Talk:List of moments of inertia

Polygon

"Thin, solid, polygon shaped plate with vertices ${\vec {P}}_{1}$, ${\vec {P}}_{2}$, ${\vec {P}}_{3}$, ..., ${\vec {P}}_{N}$ and mass $m$."

Does this work for ALL polygons? Convex/Non-convex etc. I think this should be made clear. --Smremde 10:16, 21 June 2007 (UTC)

If you consider a thin, flat sheet of material then it should work, since finding the moment of inertia is easily done if you know the surface area of the material. The formula given should have considered the formula for the surface area of an n-sided polygon. Anyway, if n → infinity the moment of inertia should tend towards that of the this flat circular disc. AquaDTRS (talk) 16:39, 22 March 2008 (UTC)

I'm going to fix some mistakes in the statement of the conditions for this one. It's only valid for rotation about an axis perpendicular to the plane, since the moment of inertia would otherwise have to depend on the orientation of the axis, and the expression given is independent of that orientation. It's also not necessary for it to be a regular polygon, since the expression is just an area-weighted sum of the r^2's of the centers of mass of the triangles. It is necessary for it to be a plane polygon, but it's not necessary for it to be thin.--207.233.88.250 (talk) 00:16, 12 December 2008 (UTC)

Do there really need to be norms around the vector products? In 2d they are scalars anyway and I believe the norms (getting absolute value functions) to be wrong. —Preceding unsigned comment added by 131.246.120.30 (talk) 14:28, 16 April 2010 (UTC)

Although the shapes are 2D, the formula holds for shapes embedded in a 3D space. So the vectors might have nonzero x, y, and z components. I prefer making it explicit that we mean the scalar magnitudes of the cross products. CosineKitty (talk) 19:59, 16 April 2010 (UTC)

Shouldn't the summation be over N triangles, with P(N+1)=P(1)? Or have I missed some 'degree of freedom' type thing? 86.4.250.43 (talk) 00:25, 9 December 2013 (UTC)

I have corrected the summation to go from 1 to N. I suspect the restriction on the polygon being star-shaped (wrt the origin) can be relaxed by taking care with the placement of the modulus signs, but I don't have a reference. Also added an entry for a triangle rotating about one vertex, from which the polygon formula is an easy corollary. A picture would be nice but I don't know how to do this. David Brightly 86.4.250.43 (talk) 22:54, 11 December 2013 (UTC)

?

I know it's generally obvious from the picture, but isn't it technically necessary to specify the axis of rotation as well as the body under consideration? Hammerite 00:51, 23 October 2005 (UTC)

The ones indicated are around the center of mass. Some are the same around any axis, others the axis is obvious, and usually shown. Gah4 (talk) 00:55, 24 June 2020 (UTC)

duplicate item in area moments of inertia

There are two rows called "an axis collinear with the base", and somewhat worryingly they have different formulae. Could somebody check and delete/clarify one or the other? Bathterror 09:56, 8 March 2006 (UTC)

The two rows referred to a rectangular area and a triangular area. Should be a little clearer now. Hemmingsen 14:16, 22 April 2006 (UTC)

Thick cylinder with open ends mistake in formula?

The formula I of z axis has the multiplier 1/2 but shouldnt it be 1/8 ?

No, the first formula is OK. $\scriptstyle {{1 \over 2}m(R_{ext}^{2}+R_{int}^{2})}$ LPFR 12:39, 4 September 2006 (UTC)

This page does NOT display correctly on several on several of my machines (Mac OSX, Linux, and Windows). WHY are these colons in front of the math directive needed and why is it responsible to put in formating that causes parse errors to the casual observer of these pages?

