24:
59:
The term linear density or linear mass density is most often used when describing the characteristics of one-dimensional objects, although linear density can also be used to describe the density of a three-dimensional quantity along one particular dimension. Just as density is most often used to
510:
can be measured by many methods. The simplest one is to measure a length of material and weigh it. However, this requires a large sample and masks the variability of linear density along the thread, and is difficult to apply if the fibers are crimped or otherwise cannot lay flat relaxed. If the
60:
mean mass density, the term linear density likewise often refers to linear mass density. However, this is only one example of a linear density, as any quantity can be measured in terms of its value along one dimension.
708:
231:
948:
483:
1000:
619:
142:
878:
409:
813:
344:
880:
The linear charge density can then be understood as the derivative of the charge function with respect to the one dimension of the wire (the position along its length,
763:
290:
836:
786:
367:
317:
898:
728:
659:
639:
583:
563:
433:
255:
182:
162:
106:
86:
511:
density of the material is known, the fibers are measured individually and have a simple shape, a more accurate method is direct imaging of the fiber with a
664:
187:
903:
438:
27:
The linear density, represented by λ, indicates the amount of a quantity, indicated by m, per unit length along a single dimension.
710:
If we describe the wire as having a varying charge (one that varies as a function of position along the length of the wire,
1205:
841:
372:
1240:
515:
to measure the diameter and calculation of the linear density. Finally, linear density is directly measured with a
1105:
1111:
1099:
955:
1206:"ISO 1973:1995. Textile fibres -- Determination of linear density -- Gravimetric method and vibroscope method"
1115:
733:
260:
1053:
1037:
588:
512:
111:
1108:, a unit of measure for the linear density of fibers, defined as the mass in grams per 9,000 meters
1102:, a unit of measure for the linear density of fibers, defined as the mass in grams per 1,000 meters
17:
791:
415:
of the mass function with respect to the one dimension of the rod (the position along its length,
322:
234:
1232:
1221:
1150:
536:
524:
48:
520:
8:
43:
818:
768:
349:
299:
1187:
883:
713:
644:
624:
568:
548:
418:
240:
167:
147:
91:
71:
1236:
1191:
1118:
unit for the linear density of fibers, defined as the mass in grams per 10,000 meters
1261:
1179:
1138:
1266:
1041:
1033:, the term linear density also refers to how densely or heavily a line is drawn.
53:
1170:
Patt, D.H. (1958). "Findings and
Recommendations on the Use of the Vibroscope".
1183:
34:
is the measure of a quantity of any characteristic value per unit of length.
1255:
1228:
1143:
293:
23:
1133:
1080:
952:
Notice that these steps were exactly the same ones we took before to find
56:
per unit length) are two common examples used in science and engineering.
1088:
1069:
621:, of this one dimensional object, we can simply divide the total charge,
1087:: used in the North American railway industry for the linear density of
516:
412:
144:, of this one dimensional object, we can simply divide the total mass,
233:
If we describe the rod as having a varying mass (one that varies as a
1030:
492:
1128:
1026:
1010:
1065:
1014:
503:
496:
1084:
1075:
542:
507:
788:, is equal to the product of its linear charge density,
1006:
488:
958:
411:
The linear mass density can then be understood as the
319:, is equal to the product of its linear mass density,
1036:
The most famous abstraction of linear density is the
906:
886:
844:
821:
794:
771:
736:
716:
667:
647:
627:
591:
571:
551:
441:
421:
375:
352:
325:
302:
263:
243:
190:
170:
150:
114:
94:
74:
703:{\displaystyle {\bar {\lambda }}_{q}={\frac {Q}{L}}}
226:{\displaystyle {\bar {\lambda }}_{m}={\frac {M}{L}}}
585:. To calculate the average linear charge density,
519:. The sample is tensioned between two hard points,
1220:
994:
942:
892:
872:
830:
807:
780:
757:
722:
702:
653:
633:
613:
577:
557:
477:
427:
403:
361:
338:
311:
284:
249:
225:
176:
156:
136:
100:
80:
108:. To calculate the average linear mass density,
1253:
1223:Introduction to Electrodynamics (2nd Edition)
943:{\displaystyle \lambda _{q}={\frac {dq}{dl}}}
478:{\displaystyle \lambda _{m}={\frac {dm}{dl}}}
995:{\textstyle \lambda _{m}={\frac {dm}{dl}}}
