904:
917:
1361:
2831:
4352:
31:
3729:
3972:
4122:
3526:
2635:
4585:
2450:
3213:
Consistent with the general definition, the spin angular velocity of a frame is defined as the orbital angular velocity of any of the three vectors (same for all) with respect to its own center of rotation. The addition of angular velocity vectors for frames is also defined by the usual vector
1863:
perpendicular to the radius. When there is no radial component, the particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear
4347:{\displaystyle {\boldsymbol {\omega }}=({\dot {\alpha }}\sin \beta \sin \gamma +{\dot {\beta }}\cos \gamma ){\hat {\mathbf {i} }}+({\dot {\alpha }}\sin \beta \cos \gamma -{\dot {\beta }}\sin \gamma ){\hat {\mathbf {j} }}+({\dot {\alpha }}\cos \beta +{\dot {\gamma }}){\hat {\mathbf {k} }}}
3451:
2813:
In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction. The sign is conventionally taken to be positive if the radius vector turns counter-clockwise, and negative if clockwise. Angular velocity then may be termed a
4110:
3062:
3724:{\displaystyle {\boldsymbol {\omega }}=\left({\dot {\mathbf {e} }}_{1}\cdot \mathbf {e} _{2}\right)\mathbf {e} _{3}+\left({\dot {\mathbf {e} }}_{2}\cdot \mathbf {e} _{3}\right)\mathbf {e} _{1}+\left({\dot {\mathbf {e} }}_{3}\cdot \mathbf {e} _{1}\right)\mathbf {e} _{2},}
2455:
2281:
3925:
4453:
3137:
4746:
4001:
Euler proved that the projections of the angular velocity pseudovector on each of these three axes is the derivative of its associated angle (which is equivalent to decomposing the instantaneous rotation into three instantaneous
4420:
3188:
1122:
4659:
3198:
Given a rotating frame of three unit coordinate vectors, all the three must have the same angular speed at each instant. In such a frame, each vector may be considered as a moving particle with constant scalar radius.
3802:
2834:
The orbital angular velocity vector encodes the time rate of change of angular position, as well as the instantaneous plane of angular displacement. In this case (counter-clockwise circular motion) the vector points
4115:
This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases. For example, changing to the mobile frame:
3364:
1938:
1803:
1635:
In the general case of a particle moving in the plane, the orbital angular velocity is the rate at which the position vector relative to a chosen origin "sweeps out" angle. The diagram shows the position vector
3518:
4012:
3282:
2236:
2968:
1333:
258:
2689:
3218:. All components of the vector can be calculated as derivatives of the parameters defining the moving frames (Euler angles or rotation matrices). As in the general case, addition is commutative:
2163:
1067:
1597:
4702:
1296:
3858:
2107:
1474:
3356:
3957:
3327:
1997:
1861:
1630:
2779:
190:
4771:
2808:
1832:
1002:
282:
2928:
2898:
2019:
1656:
1527:
5701:
3873:
2747:
2718:
1968:
2960:
1731:
2276:
1431:
2059:
2630:{\displaystyle {\dot {r}}(\cos(\varphi ),\sin(\varphi ))+r{\dot {\varphi }}(-\sin(\varphi ),\cos(\varphi ))={\dot {r}}{\hat {r}}+r{\dot {\varphi }}{\hat {\varphi }}}
1494:
4580:{\displaystyle \Omega ={\begin{pmatrix}0&-\omega _{z}&\omega _{y}\\\omega _{z}&0&-\omega _{x}\\-\omega _{y}&\omega _{x}&0\\\end{pmatrix}}}
2445:{\displaystyle {\frac {d\mathbf {r} }{dt}}=({\dot {r}}\cos(\varphi )-r{\dot {\varphi }}\sin(\varphi ),{\dot {r}}\sin(\varphi )+r{\dot {\varphi }}\cos(\varphi )),}
2256:
2039:
1751:
1696:
1676:
1402:
5093:
3085:
5791:
4710:
2871:
directions perpendicular to any plane, an additional condition is necessary to uniquely specify the direction of the angular velocity; conventionally, the
1877:
4360:
3148:
1082:
4597:
2908:, so that the right-hand rule is satisfied (i.e. the instantaneous direction of angular displacement is counter-clockwise looking from the top of
3446:{\displaystyle {\dot {\boldsymbol {r}}}={\dot {{\boldsymbol {r}}_{0}}}+{\boldsymbol {\omega }}\times ({\boldsymbol {r}}-{{\boldsymbol {r}}_{0}})}
3295:, which is the direction of the angular velocity vector, and the magnitude of the angular velocity is consistent with the two-dimensional case.