It displays fine on mine (Windows XP. IE and Netscape). LPFR 12:39, 4 September 2006 (UTC)

If the colons your are talking about are " \,</math> ", this is a TeX/LaTeX directive that adds a small space in mathmode. It cannot create a parse error in any decent browser. I think that you have a problem with your machines or your browsers. I think that this page is OK and well written. But if you think that there are things that can be ameliorated, just do it. LPFR 12:58, 4 September 2006 (UTC)

No, the first formula with R1 + R2 is not OK. If the two radii are equal the wall thickness is zero, so the mass in such a case must be zero, and the inertia is zero. Do the math: for a material of density ρ, the triple integral from height -h/2 to +h/2, radius ri to ro, and direction 0 to 2π for the cylinder yields a moment of inertia of 2πρh(ro-ri). — Preceding unsigned comment added by JeffCalTech (talk • contribs) 13:13, 23 March 2014 (UTC)

Correction on the previous comment: The formula is unfortunately easily misunderstood and hence miseading in that most people think from the vantage point of making an object of a particular shape out of a particular material. They would like to vary the dimensions and see what happens to the inertia and do not expect the mass to remain constant as if it were being compressed. This confusion would be eliminated if the mass were replaced with the density yielding a moment of inertia of 2πρh(ro-ri). — Preceding unsigned comment added by JeffCalTech (talk • contribs) 13:34, 23 March 2014 (UTC)

I would have thought that, too, but it is usual to do shape and mass. So, if the two radii are the same, it is a delta function wall cylinder. Gah4 (talk) 00:59, 24 June 2020 (UTC)

Splitting the article

Would anyone have a problem if I split this into two seperate articles? I can't see any point in having these two distinct properties in the one article. I think it adds to the common confusion between these two quantities, and the article is a bit long also. Brendanfox 02:02, 25 April 2006 (UTC)

That sounds like a good idea to me. Hemmingsen 18:23, 25 April 2006 (UTC)

Well, this should be long enough time for people to voice their objections. Going ahead with the split... (to List of area moments of inertia) Hemmingsen 15:03, 30 December 2006 (UTC)

Formula for a thick cylinder

January 2010 (UTC) The correct formula is $\scriptstyle {{1 \over 2}m(R_{ext}^{2}+R_{int}^{2})}$.

Someone using IP 216.148.248.31 (CERFN California Education and Research Federation Network) wrongly "corrected" the equation changing the + sign for a minus sign. I suggest to this user to verify this formula in a physics book. If you did find this formula with the minus sign, verify in others books and correct the wrong formula.

As you do not seem to be able to calculate the inertia moment by yourself, I can demonstrate why the sign must be a + sign. If $\scriptstyle {R_{int}}$ increases, but maintaining the total mass m unchanged, this means that you are "sending" some of the mass from a small radius to a larger one. As the moment of inertia is proportional to the squared distance to axis of rotation, the moment of inertia should increase. That is, if $\scriptstyle {R_{int}}$ increases the moment of inertia increases. Then, the sign must be a plus sign. LPFR 07:59, 7 September 2006 (UTC)

The mistake is understandable - usually one would expect the mass to decrease if one made the hole larger, and if the hole was the same diameter as the outer then the mass and MoI would be zero! If the mass stays constant then the density must increase, but the positive sign is then correct. If the density stays constant (which is the practical case of most interest) then the formula becomes (I think, if my late-night integration is correct) (pi/2)*density*length*(R2^4 - R1^4). If you substitute mass/volume for density in this you get back to the correct formula given above. 217.147.104.220 10:51, 20 December 2006 (UTC)

The mass of the cylinder could be kept constant without increasing the density of the material, by increasing the length of the cylinder.Gregorydavid 11:23, 20 December 2006 (UTC)

The formula Iz = (pi/2)*density*length*(R2^4 - R1^4) couldn't possibly be correct, since R1 = R2 gives Iz = 0, if I'm not somehow mistaken. /Andreas —Preceding unsigned comment added by 129.16.53.239 (talk • contribs) 13:45, 5 April 2008 (UTC)

That formula differs from the other formula also discussed above by assuming a fixed density instead of a fixed total mass. With a fixed, finite density and R1 = R2 the mass will be zero because the volume will be zero, and a body with no mass will have no moment of inertia. The formula looks right to me. Hemmingsen 16:17, 5 April 2008 (UTC)

