Talk:List of moments of inertia

message: "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)
replies: text: 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)
thread_title:  Polygon 

message: 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)
replies: text: 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)
thread_title:  duplicate item in area moments of inertia 

message: 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)
replies: text: That sounds like a good idea to me. Hemmingsen 18:23, 25 April 2006 (UTC)
replies: text: 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)
thread_title: Splitting the article

message: January 2010 (UTC)</span></small> 
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)
replies: text: 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)

text: 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)

text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Formula for a thick cylinder 

message: 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)
thread_title:  Formula for Sheet rotated about one end or through center 

message: 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)
thread_title:  oblate spheroid error 

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

text: 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)

text: 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)
thread_title:  Flat disc formula 

message: 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)
thread_title:  MOI of various triangles perhaps? 

message: 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)
thread_title:  Explanation of product of vector terms 

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

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

text: 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)
thread_title:  Merge List of moment of inertia tensors 

message: 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)
thread_title:  Right circular cone inertia incorrect? 

message: 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)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Error in cylinder formula 

message: 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)
thread_title:  Error in spherical shell 

message: 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)
thread_title:  Solids of revolution 

message: 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)
thread_title:  Words Clipped for exported PDF 

message: 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)
thread_title:  Error in torus description 

message: 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)
replies: text: 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)

text: 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)

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

message: 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)
thread_title:  More axes of rotation for thin-walled cylinder? 

message: 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)
replies: text: I noticed the same thing and fixed the image. Mr. Neo Anderson (talk) 23:05, 15 September 2022 (UTC)
replies: text:  @Mr. Neo Anderson: Superb, good work! Thank you!  CiaPan (talk) 07:52, 16 September 2022 (UTC)

text:  @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)
replies: text:  @Mr. Neo Anderson: Did it myself.  --CiaPan (talk) 16:39, 17 September 2022 (UTC)
replies: text: Thanks, very nice! Sorry for getting to this too late. Mr. Neo Anderson (talk) 16:19, 20 September 2022 (UTC)
thread_title:  Two point masses - misleading illustration 

message: 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..."
---
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)
thread_title:  Missing diagrams list, unclear 'side' parameter 

message: 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
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 du dv dw,\,  
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}m a^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)
thread_title:  Ellipsoid