3564:
2910:
3559:{\displaystyle {\begin{aligned}I_{x,{\text{circle}}}&=\iint _{R}y^{2}\,dA=\iint _{R}\left(r\sin {\theta }\right)^{2}\,dA=\int _{0}^{2\pi }\int _{0}^{r}\left(r\sin {\theta }\right)^{2}\left(r\,dr\,d\theta \right)\\&=\int _{0}^{2\pi }\int _{0}^{r}r^{3}\sin ^{2}{\theta }\,dr\,d\theta =\int _{0}^{2\pi }{\frac {r^{4}\sin ^{2}{\theta }}{4}}\,d\theta ={\frac {\pi }{4}}r^{4}\\J_{z,{\text{circle}}}&=\iint _{R}r^{2}\,dA=\int _{0}^{2\pi }\int _{0}^{r}r^{2}\left(r\,dr\,d\theta \right)=\int _{0}^{2\pi }\int _{0}^{r}r^{3}\,dr\,d\theta \\&=\int _{0}^{2\pi }{\frac {r^{4}}{4}}\,d\theta ={\frac {\pi }{2}}r^{4}\end{aligned}}}
1747:
2347:
4976:
1891:
4321:
525:
2546:
4248:
2342:{\displaystyle {\begin{aligned}I_{x}&=\iint _{R}y^{2}\,dA=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}\int _{-{\frac {h}{2}}}^{\frac {h}{2}}y^{2}\,dy\,dx=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}{\frac {1}{3}}{\frac {h^{3}}{4}}\,dx={\frac {bh^{3}}{12}}\\I_{y}&=\iint _{R}x^{2}\,dA=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}\int _{-{\frac {h}{2}}}^{\frac {h}{2}}x^{2}\,dy\,dx=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}hx^{2}\,dx={\frac {b^{3}h}{12}}\end{aligned}}}
4971:{\displaystyle {\begin{aligned}I_{y}&={\frac {1}{12}}\sum _{i=1}^{n}\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\left(x_{i}^{2}+x_{i}x_{i+1}+x_{i+1}^{2}\right)\\I_{x}&={\frac {1}{12}}\sum _{i=1}^{n}\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\left(y_{i}^{2}+y_{i}y_{i+1}+y_{i+1}^{2}\right)\\I_{xy}&={\frac {1}{24}}\sum _{i=1}^{n}\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\left(x_{i}y_{i+1}+2x_{i}y_{i}+2x_{i+1}y_{i+1}+x_{i+1}y_{i}\right)\end{aligned}}}
4238:
1010:
3865:
33:
1725:
For more complex areas, it is often easier to divide the area into a series of "simpler" shapes. The second moment of area for the entire shape is the sum of the second moment of areas of all of its parts about a common axis. This can include shapes that are "missing" (i.e. holes, hollow shapes,
4233:{\displaystyle {\begin{aligned}J_{z}&=\iint _{R}r^{2}\,dA=\int _{0}^{2\pi }\int _{r_{1}}^{r_{2}}r^{2}\left(r\,dr\,d\theta \right)=\int _{0}^{2\pi }\int _{r_{1}}^{r_{2}}r^{3}\,dr\,d\theta \\&=\int _{0}^{2\pi }\left\,d\theta ={\frac {\pi }{2}}\left(r_{2}^{4}-r_{1}^{4}\right)\end{aligned}}}
3833:
1645:
5286:
The term second moment is more proper than the term moment of inertia, since, logically, the latter should be used only to denote integrals of mass (see Sec. 9.11). In engineering practice, however, moment of inertia is used in connection with areas as well as
2536:
3647:
1368:
For the simplicity of calculation, it is often desired to define the polar moment of area (with respect to a perpendicular axis) in terms of two area moments of inertia (both with respect to in-plane axes). The simplest case relates
1726:
etc.), in which case the second moment of area of the "missing" areas are subtracted, rather than added. In other words, the second moment of area of "missing" parts are considered negative for the method of composite shapes.
1455:
2387:
712:
4326:
3870:
2915:
1896:
919:
1000:
1168:
371:
4285:
on the XY-plane can be computed in general by summing contributions from each segment of the polygon after dividing the area into a set of triangles. This formula is related to the
311:
425:
5122:
494:
74:
This article is about the geometrical property of an area, termed the second moment of area. For the moment of inertia dealing with the rotation of an object with mass, see
5214:
5168:
5070:
5018:
788:
614:
586:
1352:
1287:
1258:
1107:
1058:
822:
4316:
vertices, numbered in counter-clockwise fashion. If polygon vertices are numbered clockwise, returned values will be negative, but absolute values will be correct.
3640:
3613:
2899:
2872:
2801:
2774:
2727:
2700:
2653:
2622:
2595:
2380:
1884:
1857:
1830:
1711:
1448:
1421:
1394:
849:
760:
645:
4275:
3858:
738:
5038:
4314:
3586:
2841:
2821:
2747:
2673:
1799:
1779:
1691:
1671:
1327:
1307:
1233:
1213:
1191:
1082:
146:
126:
5454:
3828:{\displaystyle J_{z}=J_{z,r_{2}}-J_{z,r_{1}}={\frac {\pi }{2}}r_{2}^{4}-{\frac {\pi }{2}}r_{1}^{4}={\frac {\pi }{2}}\left(r_{2}^{4}-r_{1}^{4}\right)}
5487:
108:
which reflects how its points are distributed with regard to an arbitrary axis. The second moment of area is typically denoted with either an
5437:
5383:
5341:
5316:
5279:
2675:
axis by the method of composite shapes. This polar moment of inertia is equivalent to the polar moment of inertia of a circle with radius
650:
1640:{\displaystyle J_{z}=\iint _{R}\rho ^{2}\,dA=\iint _{R}\left(x^{2}+y^{2}\right)dA=\iint _{R}x^{2}\,dA+\iint _{R}y^{2}\,dA=I_{x}+I_{y}}
924:
5361:
5409:
860:
49:
942:
5522:
5225:
1735:
82:
1746:
512:. The MOI, in this sense, is the analog of mass for rotational problems. In engineering (especially mechanical and civil),
157:
1112:
5517:
1064:
axis of the shape. However, it is often easier to derive the second moment of area with respect to its centroidal axis,
427:, where r is the distance to some reference axis). In each case the integral is over all the infinitesimal elements of
2844:
3588:
axis for an annulus is simply, as stated above, the difference of the second moments of area of a circle with radius
2531:{\displaystyle J_{z}=I_{x}+I_{y}={\frac {bh^{3}}{12}}+{\frac {hb^{3}}{12}}={\frac {bh}{12}}\left(b^{2}+h^{2}\right)}
5245:
5230:
1363:
249:
5527:
5507:
5469:
1714:
200:
188:
500:
is the distance to some potential rotation axis, and the integral is over all the infinitesimal elements of
316:
259:
524:
180:
2729:, both centered at the origin. First, let us derive the polar moment of inertia of a circle with radius
5240:
3860:
integral the first time around to reflect the fact that there is a hole. This would be done like this.
