Hello everybody!

I have a question about finding the moment of inertia of a sphere

I have solved the problem with 2 different methods.

Now, I'm trying to find it by using spherical coordinate system (3-dimension version of polar coordinate sistem)

I have found $dV$ volume differential of a sphere by spherical coordinate system as $dV=r^2\sin\phi{dr}{d\theta}{d\phi}$

By using this volume differential I took triple integral of it with ranges $0\leq{r}\leq{R}$, $0\leq{\theta}\leq{2\pi}$ and $0\leq{\phi}\leq{\pi}$.

Then I found the volume of a sphere as $\frac{4}{3}\pi{R^3}$ as you guess.

Now, I want to use that $dV$ volume differential to find the moment of inertia of a sphere.

$\rho=\frac{M}{V}=\frac{M}{\frac{4}{3}\pi{R^3}}=\frac{dM}{dV}=\frac{dM}{r^2\sin\phi{dr}{d\theta}{d\phi}}$

But I can't go any further.

Should I found the moment of inertia of the mass dM as I consider it is a rectange prism?

Then I might be able to use that moment of inertia to use in parallel axes theorem.

But I really don't have any idea :(

@ICCQBE The moment of inertia for an arbitrary body is $\int \rho(r) r^2 \mathrm{d}V$, cf. Wikipedia

Hmmm

Let me use that please

As I found $I=\int\rho(r)r^2{dV}=\int\frac{M}{\frac{4}{3}\pi{R^3}}R^2(r^2\sin\phi{dr}{d\theta}{d\phi})$

I have two questions. What are the limits of the integral and $d\theta$ and $d\phi$ are left. What can I do with them? :P

@ICCQBE The limits of the integral are the spatial extent of the sphere (do you really mean a sphere (surface) or do you mean a ball (volume)?).

I mean a ball.

It's a filled object.

I mean, there is no space inside it.

Yes, that's a ball :)

Then your limits are 0 and $R$ (radius of the ball) in the $r$ variable, 0 and $2\pi$ in the $\phi$ variable and 0 and $\pi$ in the $\theta$ variable.

Hmmm

Same limits as in finding the volume of a unit sphere (ball :P)

Let me do it, thanks!

@ICCQBE I'm being a pedant because mathematicians will assume that you just mean the surface of the ball if you say "sphere". It's better to get into the habit of distinguishing the two early :P

@ACuriousMind only if it's understood that $r$ is the distance to the axis, which is not the usual convention. (unless by moment of inertia you mean the trace of the moment of inertia tensor)

@Semiclassical Oh, right! @ICCQBE you will get the wrong result if you use that formula with spherical coordinates...

Yeah, I already got the wrong one :D

it's fine in 2D, of course, but this isn't 2D

It gave me $\frac{8}{15}$ coefficient and I was thinking like, where did I make a mistake? :p

Hmm, well..

you can deduce the correct answer from that integral, but only because spherical symmetry is great

cylindrical coordinates are usually more appropriate for moments of inertia, if you are computing the moment along an axis of symmetry of the body

right. that said, it's still quite doable

Because then you can choose the $r$ coordinate in that integral as the cylindrical radius

Yeah, I do. Tha axis passes through center of the mass of the ball.

yeah. i'll admit, my own taste is to write cylindrical coordinates at $(s,\theta,z)$ to avoid overloading $r$ and $s$

though I guess that should be $(s,\phi,z)$

By the way, is cylindrical coordinate systems same as spherical one?

noooo

they share a rotational coordinate but otherwise no

Hmm, I have to research about cylindrical coordinate system later.

I get it.

I have another question.

cylindrical coordinates = 2D polar coordinates X 1D vertical coordinate

Thank you for the brief explanation.

What does vertical coordinate looks like?

z=0 is the xy plane

I mean what is it represented by

Hmm

I'm listening to you.

in griffithms e&m, you see s=distance from vertical axis , phi=azimuthal angle, and z=vertical coordinate

there's other conventions, but that's the one I like best

Hmm, I get it

in cartesian coordinates (x,y,z), one has $s=\sqrt{x^2+y^2}$, $\phi=\arctan(y/x)$, and $z=z$

Are $s$ and $r\sin\phi$ the same?

in this convention, yes

no, wait

They look familiar in descriptiton

$s$ is the same as $r\sin\theta$ in this convention

because in this convention $\theta$ is the polar angle in spherical coordinates

Oh, I get it.