46:, the amount of mass per unit length) and
1218:
237:of position along the length of the rod,
815:, and the infinitesimal unit of length,
530:
346:, and the infinitesimal unit of length,
22:
1254:
63:
1020:
1009:unit of linear charge density is the
614:{\displaystyle {\bar {\lambda }}_{q}}
137:{\displaystyle {\bar {\lambda }}_{m}}
1169:
765:Each infinitesimal unit of charge,
491:unit of linear mass density is the
13:
68:Consider a long, thin rod of mass
14:
1278:
873:{\displaystyle dq=\lambda _{q}dl}
404:{\displaystyle dm=\lambda _{m}dl}
1212:
1198:
1163:
752:
746:
675:
599:
279:
273:
198:
122:
1:
1156:
1219:Griffiths, David J. (1989),
1054:Units of textile measurement
1038:probability density function
808:{\displaystyle \lambda _{q}}
513:scanning electron microscope
339:{\displaystyle \lambda _{m}}
7:
1122:
10:
1283:
1184:10.1177/004051755802800809
1051:
534:
15:
1172:Textile Research Journal
1047:
18:Density (disambiguation)
641:, by the total length,
164:, by the total length,
1058:Common units include:
996:
944:
894:
874:
832:
809:
782:
759:
758:{\displaystyle q=q(l)}
724:
704:
655:
635:
615:
579:
559:
541:Consider a long, thin
479:
429:
405:
363:
340:
313:
286:
285:{\displaystyle m=m(l)}
251:
227:
178:
158:
138:
102:
82:
28:
1151:Linear number density
1096:pound (mass) per inch
1093:pound (mass) per foot
997:
945:
895:
875:
833:
810:
783:
760:
725:
705:
656:
636:
616:
580:
560:
537:Linear charge density
531:Linear charge density
525:fundamental frequency
480:
430:
406:
364:
341:
314:
287:
252:
228:
179:
159:
139:
103:
83:
49:linear charge density
26:
956:
904:
884:
842:
819:
792:
769:
734:
714:
665:
645:
625:
589:
569:
549:
521:mechanical vibration
439:
419:
373:
350:
323:
300:
261:
241:
188:
168:
148:
112:
92:
72:
16:For other uses, see
523:is induced and the
64:Linear mass density
44:textile engineering
36:Linear mass density
1062:kilogram per meter
1021:Other applications
992:
940:
890:
870:
831:{\displaystyle dl}
828:
805:
781:{\displaystyle dq}
778:
755:
720:
700:
651:
631:
611:
575:
555:
502:Linear density of
475:
425:
401:
362:{\displaystyle dl}
359:
336:
312:{\displaystyle dm}
309:
282:
247:
223:
174:
154:
134:
98:
78:
29:
1074:ounce (mass) per
990:
938:
893:{\displaystyle l}
730:), we can write:
723:{\displaystyle l}
698:
678:
654:{\displaystyle L}
634:{\displaystyle Q}
602:
578:{\displaystyle L}
558:{\displaystyle Q}
473:
428:{\displaystyle l}
257:), we can write:
250:{\displaystyle l}
221:
201:
177:{\displaystyle L}
157:{\displaystyle M}
125:
101:{\displaystyle L}
81:{\displaystyle M}
1274:
1246:
1245:
1226:
1216:
1210:
1209:
1202:
1196:
1195:
1167:
1139:Columnar density
1001:
999:
998:
993:
991:
989:
981:
973:
968:
967:
949:
947:
946:
941:
939:
937:
929:
921:
916:
915:
899:
897:
896:
891:
879:
877:
876:
871:
863:
862:
837:
835:
834:
829:
814:
812:
811:
806:
804:
803:
787:
785:
784:
779:
764:
762:
761:
756:
729:
727:
726:
721:
709:
707:
706:
701:
699:
691:
686:
685:
680:
679:
671:
660:
658:
657:
652:
640:
638:
637:
632:
620:
618:
617:
612:
610:
609:
604:
603:
595:
584:
582:
581:
576:
564:
562:
561:
556:
484:
482:
481:
476:
474:
472:
464:
456:
451:
450:
434:
432:
431:
426:
410:
408:
407:
402:
394:
393:
368:
366:
365:
360:
345:
343:
342:
337:
335:
334:
318:
316:
315:
310:
291:
289:
288:
283:
256:
254:
253:
248:
232:
230:
229:
224:
222:
214:
209:
208:
203:
202:
194:
183:
181:
180:
175:
163:
161:
160:
155:
143:
141:
140:
135:
133:
132:
127:
126:
118:
107:
105:
104:
99:
87:
85:
84:
79:
1282:
1281:
1277:
1276:
1275:
1273:
1272:
1271:
1252:
1251:
1250:
1249:
1243:
1217:
1213:
1204:
1203:
1199:
1168:
1164:
1159:
1125:
1056:
1050:
1042:random variable
1023:
982:
974:
972:
963:
959:
957:
954:
953:
930:
922:
920:
911:
907:
905:
902:
901:
885:
882:
881:
858:
854:
843:
840:
839:
820:
817:
816:
799:
795:
793:
790:
789:
770:
767:
766:
735:
732:
731:
715:
712:
711:
690:
681:
670:
669:
668:
666:
663:
662:
646:
643:
642:
626:
623:
622:
605:
594:
593:
592:
590:
587:
586:
570:
567:
566:
550:
547:
546:
539:
533:
465:
457:
455:
446:
442:
440:
437:
436:
420:
417:
416:
389:
385:
374:
371:
370:
351:
348:
347:
330:
326:
324:
321:
320:
301:
298:
297:
262:
259:
258:
242:
239:
238:
213:
204:
193:
192:
191:
189:
186:
185:
169:
166:
165:
149:
146:
145:
128:
117:
116:
115:
113:
110:
109:
93:
90:
89:
73:
70:
69:
66:
54:electric charge
52:(the amount of
21:
12:
11:
5:
1280:
1270:
1269:
1264:
1248:
1247:
1241:
1227:, New Jersey:
1211:
1197:
1178:(8): 691–700.