2567:
1030:
852:
380:
5512:
5075:
450:
5173:
5127:
5043:
1650:
236:
173:
5461:
5250:
4983:
192:
5356:
Hibbeler, R. C. (2004). Statics and
Mechanics of Materials (Second ed.). Pearson Prentice Hall.
4290:
1021:. The parallel axis theorem can be used to obtain the second moment of area with respect to the
239:
deflection, due to an applied moment parallel to its cross-section, as a function of its shape.
2545:
1035:
It is sometimes necessary to calculate the second moment of area of a shape with respect to an
5433:
5379:
5357:
5337:
5312:
5308:
5275:
5235:
4247:
2902:
244:
149:
75:
199:
applied to the beam. In order to maximize the second moment of area, a large fraction of the
5300:
4286:
1084:, and use the parallel axis theorem to derive the second moment of area with respect to the
797:
196:
184:
3618:
3591:
2877:
2850:
2779:
2752:
2705:
2678:
2631:
2600:
2573:
2358:
1862:
1835:
1808:
1696:
1426:
1399:
1372:
827:
745:
630:
224:
4254:
765:
591:
563:
5400:
3840:
1332:
1267:
1238:
1087:
1038:
720:
5023:
4299:
4282:
3571:
2826:
2806:
2732:
2658:
1784:
1764:
1676:
1656:
1312:
1292:
1218:
1198:
1176:
1067:
169:
131:
111:
1009:
794:
For example, when the desired reference axis is the x-axis, the second moment of area
17:
5501:
5427:
5301:
153:
5465:
625:
228:
152:
over the object in question. Its dimension is L (length) to the fourth power. Its
148:(for an axis perpendicular to the plane). In both cases, it is calculated with a
2749:
with respect to the origin. In this case, it is easier to directly calculate
5408:(Technical report). Canadian National Defense. Technical Memorandum 87/209.
216:
81:
For a list of equations for second moments of area of standard shapes, see
2625:
1802:
1061:
1014:
208:
5488:"On the Computation of the Moments of a Polygon, with some Applications"
2628:
also lies at the origin. We can determine the polar moment of inertia,
436:
5124:
are assumed to be equal to the coordinates of the first vertex, i.e.,
204:
1886:
represents the polar moment of inertia with respect to the z-axis.
235:
second moment of area provides insight into a beam's resistance to
616:
axis is not drawn in the adjacent image; is an axis coplanar with
220:
161:
5374:
Beer, Ferdinand P. (2013). "Chapter 9.6: Parallel-axis theorem".
1859:
represents the second moment of area with respect to the y-axis;
1832:
represents the second moment of area with respect to the x-axis;
2843:
component. Instead of obtaining the second moment of area from
165:
105:
187:
is an important property used in the calculation of the beam's
5332:
Beer, Ferdinand P. (2013). "Chapter 9.8: Product of inertia".
26:
128:(for an axis that lies in the plane of the area) or with a
2702:
minus the polar moment of inertia of a circle with radius
508:, in a three-dimensional space occupied by an object
556:
The second moment of area for an arbitrary shape
453:
383:
319:
262:
46:
Other users have requested expert review on talk page.
5455:"On the Calculation of Arbitrary Moments of Polygons"
5378:(10th ed.). New York: McGraw-Hill. p. 481.
5336:(10th ed.). New York: McGraw-Hill. p. 495.
5274:(10th ed.). New York: McGraw-Hill. p. 471.
5176:
5130:
5078:
5046:
5026:
4986:
4324:
4302:
4257:
3868:
3843:
3650:
3621:
3594:
3574:
2913:
2880:
2853:
2829:
2809:
2782:
2755:
2735:
2708:
2681:
2661:
2634:
2603:
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2390:
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945:
863:
830:
800:
768:
748:
723:
653:
633:
594:
566:
215:
second moment of area provides insight into a beam's
207:
is located at the maximum possible distance from the
134:
114:
2847:
as done in the previous section, we shall calculate
4281:The second moment of area about the origin for any
5208:
5162:
5116:
5064:
5032:
5012:
4970:
4308:
4269:
4232:
3852:
3827:
3634:
3607:
3580:
3558:
2893:
2866:
2835:
2815:
2795:
2768:
2741:
2721:
2694:
2667:
2647:
2616:
2589:
2530:
2374:
2341:
1878:
1851:
1824:
1793:
1773:
1705:
1685:
1665:
1639:
1442:
1415:
1388:
1346:
1321:
1301:
1281:
1252:
1227:
1207:
1185:
1162:
1101:
1076:
1052:
994:
913:
843:
816:
782:
754:
732:
706:
639:
608:
580:
516:commonly refers to the second moment of the area.
488:
419:
365:
305:
140:
120:
3837:Alternatively, we could change the limits on the
2570:whose center is at the origin, outside radius is
707:{\displaystyle J_{BB'}=\iint _{R}{\rho }^{2}\,dA}
2352:
2624:. Because of the symmetry of the annulus, the
373:with respect to some reference plane), or the
914:{\displaystyle I_{x}=\iint _{R}y^{2}\,dx\,dy}
435:, in some two-dimensional cross-section. In
8:
4277:, notice point "7" is identical to point 1.