Thanks for not making me google "what's the diffetence between cylindrical coordinate system and spherical coordinate system?" :P

I also have another question.

it should be stressed that all of this is -heavily- dependent on conventions

As @ACuriousMind mentioned a formula $I=\int\rho(r)r^2dV$

mathematicians tend to have the opposite labels for the angles, and they'd usually replace $s\to r$ and $r\to \rho$. (but rho as a coordinate is a bad idea in the physics context)

I wonder that how they found out that formula I mean what kind of approach they did use and find out it?

for clarify, that's not how I'd write the formula

for the moment of inertia with respect to rotation around the $z$-axis, it should be $I=\int \rho(\vec{r})r_\perp^2 \,dV$ where $r_\perp$ is the distance to the z-axis.

this is a straightforward generalization of the moment of inertia for a set of point particles: $I=\sum_k m_k r_k^2$ where $r_k$ is the distance to the z-axis

just replacing summation with integratino

Hmm

How does the term $dV$ comes into the formula?

$\rho dV$ is the mass of a differential volume $dV$

$dM$, right.

Wow!

so it's as though you chopped up the volume of integration into infinitesimal particle, computed them particle-by-particle, and then re-summed them

same as you'd do for finding an electric field of a charge distribution, for instance

I really got it.

Thank you for enlightening me @Semiclassical and @ACuriousMind

I have a question.

What are you studying? :P

nothing at the moment. I have my phd

I see a big difference between you guys here and I wonder will I become like them :P

PhD in Physics?

yeah

Wow, that's cool. Which university is it?

university of minnesota

That's really nice.

Did you learn liquid dynamics in undegrad?

not really, no, and I wouldn't say I know a ton of it now

Hmm, I understand. I really want to learn about it but I see I should to learn much more things before get into that concepts

I have another question.

I'm a freshman.

In the first semester I learnt about Units, Vectors, Newton's Laws of Motion and It's application (including briefly some circular motion), Frames of References: Galilean Transform, Conservation of Energy and lastly Conservation of Linear and Angular Momentum

I got the concepts that are shown in these subjects very well

(except Frames of References, I'm still struggling with it :P)

@ICCQBE I'm no longer a student, I work as a software engineer

In the second semester we get into rotational motion from translational movement (linear movement)

@ACuriousMind what did you studied, I'm asking too much, I think I'm a curious one too :P

Later on I seen concepts named moment of inertia, behavior of angular momentum vector and so on

I was very fine at first semester but I'm struggling so much this semester.

I hear new concepts that I have never heard.

My question is, how can I overcome this? How can I understand concepts and stop the complexity of the concepts.

@ICCQBE I studied (theoretical) physics, with a focus on math and quantum field theory.

I can't get the main idea of a concept Real quick

I want to learn about the person who told this.

That sentence made me feel better, seriously.

@ICCQBE John von Neumann was a famous mathematician and physicist.

Well, I should not care about the concepts and their meaning in detail, right?

I have mixed feelings about vN these days. As a scientist, he certainly earned his prestige

My reading of that quote - and the reason I like it - is that it is often futile to expect to understand something fully when you first encounter it. You need to wrestle with the concept, apply it to different problems, have it explained to you a few times from different perspectives. And one day, sooner or later, it will "click".

But he was also part of the Manhattan project, including the choice of targets (he wanted to bomb Kyoto) and a strong supporter of nuclear armament

And that leaves me rather conflicted

A 1950 quote: "If you say why not bomb [the Soviets] tomorrow, I say, why not today? If you say today at five o'clock, I say why not one o'clock?"

@ACuriousMind. I got what you mean. When I think about myself this situation about my second semester is depends to me too, as I can see. I should to be honest to myself.

It’s hard to imagine how dangerous the Soviets were seen in that era, I suppose

Kind of interesting person.

I would like to develop rockets as well, but not for nukes :P

Sometimes, something tell me "not yet" for every action of mine.

I'm getting awful. Well.

Thanks for the chat and your helps @ACuriousMind and @Semiclassical. Have a good night/day!

Hope to see you again!

Good night :)

@ICCQBE Rockets are pretty fun, but so are all propulsion systems! If you're interested in fluid dynamics, I hope to see some good questions/answers on the main site! I need more fluids folks around here...