3737:
948:
3461:
Consider a rigid body rotating about a fixed point O. Construct a reference frame in the body consisting of an orthonormal set of vectors
2115:
1756:
4105:{\displaystyle {\boldsymbol {\omega }}={\dot {\alpha }}\mathbf {u} _{1}+{\dot {\beta }}\mathbf {u} _{2}+{\dot {\gamma }}\mathbf {u} _{3}}
537:
3464:
5086:
656:
3221:
3057:{\displaystyle {\boldsymbol {\omega }}=\omega \mathbf {u} ={\frac {d\phi }{dt}}\mathbf {u} ={\frac {v\sin(\theta )}{r}}\mathbf {u} ,}
2171:
629:
4422:
are unit vectors for the frame fixed in the moving body. This example has been made using the Z-X-Z convention for Euler angles.
1305:
214:
5079:
5024:
4998:
4937:
4889:
4861:
3520:
fixed to the body and with their common origin at O. The spin angular velocity vector of both frame and body about O is then
3206:, and special tools have been developed for it: the spin angular velocity may be described as a vector or equivalently as a
2647:
2855:
sweeps out angle (in radians per unit of time), and whose direction is perpendicular to the instantaneous plane in which
1871:
is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as:
1038:
611:
1532:
86:
75:
4676:
4923:
941:
5827:
5802:
5763:
1999:, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity
903:
1257:
277:
1343:= 42,000 km × 0.26/h ≈ 11,000 km/h. The angular velocity is positive since the satellite travels
3807:
3292:
2064:
1344:
532:
272:
3332:
1436:
6098:
5105:
4666:
3940:
3301:
1228:
2962:
in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as:
6103:
4856:. New Delhi: John Wiley & Sons Inc., authorized reprint to Wiley – India. pp. 449, 484, 485, 487.
3934:
about O, while the formula in this section applies to a frame or rigid body. In the case of a rigid body a
1973:
1837:
1299:
1147:
934:
921:
682:
605:
527:
370:
4781:
3288:
601:
402:
4955:
2752:
1602:
1246:= 360°/24 h = 15°/h (or 2π rad/24 h ≈ 0.26 rad/h) and angular velocity direction (a
380:
161:
4754:
2784:
1808:
821:
710:
636:
496:
429:
154:
978:
5734:
5539:
5393:
575:
116:
4908:
5506:
5064:
3920:{\displaystyle {\boldsymbol {\omega }}={\frac {{\boldsymbol {r}}\times {\boldsymbol {v}}}{r^{2}}},}
1433:
from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time:
1251:
851:
695:
5040:
2911:
2881:
2002:
1639:
1503:
1302:). If angle is measured in radians, the linear velocity is the radius times the angular velocity,
6108:
5517:
4591:
4440:
4432:
2840:
1373:
801:
565:
5677:
5577:
3214:
addition (composition of linear movements), and can be useful to decompose the rotation as in a
2723:
2694:
1946:
1079:), the angular rate at which the object rotates (spins or revolves). The pseudovector direction
841:
2933:
1704:
1197:
1021:
846:
663:
2261:
2113:
1875:
1407:
5652:
5522:
5260:
4896:
4444:
2044:
1224:
856:
831:
517:
335:
5974:
5899:
5527:
5299:
4978:
4795:
1479:
1235:
1151:
1133:
1029:
876:
836:
744:
740:
732:
722:
512:
505:
261:
871:
8:
5705:
5434:
5382:
5102:
5052:
4825:
3994:
The line of nodes of the moving frame with respect to the reference frame (nutation axis)
3132:{\displaystyle {\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}.}
1169:
651:
592:
570:
315:
310:
305:
205:
4909:"Units with special names and symbols; units that incorporate special names and symbols"
4741:{\displaystyle {\boldsymbol {\omega }}\times {\boldsymbol {r}}=\Omega {\boldsymbol {r}}}
4929:
2241:
2024:
1736:
1681:
1661:
1387:
1201:
1189:
1185:
1161:
781:
522:
397:
365:
325:
101:
6118:
5724:
5642:
5478:
5237:
5020:
4994:
4933:
4885:
4857:
4800:
1699:
1129:
1125:
791:
748:
705:
700:
641:
417:
407:
300:
1164:
and is independent of the choice of origin, in contrast to orbital angular velocity.