923:The second moment of the area is crucial in
3568:Now, the polar moment of inertia about the
995:{\displaystyle I_{xy}=\iint _{R}yx\,dx\,dy}
1215:is the perpendicular distance between the
42:needs attention from an expert in Geometry
5200:
5181:
5175:
5154:
5135:
5129:
5102:
5083:
5077:
5045:
5025:
5004:
4991:
4985:
4953:
4937:
4918:
4902:
4886:
4876:
4854:
4844:
4824:
4808:
4789:
4779:
4764:
4753:
4739:
4723:
4704:
4693:
4674:
4664:
4651:
4646:
4626:
4610:
4591:
4581:
4566:
4555:
4541:
4528:
4509:
4498:
4479:
4469:
4456:
4451:
4431:
4415:
4396:
4386:
4371:
4360:
4346:
4333:
4325:
4323:
4301:
4256:
4215:
4210:
4197:
4192:
4173:
4163:
4147:
4142:
4136:
4122:
4117:
4111:
4097:
4092:
4071:
4064:
4058:
4046:
4041:
4034:
4029:
4016:
4011:
3992:
3985:
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3959:
3954:
3947:
3942:
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3924:
3910:
3904:
3894:
3877:
3869:
3867:
3842:
3814:
3809:
3796:
3791:
3772:
3763:
3758:
3744:
3735:
3730:
3716:
3705:
3694:
3679:
3668:
3655:
3649:
3626:
3620:
3599:
3593:
3573:
3546:
3532:
3522:
3511:
3505:
3496:
3491:
3470:
3463:
3457:
3447:
3442:
3429:
3424:
3405:
3398:
3384:
3374:
3369:
3356:
3351:
3337:
3331:
3321:
3303:
3296:
3282:
3268:
3258:
3247:
3238:
3228:
3221:
3212:
3207:
3193:
3186:
3181:
3172:
3162:
3152:
3147:
3134:
3129:
3103:
3096:
3082:
3072:
3051:
3046:
3033:
3028:
3014:
3008:
2998:
2977:
2963:
2957:
2947:
2929:
2922:
2914:
2912:
2885:
2879:
2858:
2852:
2828:
2808:
2787:
2781:
2760:
2754:
2734:
2713:
2707:
2686:
2680:
2660:
2639:
2633:
2608:
2602:
2581:
2575:
2517:
2504:
2480:
2465:
2455:
2440:
2430:
2421:
2408:
2395:
2389:
2366:
2360:
2320:
2313:
2303:
2297:
2279:
2268:
2264:
2250:
2243:
2237:
2222:
2211:
2207:
2192:
2181:
2177:
2163:
2157:
2147:
2130:
2110:
2100:
2090:
2079:
2073:
2063:
2052:
2041:
2037:
2023:
2016:
2010:
1995:
1984:
1980:
1965:
1954:
1950:
1936:
1930:
1920:
1903:
1895:
1893:
1870:
1864:
1843:
1837:
1816:
1810:
1786:
1766:
1698:
1678:
1658:
1631:
1618:
1604:
1598:
1588:
1574:
1568:
1558:
1534:
1521:
1506:
1492:
1486:
1476:
1463:
1457:
1434:
1428:
1407:
1401:
1380:
1374:
1334:
1314:
1294:
1269:
1240:
1220:
1200:
1178:
1154:
1138:
1120:
1114:
1089:
1069:
1040:
985:
978:
966:
950:
944:
904:
897:
891:
881:
868:
862:
835:
829:
805:
799:
767:
747:
722:
697:
691:
686:
679:
658:
652:
632:
593:
565:
474:
464:
452:
410:
404:
394:
382:
353:
347:
337:
324:
318:
296:
290:
280:
267:
261:
133:
113:
4289:and can be considered a special case of
4246:
2544:
1745:
1264:A similar statement can be made about a
1008:
523:
5262:
1109:axis. The parallel axis theorem states
740:is the infinitesimal area element, and
447:with respect to distance from an axis:
366:{\textstyle I_{y}=\iint _{R}x^{2}\,dA,}
5402:Calculation of the Moments of Polygons
5307:. John Wiley & Sons, Inc. p.
306:{\textstyle I_{x}=\iint _{R}y^{2}\,dA}
52:may be able to help recruit an expert.
1742:Rectangle with centroid at the origin
1309:axis. Or, in general, any centroidal
71:Mathematical construct in engineering
7:
5415:from the original on March 23, 2020.
5303:Analysis and Design of Elastic Beams
156:of dimension, when working with the
1163:{\displaystyle I_{x'}=I_{x}+Ad^{2}}
242:Different disciplines use the term
211:of the I-beam's cross-section. The
560:with respect to an arbitrary axis
420:{\textstyle I=\iint _{R}r^{2}\,dA}
231:, as a function of its shape. The
104:, is a geometrical property of an
25:
1289:axis and the parallel centroidal
624:axes and is perpendicular to the
443:is strictly the second moment of
183:, the second moment of area of a
164:, or inches to the fourth power,
160:, is meters to the fourth power,
1649:This relationship relies on the
532:is the distance to the element d
252:. It may refer to either of the
31:
5117:{\displaystyle x_{n+1},y_{n+1}}
1761:Consider a rectangle with base
489:{\textstyle I=\int _{Q}r^{2}dm}
5376:Vector Mechanics for Engineers
5334:Vector Mechanics for Engineers
5272:Vector Mechanics for Engineers
5226:List of second moments of area
1736:list of second moments of area
256:second moments of area (often
83:List of second moments of area
1:
5209:{\displaystyle y_{n+1}=y_{1}}
5163:{\displaystyle x_{n+1}=x_{1}}
5065:{\displaystyle 1\leq i\leq n}
4296:A polygon is assumed to have
1193:is the area of the shape, and
158:International System of Units
5270:Beer, Ferdinand P. (2013).
5020:are the coordinates of the
5013:{\displaystyle x_{i},y_{i}}
44:. The specific problem is:
5544:
5299:Pilkey, Walter D. (2002).
5246:Perpendicular axis theorem
5231:List of moments of inertia
2549:Annulus with inner radius
2541:Annulus centered at origin
2353:perpendicular axis theorem
1805:is located at the origin.
1364:Perpendicular axis theorem
1361:
1358:Perpendicular axis theorem
1028:
219:due to an applied moment,
80:
73:
5426:Obregon, Joaquin (2012).
3615:and a circle with radius
762:is the distance from the
5453:Steger, Carsten (1996).
5040:-th polygon vertex, for
4251:A simple polygon. Here,
1715:linearity of integration
170:Imperial System of Units
98:quadratic moment of area
2597:, and inside radius is
377:second moment of area (
191:and the calculation of
5210:
5164:
5118:
5066:
5034:
5014:
4972:
4769:
4571:
4376:
4310:
4278:
4271:
4234:
3854:
3829:
3636:
3609:
3582:
3560:
2895:
2868:
2837:
2817:
2797:
2770:
2743:
2723:
2696:
2669:
2649:
2618:
2591:
2563:
2532:
2376:
2343:
1880:
1853:
1826:
1795:
1775:
1758:
1707:
1687:
1667:
1641:
1444:
1417:
1390:
1348:
1323:
1303:
1283:
1254:
1229:
1209:
1187:
1164:
1103:
1078:
1060:axis different to the
1054:
1026:
996:
937:product moment of area
931:Product moment of area
925:Euler–Bernoulli theory
915:
845:
818:
817:{\displaystyle I_{xx}}
784:
756:
734:
708:
641:
610:
582:
553:
490:
421:
367:
307:
181:structural engineering
168:, when working in the
142:
122:
102:area moment of inertia
100:and also known as the
18:Area moment of inertia
5523:Mechanical quantities
5399:Hally, David (1987).