Ok, I see. However, setting that expression equal to ${\frac {1}{2}}m\left({r_{1}}^{2}+{r_{2}}^{2}\right)$ seems incorrect and misleading. I think some clarification is in order as to what assumptions correspond to each expression for I_z. /Andreas —Preceding unsigned comment added by 193.11.234.85 (talk) 19:00, 5 April 2008 (UTC)

That does seem like a good point. The two sets of assumptions are equivalent in all cases except for the one where R1 = R2 and calculating the density as mass/volume would be a division by zero, but that special case does seem to be a case people are interested in. I will attempt to rewrite that part of the list slightly; let me know if you think it is insufficient or have a better idea. Hemmingsen 19:40, 5 April 2008 (UTC)

That seems to me like a good solution! /Andreas

I agree. The equation shows itself incorrect when you realize that as the interior radius approaches the exterior radius, this formula would give you a moment of inertia of 0 while the correct value, m*r^2 comes about when you use the formula with the plus sign.
Dexter411 18:03, 20 December 2006 (UTC)

I think the confusion above from the minus sign could be avoided if a extra column was made for the inertia as a function of the density, not the mass (would make easier reference as well). —Preceding unsigned comment added by 129.230.248.1 (talk) 13:03, 23 July 2008 (UTC)

The formula is unfortunately easily misunderstood and hence miseading in that most people think from the vantage point of making an object of a particular shape out of a particular material. They would like to vary the dimensions and see what happens to the inertia and do not expect the mass to remain constant as if it were being compressed. This confusion would be eliminated if the mass were replaced with the density yielding a moment of inertia of 2πρh(ro-ri). — Preceding unsigned comment added by JeffCalTech (talk • contribs) 13:27, 23 March 2014 (UTC)

Formula for Sheet rotated about one end or through center

I came here looking for this formula, it is in any intro physics text, and is rather important. Someone should add this. I don't have time now. - Anonymous

I checked the foumula by subtracting the moment of inertia of a solid cylinder of r1 from a solid cylinder of r2. I did not get the same value as presented by the foumula 12.4.26.248 (talk) 15:06, 2 November 2009 (UTC) R Friedman

oblate spheroid error

The oblate spheroid entry looks odd ... shouldn't it have different moments of inertia for rotation about the different axes? Also, the 2/3rds suggests a hollow rather than solid object - if this is the intention, the description should specify.

I think the formula for the ellipsoid is correct on both counts, assuming the formula for a solid ball is correct. The formula works for any values a, b, c of semiaxes, assuming that you are rotating around a. If you are actually rotating around b or c, you could do one of two things: change the formula to use the other two letters (other than the one you are rotating about, that is), or you could just change the variables you are using so that a becomes the one you are rotating about. If you let a=b=c, you will see that this is the same formula as for the solid ball, so this is a solid ellipsoid. I will change it to reflect that. (I'm not sure where the 2/3 is that you are talking about, though.) CosineKitty (talk) 15:40, 14 March 2010 (UTC)

Flat disc formula

-I'm not really sure where to put this, so I'll add it here, because this section's pointing out an error as well. I believe that the entry for a flat disc rotated about the x- and y-axes is incorrect. It lists the moment of inertia as $MR^{2}/3$, when it really should be $MR^{2}/4$ —Preceding unsigned comment added by 129.21.69.255 (talk) 04:25, 14 March 2010 (UTC)

[Note: I created a separate section for this issue. I will respond soon.] CosineKitty (talk) 15:31, 14 March 2010 (UTC)

Yes, you were right, and I corrected it. I had to re-derive the formula from scratch to be sure. I am also working on citing some references for this article so people can look up the formulas and confirm them. CosineKitty (talk) 16:22, 14 March 2010 (UTC)

Follow-up: I confirmed several of these formulas from my college physics book (Serway) and cited it in the article. CosineKitty (talk) 18:59, 14 March 2010 (UTC)

MOI of various triangles perhaps?

