35:) of a laminar body about an axis (z) perpendicular to its plane is the sum of its moments of inertia about two mutually perpendicular axes (x and y) in its plane, all the three axes being concurrent."
560:
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254:
732:
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295:
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139:
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416:
159:
116:
96:
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56:
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852:
823:
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axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly.
901:
330:
206:
906:
896:
700:
891:
871:
395:
260:
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273:
8:
568:
762:-axis rotation, deviation distance from the axis of rotation of a point is equal to its
121:
737:
614:
594:
401:
144:
101:
81:
61:
41:
555:{\displaystyle I_{z}=\int (x^{2}+y^{2})\,dm=\int x^{2}\,dm+\int y^{2}\,dm=I_{y}+I_{x}}
848:
841:
819:
813:
792:
25:
839:
K. F. Riley, M. P. Hobson & S. J. Bence (2006). "Ch. 6: Multiple
Integrals".
324:
are equal, then the perpendicular axes theorem provides the useful relationship:
783:
Paul A. Tipler (1976). "Ch. 12: Rotation of a Rigid Body about a Fixed Axis".
885:
785:
876:
838:
264:
200:
respectively. Then the perpendicular axis theorem states that
591:, so these two terms are the moments of inertia about the
267:
to find polar moments of inertia for a variety of shapes.
740:
703:
640:
617:
597:
571:
427:
404:
398:, the moment of inertia of the planar body about the
333:
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276:
270:
If a planar object has rotational symmetry such that
209:
161:
axis is perpendicular to the plane of the body. Let
147:
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840:
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153:
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50:
883:
843:Mathematical Methods for Physics and Engineering
782:
690:{\displaystyle \int x^{2}\,dm=I_{y}\neq I_{x}}
717:
654:
519:
496:
473:
811:
884:
379:{\displaystyle I_{z}=2I_{x}=2I_{y}}
13:
259:This rule can be applied with the
14:
918:
249:{\displaystyle I_{z}=I_{x}+I_{y}}
188:be moments of inertia about axis
754:measures the distance from the
118:) so that the body lies in the
847:. Cambridge University Press.
832:
805:
776:
727:{\displaystyle \int r^{2}\,dm}
470:
444:
1:
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389:
7:
865:
10:
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38:Define perpendicular axes
18:perpendicular axis theorem
812:Obregon, Joaquin (2012).
791:. Worth Publishers Inc.
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98:(which meet at origin
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872:Parallel axis theorem
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729:
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413:
396:Cartesian coordinates
381:
319:
317:{\displaystyle I_{y}}
292:
290:{\displaystyle I_{x}}
261:parallel axis theorem
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24:) states that, "The
22:plane figure theorem
902:Classical mechanics
815:Mechanical Simmetry
584:{\displaystyle z=0}
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418:axis is given by:
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134:{\displaystyle xy}
131:
108:
88:
68:
48:
854:978-0-521-67971-8
825:978-1-4772-3372-6
747:{\displaystyle r}
624:{\displaystyle y}
604:{\displaystyle x}
411:{\displaystyle z}
154:{\displaystyle z}
111:{\displaystyle O}
91:{\displaystyle z}
71:{\displaystyle y}
51:{\displaystyle x}
26:moment of inertia
914:
907:Moment (physics)
897:Physics theorems
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818:. Author House.
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756:axis of rotation
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141:plane, and the
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565:On the plane,
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892:Rigid bodies
877:Stretch rule
842:
834:
814:
807:
786:
778:
766:coordinate.
763:
759:
755:
633:
564:
393:
269:
265:stretch rule
258:
197:
193:
189:
184:
180:
175:
171:
166:
162:
37:
29:
21:
17:
15:
758:, so for a
697:because in
394:Working in
886:Categories
770:References
634:Note that
390:Derivation
705:∫
675:≠
642:∫
507:∫
484:∫
442:∫
866:See also
263:and the
787:Physics
851:
822:
795:
78:, and
849:ISBN
820:ISBN
793:ISBN
611:and
297:and
179:and
20:(or
16:The
888::
734:,
196:,
192:,
170:,
58:,
857:.
828:.
801:.
764:x
760:y
742:r
722:m
719:d
713:2
709:r
683:x
679:I
670:y
666:I
662:=
659:m
656:d
650:2
646:x
619:y
599:x
579:0
576:=
573:z
548:x
544:I
540:+
535:y
531:I
527:=
524:m
521:d
515:2
511:y
504:+
501:m
498:d
492:2
488:x
481:=
478:m
475:d
471:)
466:2
462:y
458:+
453:2
449:x
445:(
439:=
434:z
430:I
406:z
372:y
368:I
364:2
361:=
356:x
352:I
348:2
345:=
340:z
336:I
310:y
306:I
283:x
279:I
242:y
238:I
234:+
229:x
225:I
221:=
216:z
212:I
198:z
194:y
190:x
185:z
181:I
176:y
172:I
167:x
163:I
149:z
129:y
126:x
106:O
86:z
66:y
46:x
32:z
30:I
28:(
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