1161:
1160:
1158:
1155:
1154:
1153:
1148:
1147:
1146:
1141:
1136:
1124:
1121:
1120:
1119:
1109:
1103:
1097:
1094:
1091:
1078:
1072:
1063:
1049:
1046:
1022:
1019:
988:
985:
980:
977:
971:
966:
962:
936:
933:
928:
925:
919:
914:
910:
889:
869:
866:
861:
857:
853:
850:
847:
827:
824:
802:
798:
777:
774:
754:
751:
748:
745:
742:
739:
719:
697:
694:
689:
684:
677:
674:
650:
630:
608:
601:
598:
574:
554:
535:Main article:
532:
529:
471:
468:
463:
460:
454:
449:
445:
424:
400:
397:
392:
388:
384:
381:
378:
358:
355:
333:
329:
308:
305:
296:unit of mass,
281:
278:
275:
272:
269:
266:
246:
220:
217:
212:
207:
200:
197:
173:
153:
131:
124:
121:
97:
77:
65:
62:
32:Linear density
9:
6:
4:
3:
2:
1279:
1268:
1265:
1263:
1260:
1259:
1257:
1244:
1242:0-13-481367-7
1238:
1234:
1230:
1229:Prentice Hall
1225:
1224:
1215:
1207:
1201:
1193:
1189:
1185:
1181:
1177:
1173:
1166:
1162:
1152:
1149:
1145:
1144:Paper density
1142:
1140:
1137:
1135:
1132:
1131:
1130:
1127:
1126:
1117:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1090:
1086:
1082:
1079:
1077:
1073:
1071:
1067:
1064:
1061:
1060:
1059:
1055:
1045:
1043:
1039:
1034:
1032:
1028:
1018:
1016:
1012:
1008:
1003:
986:
983:
978:
975:
969:
964:
960:
950:
934:
931:
926:
923:
917:
912:
908:
887:
867:
864:
859:
855:
851:
848:
845:
825:
822:
800:
796:
775:
772:
749:
743:
740:
737:
717:
695:
692:
687:
682:
672:
648:
628:
606:
596:
572:
552:
544:
538:
528:
527:is measured.
526:
522:
518:
514:
509:
505:
500:
498:
494:
490:
485:
469:
466:
461:
458:
452:
447:
443:
422:
414:
398:
395:
390:
386:
382:
379:
376:
356:
353:
331:
327:
306:
303:
295:
294:infinitesimal
276:
270:
267:
264:
244:
236:
218:
215:
210:
205:
195:
171:
151:
129:
119:
95:
75:
61:
57:
55:
51:
50:
45:
41:
37:
33:
25:
19:
1222:
1214:
1200:
1175:
1171:
1165:
1134:Area density
1114:(dtex), the
1057:
1040:of a single
1035:
1024:
1004:
951:
540:
501:
486:
67:
58:
47:
39:
35:
31:
30:
1231:, pp.
1083:(mass) per
1068:(mass) per
565:and length
88:and length
1256:Categories
1157:References
1052:See also:
545:of charge
517:vibroscope
413:derivative
1192:137534752
961:λ
909:λ
856:λ
797:λ
676:¯
673:λ
600:¯
597:λ
444:λ
387:λ
328:λ
199:¯
196:λ
123:¯
120:λ
1123:See also
1031:printing
499:(kg/m).
493:kilogram
235:function
1262:Density
1129:Density
1112:decitex
1027:drawing
1017:(C/m).
1011:coulomb
1267:Length
1239:
1190:
1106:denier
504:fibers
1188:S2CID
1089:rails
1081:pound
1066:ounce
1048:Units
1015:meter
508:yarns
497:meter
292:Each
40:titer
1237:ISBN
1085:yard
1076:inch
1070:foot
1013:per
1005:The
543:wire
506:and
495:per
487:The
1180:doi
1100:tex
1029:or
1025:In
42:in
1258::
1235:,
1233:64
1186:.
1176:28
1174:.
1116:SI
1044:.
1007:SI
1002:.
900:)
838::
661::
489:SI
435:)
369::
184::
1208:.
1194:.
1182::
987:l
984:d
979:m
976:d
970:=
965:m
935:l
932:d
927:q
924:d
918:=
913:q
888:l
868:l
865:d
860:q
852:=
849:q
846:d
826:l
823:d
801:q
776:q
773:d
753:)
750:l
747:(
744:q
741:=
738:q
718:l
696:L
693:Q
688:=
683:q
649:L
629:Q
607:q
573:L
553:Q
470:l
467:d
462:m
459:d
453:=
448:m
423:l
399:l
396:d
391:m
383:=
380:m
377:d
357:l
354:d
332:m
307:m
304:d
280:)
277:l
274:(
271:m
268:=
265:m
245:l
219:L
216:M
211:=
206:m
172:L
152:M
130:m
96:L
76:M
38:(
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