Would it be possible to source any formulas for the MOI of a triangle object or frame?

I know there is the general vector form for any polygon, but triangles are relatively common objects to come across (at least in text book material) and hence I think it is necessary to have it. I suppose a formula would make sense for a right-angle triangle, equilateral triangle and isosceles triangle.

Anyone know how difficult it may be to obtain the formulas? (My text book doesn't indicate any formulas)

Discussion much appreciated! (Eug.galeotti (talk) 07:02, 28 June 2010 (UTC))

Explanation of product of vector terms

The equation for the calculation of the moment of inertia for a general polygon consists of terms such as P^2; ie the square of a vector. I'm familiar with the dot and cross products of vectors but what does the square of a vector mean? —Preceding unsigned comment added by 81.187.174.42 (talk) 19:12, 4 August 2010 (UTC)

Merge List of moment of inertia tensors

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

The result of the discussion was no consensus.

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I suggest to merge List of moment of inertia tensors into the current article; since all tensors in the principal axes have only three non-zero components, and this information is already listed in this article.  // stpasha »  16:37, 17 August 2010 (UTC)

If the resulting article is not too long, I would think this a good suggestion. Baccyak4H (Yak!) 17:44, 17 August 2010 (UTC)

I don't think this is a good suggestions. Tensors are too confusing.Eregli bob (talk) 15:43, 13 October 2010 (UTC)

The article should only be merged if the information about the tensors don't overtake the simple information given in this article. Bande-Ali (talk) 02:02, 1 May 2011 (UTC)

Perhaps they tensors could be added as optional additional info, that way someone who is only looking for the basic formula will not be confused by a scary looking matrix — Preceding unsigned comment added by 147.4.36.65 (talk) 15:54, 30 April 2012 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Right circular cone inertia incorrect?

The inertia is not around the center of mass but around the apex I believe. This should be stated in the notes. 132.239.186.132 (talk) 00:37, 28 January 2011 (UTC)

Error in cylinder formula

As far as I can tell the formula for a solid cylinder in Ix and Iy has an error. The displayed formula Ix = Iy = (1/12)*m*(3r^2 + 4h^2) should not have a 4*h I think. It should read Ix = Iy = (1/12)*m*(3r^2 + h^2) anyone agree? — Preceding unsigned comment added by 62.189.28.130 (talk) 15:03, 27 July 2011 (UTC)

I found the original error when I was working through the integrals for the moment of inertia. I then verified this with wolfram alpha. The h^2 should be 4h^2. http://www.wolframalpha.com/input/?i=moment+of+inertia+of+a+cylinder. If my math is wrong and wolfram alpha is wrong, then my update is wrong. Otherwise this newer version with 4h^2 is the right one. Hanavi (talk) 07:51, 29 July 2011 (UTC)

Also, someone should check the thick walled cylinder equation. I think it has the same error, but I never did the integral for that one. I am betting that it should also have the 4h^2 Hanavi (talk) 07:57, 29 July 2011 (UTC)

Compare with a rod, and see http://scienceworld.wolfram.com/physics/MomentofInertiaCylinder.html: it looks like it was correct without the 4, and http://www.wolframalpha.com/input/?i=moment+of+inertia+of+a+cylinder is wrong.--Patrick (talk) 08:24, 29 July 2011 (UTC)

Ok, I reworked the integral and found the mistake. I had mistakenly used the limits of integration from 0 to h and it should have been from -h/2 to h/2. The way I had done it would give the moments of inertia around the end of the cylinder (as can be seen by comparing the rod formulae). The original and now current versions are correct. Hanavi (talk) 21:31, 29 July 2011 (UTC)

I would also recomend updating the entry on the solid cylinder to include the moment of inertia about the z axis. MSWurmstein (talk) 00:59, 12 April 2012 (UTC)

Error in spherical shell

Article says I about axis of spherical shell is 3/5 MR^2.