5241:Parallel axis theorem
5211:
5165:
5119:
5067:
5035:
5015:
4973:
4749:
4551:
4356:
4311:
4272:
4250:
4235:
3855:
3830:
3637:
3635:{\displaystyle r_{1}}
3610:
3608:{\displaystyle r_{2}}
3583:
3561:
2896:
2894:{\displaystyle J_{z}}
2869:
2867:{\displaystyle I_{x}}
2845:Cartesian coordinates
2838:
2818:
2798:
2796:{\displaystyle r^{2}}
2771:
2769:{\displaystyle J_{z}}
2744:
2724:
2722:{\displaystyle r_{1}}
2697:
2695:{\displaystyle r_{2}}
2670:
2650:
2648:{\displaystyle J_{z}}
2619:
2617:{\displaystyle r_{1}}
2592:
2590:{\displaystyle r_{2}}
2548:
2533:
2377:
2375:{\displaystyle J_{z}}
2344:
1881:
1879:{\displaystyle J_{z}}
1854:
1852:{\displaystyle I_{y}}
1827:
1825:{\displaystyle I_{x}}
1796:
1776:
1749:
1708:
1706:{\displaystyle \rho }
1688:
1668:
1642:
1445:
1443:{\displaystyle I_{y}}
1418:
1416:{\displaystyle I_{x}}
1391:
1389:{\displaystyle J_{z}}
1349:
1324:
1304:
1284:
1255:
1230:
1210:
1188:
1165:
1104:
1079:
1055:
1031:Parallel axis theorem
1012:
1005:Parallel axis theorem
997:
916:
853:Cartesian coordinates
851:) can be computed in
846:
844:{\displaystyle I_{x}}
819:
785:
757:
755:{\displaystyle \rho }
735:
709:
642:
640:{\displaystyle \rho }
611:
583:
528:An arbitrary shape.
527:
491:
422:
368:
308:
227:perpendicular to its
217:resistance to bending
143:
123:
90:second moment of area
5174:
5128:
5076:
5044:
5024:
4984:
4322:
4300:
4255:
3866:
3841:
3648:
3619:
3592:
3572:
2911:
2878:
2851:
2827:
2807:
2803:, which has both an
2780:
2753:
2733:
2706:
2679:
2659:
2632:
2601:
2574:
2388:
2359:
2355:we get the value of
1892:
1863:
1836:
1809:
1785:
1765:
1750:Rectangle with base
1697:
1677:
1657:
1456:
1427:
1400:
1373:
1333:
1329:axis and a parallel
1313:
1293:
1268:
1239:
1219:
1199:
1177:
1113:
1088:
1068:
1039:
943:
935:More generally, the
861:
828:
798:
766:
746:
721:
651:
631:
592:
564:
451:
381:
317:
260:
201:cross-sectional area
132:
112:
50:WikiProject Geometry
5518:Structural analysis
5429:Mechanical Simmetry
4709:
4656:
4514:
4461:
4270:{\displaystyle n=6}
4220:
4202:
4152:
4127:
4105:
4053:
4024:
3966:
3937:
3819:
3801:
3768:
3740:
3504:
3452:
3437:
3379:
3364:
3220:
3157:
3142:
3056:
3041:
2776:as we already have
2289:
2232:
2202:
2062:
2005:
1975:
1651:Pythagorean theorem
783:{\displaystyle BB'}
609:{\displaystyle BB'}
581:{\displaystyle BB'}
536:, with projections
174:US customary system
5251:Radius of gyration
5206:
5160:
5114:
5062:
5030:
5010:
4968:
4966:
4689:
4642:
4494:
4447:
4306:
4279:
4267:
4230:
4228:
4206:
4188:
4138:
4113:
4088:
4025:
4007:
3938:
3920:
3853:{\displaystyle dr}
3850:
3825:
3805:
3787:
3754:
3726:
3632:
3605:
3578:
3556:
3554:
3487:
3438:
3420:
3365:
3347:
3203:
3143:
3125:
3042:
3024:
2891:
2864:
2833:
2813:
2793:
2766:
2739:
2719:
2692:
2665:
2645:
2614:
2587:
2564:
2528:
2372:
2339:
2337:
2260:
2203:
2173:
2033:
1976:
1946:
1876:
1849:
1822:
1791:
1771:
1759:
1738:for other shapes.
1703:
1683:
1663:
1637:
1440:
1413:
1386:
1347:{\displaystyle B'}
1344:
1319:
1299:
1282:{\displaystyle y'}
1279:
1253:{\displaystyle x'}
1250:
1225:
1205:
1183:
1160:
1102:{\displaystyle x'}
1099:
1074:
1053:{\displaystyle x'}
1050:
1027:
992:
927:of slender beams.
911:
841:
824:(often denoted as
814:
780:
752:
733:{\displaystyle dA}
730:
704:
637:
606:
578:
554:
486:
417:
363:
303:
248:(MOI) to refer to
138:
118:
94:second area moment
5486:Soerjadi, Ir. R.
5439:978-1-4772-3372-6
5385:978-0-07-339813-6
5343:978-0-07-339813-6
5318:978-0-471-38152-5
5281:978-0-07-339813-6
5236:Moment of inertia
5033:{\displaystyle i}
4747:
4549:
4354:
4309:{\displaystyle n}
4181:
4156:
4131:
3780:
3752:
3724:
3581:{\displaystyle z}
3540:
3520:
3306:
3276:
3256:
2932:
2903:polar coordinates
2836:{\displaystyle y}
2816:{\displaystyle x}
2742:{\displaystyle r}
2668:{\displaystyle z}
2556:and outer radius
2493:
2475:
2450:
2333:
2287:
2276:
2230:
2219:
2200:
2189:
2120:
2088:
2071:
2060:
2049:
2003:
1992:
1973:
1962:
1794:{\displaystyle h}
1774:{\displaystyle b}
1686:{\displaystyle y}
1666:{\displaystyle x}
1322:{\displaystyle B}
1302:{\displaystyle y}
1228:{\displaystyle x}
1208:{\displaystyle d}
1186:{\displaystyle A}
1077:{\displaystyle x}
514:moment of inertia
441:moment of inertia
250:different moments
245:moment of inertia
223:, or distributed
150:multiple integral
141:{\displaystyle J}
121:{\displaystyle I}
76:Moment of inertia
67:
66:
16:(Redirected from
5535:
5528:Moment (physics)
5508:Applied geometry
5492:
5491:
5483:
5477:
5476:
5474:
5468:. Archived from
5459:
5450:
5444:
5443:
5423:
5417:
5416:
5414:
5407:
5396:
5390:
5389:
5371:
5365:
5354:
5348:
5347:
5329:
5323:
5322:
5306:
5296:
5290:
5289:
5267:
5215:
5213:
5212:
5207:
5205:
5204:
5192:
5191:
5169:
5167:
5166:
5161:
5159:
5158:
5146:
5145:
5123:
5121:
5120:
5115:
5113:
5112:
5094:
5093:
5071:
5069:
5068:
5063:
5039:
5037:
5036:
5031:
5019:
5017:
5016:
5011:
5009:
5008:
4996:
4995:
4977:
4975:
4974:
4969:
4967:
4963:
4959:
4958:
4957:
4948:
4947:
4929:
4928:
4913:
4912:
4891:
4890:
4881:
4880:
4865:
4864:
4849:
4848:
4834:
4830:
4829:
4828:
4819:
4818:
4800:
4799:
4784:
4783:
4768:
4763:
4748:
4740:
4731:
4730:
4714:
4710:
4708:
4703:
4685:
4684:
4669:
4668:
4655:
4650:
4636:
4632:
4631:
4630:
4621:
4620:
4602:
4601:
4586:
4585:
4570:
4565:
4550:
4542:
4533:
4532:
4519:
4515:
4513:
4508:
4490:
4489:
4474:
4473:
4460:
4455:
4441:
4437:
4436:
4435:
4426:
4425:
4407:
4406:
4391:
4390:
4375:
4370:
4355:
4347:
4338:
4337:
4315:
4313:
4312:
4307:
4287:shoelace formula
4276:
4274:
4273:
4268:
4239:
4237:
4236:
4231:
4229:
4225:
4221:
4219:
4214:
4201:
4196:
4182:
4174:
4162:
4158:
4157:
4151:
4146:
4137:
4132:
4126:
4121:
4112:
4104:
4096:
4081:
4063:
4062:
4052:
4051:
4050:
4040:
4039:
4038:
4023:
4015:
4003:
3999:
3976:
3975:
3965:
3964:
3963:
3953:
3952:
3951:
3936:
3928:
3909:
3908:
3899:
3898:
3882:
3881:
3859:
3857:
3856:
3851:
3834:
3832:
3831:
3826:
3824:
3820:
3818:
3813:
3800:
3795:
3781:
3773:
3767:
3762:
3753:
3745:
3739:
3734:
3725:
3717:
3712:
3711:
3710:
3709:
3686:
3685:
3684:
3683:
3660:
3659:
3641:
3639:
3638:
3633:
3631:
3630:
3614:
3612:
3611:
3606:
3604:
3603:
3587:
3585:
3584:
3579:
3565:
3563:
3562:
3557:
3555:
3551:
3550:
3541:
3533:
3521:
3516:
3515:
3506:
3503:
3495:
3480:
3462:
3461:
3451:
3446:
3436:
3428:
3416:
3412:
3389:
3388:
3378:
3373:
3363:
3355:
3336:
3335:
3326:
3325:
3309:
3308:
3307:
3304:
3287:
3286:
3277:
3269:
3257:
3252:
3251:
3243:
3242:
3233:
3232:
3222:
3219:
3211:
3185:
3177:
3176:
3167:
3166:
3156:
3151:
3141:
3133:
3118:
3114:
3110:
3087:
3086:
3081:
3077:
3076:
3055:
3050:
3040:
3032:
3013:
3012:
3007:
3003:
3002:
2982:
2981:
2962:
2961:
2952:
2951:
2935:
2934:
2933:
2930:
2900:
2898:
2897:
2892:
2890:
2889:
2873:
2871:
2870:
2865:
2863:
2862:
2842:
2840:
2839:
2834:
2822:
2820:
2819:
2814:
2802:
2800:
2799:
2794:
2792:
2791:
2775:
2773:
2772:
2767:
2765:
2764:
2748:
2746:
2745:
2740:
2728:
2726:
2725:
2720:
2718:
2717:
2701:
2699:
2698:
2693:
2691:
2690:
2674:
2672:
2671:
2666:
2654:
2652:
2651:
2646:
2644:
2643:
2623:
2621:
2620:
2615:
2613:
2612:
2596:
2594:
2593:
2588:
2586:
2585:
2537:
2535:
2534:
2529:
2527:
2523:
2522:
2521:
2509:
2508:
2494:
2489:
2481:
2476:
2471:
2470:
2469:
2456:
2451:
2446:
2445:
2444:
2431:
2426:
2425:
2413:
2412:
2400:
2399:
2381:
2379:
2378:
2373:
2371:
2370:
2348:
2346:
2345:
2340:
2338:
2334:
2329:
2325:
2324:
2314:
2302:
2301:
2288:
2280:
2278:
2277:
2269:
2242:
2241:
2231:
2223:
2221:
2220:
2212:
2201:
2193:
2191:
2190:
2182:
2162:
2161:
2152:
2151:
2135:
2134:
2121:
2116:
2115:
2114:
2101:
2089:
2084:
2083:
2074:
2072:
2064:
2061:
2053:
2051:
2050:
2042:
2015:
2014:
2004:
1996:
1994:
1993:
1985:
1974:
1966:
1964:
1963:
1955:
1935:
1934:
1925:
1924:
1908:
1907:
1885:
1883:
1882:
1877:
1875:
1874:
1858:
1856:
1855:
1850:
1848:
1847:
1831:
1829:
1828:
1823:
1821:
1820:
1800:
1798:
1797:
1792:
1780:
1778:
1777:
1772:
1721:Composite shapes
1712:
1710:
1709:
1704:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1646:
1644:
1643:
1638:
1636:
1635:
1623:
1622:
1603:
1602:
1593:
1592:
1573:
1572:
1563:
1562:
1544:
1540:
1539:
1538:
1526:
1525:
1511:
1510:
1491:
1490:
1481:
1480:
1468:
1467:
1449:
1447:
1446:
1441:
1439:
1438:
1422:
1420:
1419:
1414:
1412:
1411:
1395:
1393:
1392:
1387:
1385:
1384:
1353:
1351:
1350:
1345:
1343:
1328:
1326:
1325:
1320:
1308:
1306:
1305:
1300:
1288:
1286:
1285:
1280:
1278:
1259:
1257:
1256:
1251:
1249:
1234:
1232:
1231:
1226:
1214:
1212:
1211:
1206:
1192:
1190:
1189:
1184:
1169:
1167:
1166:
1161:
1159:
1158:
1143:
1142:
1130:
1129:
1128:
1108:
1106:
1105:
1100:
1098:
1083:
1081:
1080:
1075:
1059:
1057:
1056:
1051:
1049:
1001:
999:
998:
993:
971:
970:
958:
957:
920:
918:
917:
912:
896:
895:
886:
885:
873:
872:
850:
848:
847:
842:
840:
839:
823:
821:
820:
815:
813:
812:
789:
787:
786:
781:
779:
761:
759:
758:
753:
739:
737:
736:
731:
713:
711:
710:
705:
696:
695:
690:
684:
683:
671:
670:
669:
647:) is defined as
646:
644:
643:
638:
615:
613:
612:
607:
605:
587:
585:
584:
579:
577:
559:
511:
495:
493:
492:
487:
479:
478:
469:
468:
426:
424:
423:
418:
409:
408:
399:
398:
372:
370:
369:
364:
352:
351:
342:
341:
329:
328:
312:
310:
309:
304:
295:
294:
285:
284:
272:
271:
147:
145:
144:
139:
127:
125:
124:
119:
62:
59:
53:
35:
34:
27:
21:
5543:
5542:
5538:
5537:
5536:
5534:
5533:
5532:
5498:
5497:
5496:
5495:
5485:
5484:
5480:
5472:
5457:
5452:
5451:
5447:
5440:
5432:. AuthorHouse.
