nothing and you are disappointed.

I'm not going to take your advice and stop helping people. I have been helping other people with problems in math, physics, and engineering for over 50 years now, and just because you are a sore head does not mean I should stop. However, if you should wish to return to your native country to study there, you will not get any objection from me.

You really might want to think twice before you decide to tell off someone who has tried to help you. It reflects badly on you.

 

[sberxa]

Again correcting my grammar. You asked me a question 'what is B' and the rest of that paragraph was dedicated to explaining that I was using B to represent rho. What do you think the subject of that sentence was?

But I'm glad you have so much time to dedicate to helping others. Are you a professor? I'm seriously interested because only professor's and "geniuses" have ever been so vague with their "help".

But contrary to your belief, I have been working on this handout for the last 10 hours and I will continue to work on it for the next week. No offense, but if I were really the kind of person who relied on a website to solve my problems, don't you think I would have a lot more posts considering all of the math courses I have taken to reach spherical coordinates. Not to mention, had I simply been waiting for your responses, I think I would have answered a lot more quickly than I did.

But alas, just like you, I seem to have misused this website and for that I do apologize. I will not be wasting any more time arguing with you over something so silly. This is meant to help and none of this back and forth will prove helpful to anybody who might have a similar question (although it might prove quite entertaining).

I wish you luck in all of your endeavors and I would like to thank you for at least trying to help. Obviously I did not understand your attempts seeing as I am so 'thick,' but let's just part and say that we obviously don't connect on a 'helper'-'helpee' standpoint.

Have a good life! :)

 

[djeitnstine]

Homework Statement

Find I(z) (moment of inertia about the z-axis) bounded by the cone z=sqrt(3(X^2 + y^2)) and the sphere with radius a. Density is inversely proportional to the distance from the z-axis.

Homework Equations

double integration

The Attempt at a Solution

basicall, I set up my integral, solved it and got a negative number and was hoping somebody could let me know if my original integral is wrong.

it is:

$$\int\int\int(B^3)(sin\Phi)^3$$ d$$B$$d$$\Phi$$d$$\Theta$$

please ignore the exponent in the above intergral. I can't figure out this website! It's supposed to be on the same level as the rest of the integral to show you the order I am intergrating with respect to. Thanks!

I put the limits of B from 0 to a, of Phi from 0 to (pi/6) and the limits of Theta from 0 to 2(pi). Intergrate first with respect to B then, phi, then theta.

I'm basically trying to use spherical coordinates to solve this integral.

Thanks!

Your limits look fine. But you should show us your steps to getting your function into spherical. Also as Dr. D said I'm not sure what your variables are. There should be no guessing as to what they mean. I'm guessing B is the radius and judging by the order and the limits I understand phi and theta.

 

[sberxa]

Thank you very much for your help.
I apologize if my limits were not clear. I did try to explain B meant rho and I tried to define my limits.
Anyways, thank you for verifying my limits. It must be something in the body of the integral I am doing wrong. Now I have somewhere to go from again. Thank you!
:)

 

[djeitnstine]

Thank you very much for your help.
I apologize if my limits were not clear. I did try to explain B meant rho and I tried to define my limits.
Anyways, thank you for verifying my limits. It must be something in the body of the integral I am doing wrong. Now I have somewhere to go from again. Thank you!
:)

No problem, I suggest just typing out what steps you take/took to get to your integrating function. So we can help you from there.