I'm no physics PhD -- just a high school teacher -- but when I do out the integrals I get 2/3 not 3/5.  

My text book agrees with me. (Halliday, Resnick, Walker, Fundamentals of Physics 9th ed, p255)

Is this just a typo? and if so -- I'm not a regular wikipedian -- can I just correct this?

Dstras (talk) 00:48, 3 February 2013 (UTC)david strasburger strasburger@nobles.edu

Also DPHutchins Finds that: When r₁ = r₂, $\left({\frac {r_{2}^{5}-r_{1}^{5}}{r_{2}^{3}-r_{1}^{3}}}\right)={\frac {5}{3}}r_{2}^{2}$, seems to be in error as it does not match the units used in all of the other examples. 2001:5B0:2B13:E778:0:0:0:1003 (talk) 05:41, 15 May 2019 (UTC)

Solids of revolution

I found this paper describing how to get the mass moment of inertia for solid of revolution about the revolving axis as well as the perpendicular axis.

Maybe these formulas need to be included in the list. This theory would be applicable when one is looking for the moment of inertia of tapered rolling element, or a parabolic shape.

iou (talk) 17:06, 16 August 2013 (UTC)

Words Clipped for exported PDF

The styling of the webpage requires a certain width for each of the cells. When sent to the pdf, the size of the margins clips off a number of words from the Comments and Description column. — Preceding unsigned comment added by 132.194.3.169 (talk) 17:23, 9 November 2013 (UTC)

Error in torus description

The part of article about torus below:

Torus of tube radius a, cross-sectional radius b and mass m. About the VERTICAL axis: \left(a^2 + \frac{3}{4}b^2\right)m [4]

has an error: The sited source [4] is correct, but has a different picture with different orientation. Axis that is vertical on the source is HORIZONTAL on the wikipicture, therefore the leftmost description about the axis used "VERTICAL" is wrong. The left most formula is correct for an axis going through the torus, and that is horizontal axis. Some native english speaker should correct that in unambigious way.86.50.116.35 (talk) 15:03, 10 June 2014 (UTC)

Errors in entry "Rod of length L and mass m, axis of rotation at the end of the rod"

The Description speaks of "Length L", but the Figure formulas use the variable "r". It's not clear how "r" (radius?) would affect formulas I(x) and I(z), where the rod is assumed in Description to be "infinitely thin". Also, the Figure shows I(y)= m * r^2 / 12, which is similar to the formula for "Rod of Length L and mass m" in the Figure immediately below in which the axis of rotation is at the center of mass of the rod (this latter case is derived in Section "Example Calculation of moment of inertia" of article Moment of inertia. It seems for the "end of rod" case that an additional term needs to be added to the formula per Parallel axis theorem, so it reads "I(y) = m * L^2 / 12 + m * (L / 2)" or equivalent. MeraNaamJim (talk) 23:54, 28 November 2014 (UTC)

This is a correction to my discussion above. I(x) and I(y) refer to rotation about the end of the rod (not I(y)), and are correct if "L" is used instead of "r" in the formulas, to be consistent with the Description: From the "Rod of length L and mass m" and the parallel axis theorem, I(x) = I(z) = m * L^2 / 12 + m (L/2)^2 = m * L^2 / 3. I(y) appears to refer to rotation along the axis of the rod of radius "r"; if so, then "r" needs to be defined in the Description and reconciled with the "infinitely thin wire" assumption in the Description. Then, either I(y) would appear to be a case of "Solid cylinder of radius r, height h, and mass m" on the page, so I(y) = m * r^2 / 2, or the "I(y)" formula should just be removed from the entry. The reference to "a special case of the thin rectangular plate with axis of rotation at the end of the plate [where there is no rotation addressed in the plane of the plate], with h=L and w=0" in the Description is also an argument for just removing the "I(y)" formula. MeraNaamJim (talk) 18:37, 29 November 2014 (UTC)