5425:
5424:
5420:
5412:
5405:
5398:
5397:
5393:
5386:
5373:
5372:
5368:
5355:
5351:
5344:
5331:
5330:
5326:
5319:
5298:
5297:
5293:
5282:
5269:
5268:
5264:
5259:
5222:
5196:
5177:
5172:
5171:
5150:
5131:
5126:
5125:
5098:
5079:
5074:
5073:
5042:
5041:
5022:
5021:
5000:
4987:
4982:
4981:
4965:
4964:
4949:
4933:
4914:
4898:
4882:
4872:
4850:
4840:
4839:
4835:
4820:
4804:
4785:
4775:
4774:
4770:
4732:
4719:
4716:
4715:
4670:
4660:
4641:
4637:
4622:
4606:
4587:
4577:
4576:
4572:
4534:
4524:
4521:
4520:
4475:
4465:
4446:
4442:
4427:
4411:
4392:
4382:
4381:
4377:
4339:
4329:
4320:
4319:
4298:
4297:
4291:Green's theorem
4253:
4252:
4245:
4227:
4226:
4187:
4183:
4110:
4106:
4079:
4078:
4054:
4042:
4030:
3981:
3977:
3967:
3955:
3943:
3900:
3890:
3883:
3873:
3864:
3863:
3839:
3838:
3786:
3782:
3701:
3690:
3675:
3664:
3651:
3646:
3645:
3622:
3617:
3616:
3595:
3590:
3589:
3570:
3569:
3553:
3552:
3542:
3507:
3478:
3477:
3453:
3394:
3390:
3380:
3327:
3317:
3310:
3292:
3289:
3288:
3278:
3234:
3224:
3223:
3168:
3158:
3116:
3115:
3092:
3088:
3062:
3058:
3057:
2988:
2984:
2983:
2973:
2953:
2943:
2936:
2918:
2909:
2908:
2901:directly using
2881:
2876:
2875:
2854:
2849:
2848:
2825:
2824:
2805:
2804:
2783:
2778:
2777:
2756:
2751:
2750:
2731:
2730:
2709:
2704:
2703:
2682:
2677:
2676:
2657:
2656:
2635:
2630:
2629:
2604:
2599:
2598:
2577:
2572:
2571:
2562:
2555:
2543:
2513:
2500:
2499:
2495:
2482:
2461:
2457:
2436:
2432:
2417:
2404:
2391:
2386:
2385:
2362:
2357:
2356:
2336:
2335:
2316:
2315:
2293:
2233:
2153:
2143:
2136:
2126:
2123:
2122:
2106:
2102:
2075:
2006:
1926:
1916:
1909:
1899:
1890:
1889:
1866:
1861:
1860:
1839:
1834:
1833:
1812:
1807:
1806:
1783:
1782:
1763:
1762:
1744:
1732:
1723:
1695:
1694:
1675:
1674:
1655:
1654:
1627:
1614:
1594:
1584:
1564:
1554:
1530:
1517:
1516:
1512:
1502:
1482:
1472:
1459:
1454:
1453:
1430:
1425:
1424:
1403:
1398:
1397:
1376:
1371:
1370:
1366:
1360:
1336:
1331:
1330:
1311:
1310:
1291:
1290:
1271:
1266:
1265:
1242:
1237:
1236:
1217:
1216:
1197:
1196:
1175:
1174:
1150:
1134:
1121:
1116:
1111:
1110:
1091:
1086:
1085:
1066:
1065:
1042:
1037:
1036:
1033:
1007:
962:
946:
941:
940:
933:
887:
877:
864:
859:
858:
831:
826:
825:
801:
796:
795:
772:
764:
763:
744:
743:
719:
718:
685:
675:
662:
654:
649:
648:
629:
628:
598:
590:
589:
570:
562:
561:
557:
522:
509:
470:
460:
449:
448:
400:
390:
379:
378:
343:
333:
320:
315:
314:
286:
276:
263:
258:
257:
130:
129:
110:
109:
86:
79:
72:
63:
57:
54:
48:
36:
32:
23:
22:
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5494:
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5475:on 2018-10-03.
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4332:
4328:
4327:
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4283:simple polygon
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2267:
2263:
2259:
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2240:
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2229:
2226:
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2210:
2206:
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2196:
2188:
2185:
2180:
2176:
2172:
2169:
2166:
2160:
2156:
2150:
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2139:
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2070:
2067:
2059:
2056:
2048:
2045:
2040:
2036:
2032:
2029:
2026:
2022:
2019:
2013:
2009:
2002:
1999:
1991:
1988:
1983:
1979:
1972:
1969:
1961:
1958:
1953:
1949:
1945:
1942:
1939:
1933:
1929:
1923:
1919:
1915:
1912:
1910:
1906:
1902:
1898:
1897:
1873:
1869:
1846:
1842:
1819:
1815:
1790:
1770:
1743:
1740:
1731:
1728:
1722:
1719:
1702:
1682:
1662:
1653:which relates
1634:
1630:
1626:
1621:
1617:
1613:
1610:
1607:
1601:
1597:
1591:
1587:
1583:
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1577:
1571:
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1561:
1557:
1553:
1550:
1547:
1543:
1537:
1533:
1529:
1524:
1520:
1515:
1509:
1505:
1501:
1498:
1495:
1489:
1485:
1479:
1475:
1471:
1466:
1462:
1437:
1433:
1410:
1406:
1383:
1379:
1362:Main article:
1359:
1356:
1342:
1339:
1318:
1298:
1277:
1274:
1262:
1261:
1248:
1245:
1224:
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1182:
1157:
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1149:
1146:
1141:
1137:
1133:
1127:
1124:
1119:
1097:
1094:
1073:
1048:
1045:
1029:Main article:
1006:
1003:
991:
988:
984:
981:
977:
974:
969:
965:
961:
956:
953:
949:
939:is defined as
932:
929:
910:
907:
903:
900:
894:
890:
884:
880:
876:
871:
867:
838:
834:
811:
808:
804:
792:
791:
778:
775:
771:
751:
741:
729:
726:
703:
700:
694:
689:
682:
678:
674:
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665:
661:
657:
636:
604:
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569:
521:
518:
485:
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477:
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403:
397:
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389:
386:
362:
359:
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350:
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336:
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327:
323:
302:
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293:
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283:
279:
275:
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266:
137:
117:
70:
65:
64:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5540:
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5526:
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5519:
5516:
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5506:
5505:
5503:
5489:
5482:
5479:
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5467:
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5456:
5449:
5446:
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5435:
5431:
5430:
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5419:
5411:
5404:
5403:
5395:
5392:
5387:
5381:
5377:
5370:
5367:
5363:
5362:0-13-028127-1
5359:
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5350:
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5339:
5335:
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5325:
5320:
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5304:
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5288:
5283:
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3130:
3126:
3122:
3120:
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3104:
3100:
3097:
3093:
3089:
3083:
3078:
3073:
3069:
3066:
3063:
3059:
3052:
3047:
3043:
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2978:
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2076:
2068:
2065:
2057:
2054:
2046:
2043:
2038:
2034:
2030:
2027:
2024:
2020:
2017:
2011:
2007:
2000:
1997:
1989:
1986:
1981:
1977:
1970:
1967:
1959:
1956:
1951:
1947:
1943:
1940:
1937:
1931:
1927:
1921:
1917:
1913:
1911:
1904:
1900:
1887:
1871:
1867:
1844:
1840:
1817:
1813:
1804:
1788:
1768:
1757:
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1589:
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1435:
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1404:
1381:
1377:
1365:
1357:
1355:
1340:
1337:
1316:
1296:
1275:
1272:
1246:
1243:
1222:
1202:
1195:
1180:
1173:
1172:
1171:
1155:
1151:
1147:
1144:
1139:
1135:
1131:
1125:
1122:
1117:
1095:
1092:
1071:
1063:
1046:
1043:
1032:
1024:
1020:
1016:
1013:A shape with
1011:
1004:
1002:
989:
986:
982:
979:
975:
972:
967:
963:
959:
954:
951:
947:
938:
930:
928:
926:
921:
908:
905:
901:
898:
892:
888:
882:
878:
874:
869:
865:
856:
854:
836:
832:
809:
806:
802:
776:
773:
769:
749:
742:
727:
724:
717:
716:
715:
701:
698:
692:
687:
680:
676:
672:
666:
663:
659:
655:
634:
627:
623:
619:
602:
599:
595:
574:
571:
567:
551:
547:
543:
539:
535:
531:
526:
519:
517:
515:
507:
503:
499:
483:
480:
475:
471:
465:
461:
457:
454:
446:
442:
438:
434:
430:
414:
411:
405:
401:
395:
391:
387:
384:
376:
360:
357:
354:
348:
344:
338:
334:
330:
325:
321:
300:
297:
291:
287:
281:
277:
273:
268:
264:
255:
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234:
230:
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218:
214:
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206:
202:
198:
194:
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177:
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171:
167:
163:
159:
155:
151:
135:
115:
107:
103:
99:
95:
91:
84:
77:
69:
61:
51:
47:
43:
40:This article
38:
29:
28:
19:
5481:
5470:the original
5448:
5428:
5421:
5401:
5394:
5375:
5369:
5352:
5333:
5327:
5302:
5294:
5285:
5271:
5265:
4979:
4318:
4295:
4280:
3862:
3836:
3644:
3567:
2907:
2655:, about the
2566:Consider an
2565:
2557:
2550:
2384:
2350:
1888:
1760:
1755:
1751:
1733:
1724:
1648:
1452:
1367:
1263:
1034:
1022:
1018:
936:
934:
922:
857:
793:
626:line segment
621:
617:
555:
549:
545:
541:
537:
533:
529:
513:
505:
501:
497:
444:
440:
432:
428:
374:
253:
243:
241:
232:
229:neutral axis
212:
195:caused by a
178:
101:
97:
93:
89:
87:
68:
55:
45:
41:
5513:Beam theory
4243:Any polygon
1781:and height
1754:and height
1713:and on the
58:August 2023
5502:Categories
5257:References
2351:Using the
1062:centroidal
1015:centroidal
520:Definition
189:deflection
5057:≤
5051:≤
4802:−
4751:∑
4604:−
4553:∑
4409:−
4358:∑
4204:−
4176:π
4168:θ
4134:−
4102:π
4090:∫
4076:θ
4027:∫
4021:π
4009:∫
3997:θ
3940:∫
3934:π
3922:∫
3892:∬
3803:−
3775:π
3747:π
3742:−
3719:π
3688:−
3535:π
3527:θ
3501:π
3489:∫
3475:θ
3440:∫
3434:π
3422:∫
3410:θ
3367:∫
3361:π
3349:∫
3319:∬
3271:π
3263:θ
3249:θ
3245:
3217:π
3205:∫
3198:θ
3183:θ
3179:
3145:∫
3139:π
3127:∫
3108:θ
3074:θ
3070:
3044:∫
3038:π
3026:∫
3000:θ
2996:
2975:∬
2945:∬
2266:−
2262:∫
2209:−
2205:∫
2179:−
2175:∫
2145:∬
2039:−
2035:∫
1982:−
1978:∫
1952:−
1948:∫
1918:∬
1701:ρ
1586:∬
1556:∬
1504:∬
1484:ρ
1474:∬
964:∬
879:∬
750:ρ
688:ρ
677:∬
635:ρ
462:∫
392:∬
335:∬
278:∬
237:torsional
5466:17506973
5410:Archived
5220:See also
5072:. Also,
2626:centroid
1803:centroid
1730:Examples
1341:′
1276:′
1247:′
1126:′
1096:′
1047:′
777:′
667:′
603:′
575:′
496:, where
209:centroid
5287:masses.
2568:annulus
544:on the
437:physics
172:or the
5464:
5436:
5382:
5360:
5340:
5315:
5278:
4980:where
3305:circle
2931:circle
1801:whose
1354:axis.
1170:where
714:where
254:planar
213:planar
205:I-beam
203:of an
197:moment
193:stress
5473:(PDF)
5462:S2CID
5458:(PDF)
5413:(PDF)
5406:(PDF)
1260:axes.
1025:axis.
1017:axis
790:axis.
552:axes.
375:polar
233:polar
221:force
96:, or
92:, or
5434:ISBN
5380:ISBN
5358:ISBN
5338:ISBN
5313:ISBN
5276:ISBN
5170:and
2874:and
2823:and
1734:See
1673:and
1423:and
1235:and
620:and
548:and
540:and
502:mass
445:mass
429:area
225:load
185:beam
154:unit
106:area
88:The
3236:sin
3170:sin
3067:sin
2993:sin
1693:to
1396:to
855:as
313:or
179:In
5504::
5460:.
5311:.
5309:15
5284:.
5216:.
4745:24
4547:12
4352:12
4293:.
3642:.
2905:.
2491:12
2473:12
2448:12
2382:.
2331:12
2118:12
1717:.
1450:.
1023:x'
506:dm
504:,
439:,
433:dA
431:,
176:.
166:in
5490:.
5442:.
5388:.
5364:.
5346:.
5321:.
