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)

?

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)

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)

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)

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

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)