The variable "r" has no definition making the formulas for I(x), I(y), and I(z) consistent with the case Description. My comment above about I(y) as rotation about the axis of the rod is incorrect. Looking at the "thin rectangular plate with axis of rotation at the end of the plate" case referenced in the Description, Iend = m * h^2 / 3 + m * w^2 / 12, the first term gives m * L^2 / 3 when "L" is substituted for "h" (which looks like the I(x) and I(z) values of our rod case, where "r" = "L"). The second term looks like the I(y) value of our rod case, where "w" = "r" - inconsistent with "r" already being "L". (Also, w = 0 makes the term "m * w^2 / 12" = 0, which would make I(y) = 0.) MeraNaamJim (talk) 20:21, 4 December 2014 (UTC)

I am reverting the edit of "19:28, 2014 November 24" made by "97.65.103.250", which modified the previous scaler moment of inertia formula for "Rod of length L and mass m, axis of rotation at the end of the rod" where the rod is assumed to be an infinitely thin rigid wire. The edit replaced the original variables "L" and "w" in the Description with an undefined variable "r", and expressed moments of inertia in terms of this variable "r" with respect to three rotation axes (x, y, z) not shown or described in the corresponding Figure or Description. Looking at the "...special case of the thin rectangular plate with axis of rotation at the end of the plate, with h = L and w = 0" (the 19th list entry) noted in the Description, Iend = m * h^2 / 3 + m * w^2 / 12. Applying the Description assumptions to the plate case gives Iend = m * L^2 / 3, possibly the intended I(x) and I(z) values of the edit, and sets the second term to 0, possibly the actual value of I(y) in the edit. MeraNaamJim (talk) 21:01, 4 December 2014 (UTC)

More axes of rotation for thin-walled cylinder?

It's annoying that the moments for the x/y rotation axes for the thin-walled cylinder are not present, while the x/y rotation axes for the solid cylinder is present. Could someone add this information, and create a new figure to match? Thanks, Ws04 (talk) 05:31, 22 January 2016 (UTC)

tensor

The section giving tensor values is nice, but all are diagonal. Also, many have the same entry in all diagonal elements so might as well be scalar. Also, could we have at least one entry with off-diagonal values? Maybe a thin rod rotated around an axis not perpendicular, but with an angle (theta) to the rotation axis? Gah4 (talk) 01:10, 24 June 2020 (UTC)

The useful tensors are the diagonal ones that align with principle axes. If you want a non-principal axis, you use the Parallel axis theorem and the Perpendicular axis theorem, or you compute it from scratch because your rotation axis is too weird to bother summarizing in a table like this, like what's the moment of inertia of a cone when spun around a line at some angle off the main axis going through the center of mass. These things have answers, which you get through

$I_{P}=\iiint \limits _{Q}\rho \left(x,y,z\right)\left\|\mathbf {r} \right\|^{2}\mathrm {d} V$

but these aren't the things you need in a table like this. Headbomb {t · c · p · b} 04:47, 24 June 2020 (UTC)

Which you could also probably obtain by manipulating the tensors via rotation or similar, I'm sure. Headbomb {t · c · p · b} 04:51, 24 June 2020 (UTC)

Yes you get them by tensor rotation, but it isn't something that people are used to doing. I hope a good fraction of people know how to rotate a vector, though I could be surprised there, but I suspect most can get through a good engineering school without ever thinking about tensor rotation. How many people even know that there are principle axes? One example of rotation around another axis would get people started thinking about tensor rotation, after which they could figure out the other ones. Gah4 (talk) 09:09, 24 June 2020 (UTC)

If you know of moments of inertia tensors, you know of tensor rotations. Headbomb {t · c · p · b} 10:35, 24 June 2020 (UTC)