5202:1
5198:y
5194:=
5189:1
5186:+
5183:n
5179:y
5156:1
5152:x
5148:=
5143:1
5140:+
5137:n
5133:x
5110:1
5107:+
5104:n
5100:y
5096:,
5091:1
5088:+
5085:n
5081:x
5060:n
5054:i
5048:1
5028:i
5006:i
5002:y
4998:,
4993:i
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4961:)
4955:i
4951:y
4945:1
4942:+
4939:i
4935:x
4931:+
4926:1
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4920:i
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4907:+
4904:i
4900:x
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4893:+
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4884:y
4878:i
4874:x
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4867:+
4862:1
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4856:i
4852:y
4846:i
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4837:(
4832:)
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4822:y
4816:1
4813:+
4810:i
4806:x
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4794:+
4791:i
4787:y
4781:i
4777:x
4772:(
4766:n
4761:1
4758:=
4755:i
4742:1
4737:=
4728:y
4725:x
4721:I
4712:)
4706:2
4701:1
4698:+
4695:i
4691:y
4687:+
4682:1
4679:+
4676:i
4672:y
4666:i
4662:y
4658:+
4653:2
4648:i
4644:y
4639:(
4634:)
4628:i
4624:y
4618:1
4615:+
4612:i
4608:x
4599:1
4596:+
4593:i
4589:y
4583:i
4579:x
4574:(
4568:n
4563:1
4560:=
4557:i
4544:1
4539:=
4530:x
4526:I
4517:)
4511:2
4506:1
4503:+
4500:i
4496:x
4492:+
4487:1
4484:+
4481:i
4477:x
4471:i
4467:x
4463:+
4458:2
4453:i
4449:x
4444:(
4439:)
4433:i
4429:y
4423:1
4420:+
4417:i
4413:x
4404:1
4401:+
4398:i
4394:y
4388:i
4384:x
4379:(
4373:n
4368:1
4365:=
4362:i
4349:1
4344:=
4335:y
4331:I
4304:n
4265:6
4262:=
4259:n
4223:)
4217:4
4212:1
4208:r
4199:4
4194:2
4190:r
4185:(
4179:2
4171:=
4165:d
4160:]
4154:4
4149:4
4144:1
4140:r
4129:4
4124:4
4119:2
4115:r
4108:[
4099:2
4094:0
4086:=
4073:d
4069:r
4066:d
4060:3
4056:r
4048:2
4044:r
4036:1
4032:r
4018:2
4013:0
4005:=
4001:)
3994:d
3990:r
3987:d
3983:r
3979:(
3973:2
3969:r
3961:2
3957:r
3949:1
3945:r
3931:2
3926:0
3918:=
3915:A
3912:d
3906:2
3902:r
3896:R
3888:=
3879:z
3875:J
3848:r
3845:d
3822:)
3816:4
3811:1
3807:r
3798:4
3793:2
3789:r
3784:(
3778:2
3770:=
3765:4
3760:1
3756:r
3750:2
3737:4
3732:2
3728:r
3722:2
3714:=
3707:1
3703:r
3699:,
3696:z
3692:J
3681:2
3677:r
3673:,
3670:z
3666:J
3662:=
3657:z
3653:J
3628:1
3624:r
3601:2
3597:r
3576:z
3548:4
3544:r
3538:2
3530:=
3524:d
3518:4
3513:4
3509:r
3498:2
3493:0
3485:=
3472:d
3468:r
3465:d
3459:3
3455:r
3449:r
3444:0
3431:2
3426:0
3418:=
3414:)
3407:d
3403:r
3400:d
3396:r
3392:(
3386:2
3382:r
3376:r
3371:0
3358:2
3353:0
3345:=
3342:A
3339:d
3333:2
3329:r
3323:R
3315:=
3301:,
3298:z
3294:J
3284:4
3280:r
3274:4
3266:=
3260:d
3254:4
3240:2
3230:4
3226:r
3214:2
3209:0
3201:=
3195:d
3191:r
3188:d
3174:2
3164:3
3160:r
3154:r
3149:0
3136:2
3131:0
3123:=
3112:)
3105:d
3101:r
3098:d
3094:r
3090:(
3084:2
3079:)
3064:r
3060:(
3053:r
3048:0
3035:2
3030:0
3022:=
3019:A
3016:d
3010:2
3005:)
2990:r
2986:(
2979:R
2971:=
2968:A
2965:d
2959:2
2955:y
2949:R
2941:=
2927:,
2924:x
2920:I
2887:z
2883:J
2860:x
2856:I
2831:y
2811:x
2789:2
2785:r
2762:z
2758:J
2737:r
2715:1
2711:r
2688:2
2684:r
2663:z
2641:z
2637:J
2610:1
2606:r
2583:2
2579:r
2561:2
2558:r
2554:1
2551:r
2525:)
2519:2
2515:h
2511:+
2506:2
2502:b
2497:(
2487:h
2484:b
2478:=
2467:3
2463:b
2459:h
2453:+
2442:3
2438:h
2434:b
2428:=
2423:y
2419:I
2415:+
2410:x
2406:I
2402:=
2397:z
2393:J
2368:z
2364:J
2327:h
2322:3
2318:b
2311:=
2308:x
2305:d
2299:2
2295:x
2291:h
2285:2
2282:b
2274:2
2271:b
2258:=
2255:x
2252:d
2248:y
2245:d
2239:2
2235:x
2228:2
2225:h
2217:2
2214:h
2198:2
2195:b
2187:2
2184:b
2171:=
2168:A
2165:d
2159:2
2155:x
2149:R
2141:=
2132:y
2128:I
2112:3
2108:h
2104:b
2098:=
2095:x
2092:d
2086:4
2081:3
2077:h
2069:3
2066:1
2058:2
2055:b
2047:2
2044:b
2031:=
2028:x
2025:d
2021:y
2018:d
2012:2
2008:y
2001:2
1998:h
1990:2
1987:h
1971:2
1968:b
1960:2
1957:b
1944:=
1941:A
1938:d
1932:2
1928:y
1922:R
1914:=
1905:x
1901:I
1872:z
1868:J
1845:y
1841:I
1818:x
1814:I
1789:h
1769:b
1756:h
1752:b
1681:y
1661:x
1633:y
1629:I
1625:+
1620:x
1616:I
1612:=
1609:A
1606:d
1600:2
1596:y
1590:R
1582:+
1579:A
1576:d
1570:2
1566:x
1560:R
1552:=
1549:A
1546:d
1542:)
1536:2
1532:y
1528:+
1523:2
1519:x
1514:(
1508:R
1500:=
1497:A
1494:d
1488:2
1478:R
1470:=
1465:z
1461:J
1436:y
1432:I
1409:x
1405:I
1382:z
1378:J
1338:B
1317:B
1297:y
1273:y
1244:x
1223:x
1203:d
1181:A
1156:2
1152:d
1148:A
1145:+
1140:x
1136:I
1132:=
1123:x
1118:I
1093:x
1072:x
1044:x
1019:x
990:y
987:d
983:x
980:d
976:x
973:y
968:R
960:=
955:y
952:x
948:I
909:y
906:d
902:x
899:d
893:2
889:y
883:R
875:=
870:x
866:I
837:x
833:I
810:x
807:x
803:I
774:B
770:B
728:A
725:d
702:A
699:d
693:2
681:R
673:=
664:B
660:B
656:J
622:y
618:x
600:B
596:B
588:(
572:B
568:B
558:R
550:y
546:x
542:y
538:x
534:A
530:ρ
510:Q
498:r
484:m
481:d
476:2
472:r
466:Q
458:=
455:I
415:A
412:d
406:2
402:r
396:R
388:=
385:I
361:,
358:A
355:d
349:2
345:x
339:R
331:=
326:y
322:I
301:A
298:d
292:2
288:y
282:R
274:=
269:x
265:I
162:m
136:J
116:I
85:.
78:.
60:)
56:(
20:)
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