1028:(spins or revolves) around an axis of rotation and how fast the axis itself changes
6113:
5811:
5781:
5747:
5422:
5377:
4820:
4805:
2819:
1336:
1017:
886:
866:
811:
806:
727:
582:
440:
385:
360:
3979:
The components of the spin angular velocity pseudovector were first calculated by
6056:
6016:
5012:
4879:
4851:
4774:
4415:{\displaystyle {\hat {\mathbf {i} }},{\hat {\mathbf {j} }},{\hat {\mathbf {k} }}}
4003:
2872:
1216:
881:
826:
776:
771:
690:
5819:
5755:
3183:{\displaystyle \mathbf {v} _{\perp }={\boldsymbol {\omega }}\times \mathbf {r} }
1227:(multiplication by −1) leaves the magnitude unchanged but flips the axis in the
6093:
6000:
5964:
5889:
5670:
5557:
5447:
5348:
5071:
4810:
3980:
1500:, the arc-length from the positive x-axis around the circle to the particle is
908:
816:
717:
434:
1335:. With orbital radius 42,000 km from the Earth's center, the satellite's
1117:{\displaystyle {\hat {\boldsymbol {\omega }}}={\boldsymbol {\omega }}/\omega }
6087:
5874:
5742:
5456:
4671:
4654:{\displaystyle {\boldsymbol {\omega }}=(\omega _{x},\omega _{y},\omega _{z})}
1071:
796:
623:
5949:
5935:
5835:
5771:
5496:
5279:
5190:
3984:
3193:
2848:
2815:
1013:
861:
786:
475:
355:
63:
3456:
1242:(360 degrees per 24 hours) has angular velocity magnitude (angular speed)
5317:
5060:
3797:{\displaystyle {\dot {\mathbf {e} }}_{i}={\frac {d\mathbf {e} _{i}}{dt}}}
2638:
1347:
with the Earth's rotation (the same direction as the rotation of Earth).
1247:
1200:, thus the SI units of angular velocity are dimensionally equivalent to
5008:
3203:
1834:
parallel to the radius, and the cross-radial (or tangential) component
1150:, i.e. the time rate of change of its angular position relative to the
646:
320:
93:
5410:
5337:
5325:
4830:
3142:
From the above equation, one can recover the tangential velocity as:
2749:
is a radial unit vector; and the perpendicular component is given by
1933:{\displaystyle \omega ={\frac {d\phi }{dt}}={\frac {v_{\perp }}{r}}.}
1220:
668:
2691:, we conclude that the radial component of the velocity is given by
1798:{\displaystyle \mathbf {v} =\mathbf {v} _{\|}+\mathbf {v} _{\perp }}
1360:
5714:
5613:
5367:
5226:
5180:
4815:
2830:
2258:
a function of the distance to the origin with respect to time, and
1177:
1173:
587:
470:
445:
3513:{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}}
6040:
2278:
a function of the angle between the vector and the x axis. Then:
1239:
1215:
The sense of angular velocity is conventionally specified by the
1181:
962:
560:
413:
330:
52:
5954:
5923:
5879:
5857:
5311:
5210:
5175:
3277:{\displaystyle \omega _{1}+\omega _{2}=\omega _{2}+\omega _{1}}
3215:
3207:
1497:
1350:
1193:
1025:
619:
465:
375:
30:
4979:
K.S.HEDRIH: Leonhard Euler (1707–1783) and rigid body dynamics
2231:{\displaystyle \mathbf {r} =(r\cos(\varphi ),r\sin(\varphi ))}
5988:
5913:
5847:
5451:
5357:
5352:
5289:
5264:
3971:
1209:
1005:
455:
450:
392:
1160:
refers to how fast a rigid body rotates with respect to its
6070:
6030:
5603:
5269:
5200:
5165:
1024:
of an object changes with time, i.e. how quickly an object
460:
423:
3997:
One axis of the moving frame (the intrinsic rotation axis)
2847:
of a moving particle. Here, orbital angular velocity is a
1328:{\displaystyle {\boldsymbol {v}}=r{\boldsymbol {\omega }}}
1204:, s, although rad/s is preferable to avoid confusion with
3194:
Spin angular velocity of a rigid body or reference frame
2900:
be the unit vector perpendicular to the plane spanned by
253:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
4932:, New Jersey: Pearson Prentice Hall. pp. 314, 153.
4779:
When multiplied by a time difference, it results in the
2822:, such as inverting one axis or switching the two axes.
4884:(revised 2008 ed.). DIANE Publishing. p. 27.