Two point masses - misleading illustration

A picture in the second row of the table (two point masses rotating about the center of mass of the system) is misleading. The red curve suggests both masses travel the same elliptic orbit, while the formula given represents a constant configuration, that is a case of each mass on its own circular orbit, both orbits centered at the center of mass of the system. Anybody able to fix the graphics? --CiaPan (talk) 10:27, 6 April 2021 (UTC)

I noticed the same thing and fixed the image. Mr. Neo Anderson (talk) 23:05, 15 September 2022 (UTC)

@Mr. Neo Anderson: Superb, good work! Thank you! CiaPan (talk) 07:52, 16 September 2022 (UTC)

@Mr. Neo Anderson: Could you also replace 'r' with 'x' in the image, as used in the formula on the right? The distance marked is not a radius, so the symbol 'r' can be a bit misleading. --CiaPan (talk) 07:57, 16 September 2022 (UTC)

@Mr. Neo Anderson: Did it myself. --CiaPan (talk) 16:39, 17 September 2022 (UTC)

Thanks, very nice! Sorry for getting to this too late. Mr. Neo Anderson (talk) 16:19, 20 September 2022 (UTC)

Derekleungtszhei, the file File:MOIaboutCOMaxis2.svg I talk about above is a simple but incorrect modification of your File:PointInertia.svg. Do you think you could expand the latter to create a correct version for the former...? --CiaPan (talk) 07:37, 6 May 2021 (UTC)

Missing diagrams list, unclear 'side' parameter

The following diagrams are missing:

1. "Regular dodecahedron of side s and mass m"

2. "Regular icosahedron of side s and mass m"

3. "Thin rectangular plate of radius r and mass m"

4. "Triangle with vertices at the origin and at P and Q, with mass m, rotating about an axis perpendicular to the plane and passing through the origin."

5. "Plane regular polygon with n-vertices and mass m uniformly distributed on its interior..."

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Additionally, the use of the word "side" is unclear for the dodecahedron/icosahedron cases. What _exactly_ does a side mean? Edge length? Raleighlittles (talk) 17:27, 20 December 2022 (UTC)

Ellipsoid

For a rotation around the x-axis the moment of inertia is: $I_{x}=\int _{V}(y^{2}+z^{2})\rho dV=\int _{V}y^{2}\rho dV+\int _{V}z^{2}\rho dV$

Homogenous ellipsoid with $(x/a)^{2}+(y/b)^{2}+(z/c)^{2}\leq 1$ has the principal moments of inertia $(I_{x},I_{y},I_{z})=(b^{2}+c^{2},a^{2}+c^{2},a^{2}+b^{2})\cdot m/5\,$

Set: $u=x/a,\,v=y/b,\,w=z/c,\,$ and therefore: $\int _{V}z^{2}\rho dV=\rho \cdot abc\cdot c^{2}\cdot K,\,$ with $K=\int _{u=-1}^{1}\int _{v=-{\sqrt {1-u^{2}}}}^{\sqrt {1-u^{2}}}\int _{w=-{\sqrt {1-u^{2}-v^{2}}}}^{\sqrt {1-u^{2}-v^{2}}}w^{2}dudvdw,\,$ and likewise $\int _{V}y^{2}\rho dV=\rho \cdot abc\cdot b^{2}\cdot K,\,$ hence $I_{x}=\rho \cdot abc\cdot (c^{2}+b^{2})\cdot K={\frac {3m}{4\pi }}(c^{2}+b^{2})\cdot K,\,$ where $m=\rho \cdot V=\rho \cdot {\frac {4\pi }{3}}\cdot abc$

A solid sphere with mit a=b=c has $I_{x}={\frac {3m}{4\pi }}\cdot 2a^{2}\cdot K$. That compares with the result for the sphere $I_{x}={\frac {2}{5}}ma^{2}$, and so: $K=4\pi /15$.

Hence: $I_{x}={\frac {m}{5}}(b^{2}+c^{2})\,$ and likewise for the other two principal moments of inertia.--Mominert (talk) 08:40, 30 April 2023 (UTC)