4849:
3457:
Components from the basis vectors of a body-fixed frame
4468:
2650:
1605:
1535:
1439:
5680:
4757:
4713:
4679:
4600:
4456:
4363:
4125:
4015:
3991:
One axis of the reference frame (the precession axis)
3943:
3876:
3863:
This formula is incompatible with the expression for
3810:
3740:
3529:
3467:
3367:
3335:
3304:
3224:
3151:
3088:
2971:
2936:
2914:
2884:
2787:
2755:
2726:
2697:
2684:{\textstyle {\frac {d\mathbf {r} }{dt}}=\mathbf {v} }
2458:
2284:
2264:
2244:
2174:
2118:
2067:
2047:
2027:
2005:
1976:
1949:
1880:
1840:
1811:
1759:
1739:
1707:
1684:
1664:
1642:
1506:
1482:
1410:
1390:
1308:
1260:
1085:
1041:
981:
217:
164:
5007:
2158:{\displaystyle \omega ={\frac {v\sin(\theta )}{r}}.}
1172:
of angle per unit time; this is analogous to linear
1062:{\displaystyle \omega =\|{\boldsymbol {\omega }}\|}
5695:
4765:
4740:
4696:
4653:
4579:
4414:
4346:
4104:
3951:
3919:
3852:
3796:
3723:
3512:
3445:
3350:
3321:
3276:
3182:
3131:
3056:
2954:
2922:
2892:
2818:, a numerical quantity which changes sign under a
2802:
2773:
2741:
2712:
2683:
2629:
2444:
2270:
2250:
2230:
2157:
2101:
2053:
2033:
2013:
1991:
1962:
1932:
1855:
1826:
1797:
1745:
1725:
1690:
1670:
1650:
1624:
1592:{\textstyle v(t)={\frac {d\ell }{dt}}=r\omega (t)}
1591:
1521:
1488:
1468:
1425:
1404:, with position given by the angular displacement
1396:
1384:In the simplest case of circular motion at radius
1327:
1290:
1116:
1061:
996:
252:
184:
1753:.) The particle has linear velocity splitting as
6085:
5101:
5053:"ω Speed of Rotation [Angular Velocity]"
4697:{\displaystyle ({\boldsymbol {\omega }}\times )}
1238:satellite completes one orbit per day above the
1223:rotations (as viewed on the plane of rotation);
3966:
3930:as that formula defines angular velocity for a
3804:is the time rate of change of the frame vector
2825:
2061:relative to the radius vector; in these terms,
5087:
3202:The rotating frame appears in the context of
1355:
1351:Orbital angular velocity of a point particle
942:
4590:The scalar elements above correspond to the
2859:sweeps out angle (i.e. the plane spanned by
1819:
1775:
1056:
1048:
1291:{\displaystyle {\hat {\omega }}={\hat {Z}}}
5094:
5080:
3079:. In terms of the cross product, this is:
1864:velocity contributes to angular velocity.
949:
935:
29:
5050:
4850:Cummings, Karen; Halliday, David (2007).
3853:{\displaystyle \mathbf {e} _{i},i=1,2,3,}
2102:{\displaystyle v_{\perp }=v\sin(\theta )}
1469:{\textstyle \omega ={\frac {d\phi }{dt}}}
1139:There are two types of angular velocity:
4921:
3970:
3351:{\displaystyle {\dot {\boldsymbol {r}}}}
2829:
1364:The angular velocity of the particle at
1359:
4759:
4734:
4723:
4715:
4684:
4602:
4127:
4017:
3952:{\displaystyle {\boldsymbol {\omega }}}
3945:
3897:
3889:
3878:
3531:
3429:
3419:
3408:
3388:
3371:
3339:
3329:fixed in the rigid body, the velocity
3322:{\displaystyle {{\boldsymbol {r}}_{0}}}
3308:
3168:
3090:
2973:
1733:. (All variables are functions of time
1321:
1310:
1102:
1089:
1052:
6086:
4877:
3987:and the use of an intermediate frame:
3358:of any point in the body is given by
168:
5075:
4988:
2851:whose magnitude is the rate at which
1992:{\displaystyle \mathbf {v} _{\perp }}
1856:{\displaystyle \mathbf {v} _{\perp }}
631:Newton's law of universal gravitation
3975:Diagram showing Euler frame in green
2843:, we again have the position vector
2168:These formulas may be derived doing
1625:{\textstyle \omega ={\frac {v}{r}}}
1035:The magnitude of the pseudovector,
612:Mechanics of planar particle motion
220:
13:
4915:
4881:International System of Units (SI)
4843:
4730:
4457:
3291:, any rotating frame possesses an
2774:{\displaystyle r{\dot {\varphi }}}
2112:
1874:
1146:refers to how fast a point object
185:{\displaystyle {\mathsf {T}}^{-1}}
14:
6130:
5034:
4766:{\displaystyle {\boldsymbol {r}}}
4670:. The linear mapping Ω acts as a
3959:has to account for the motion of
2803:{\displaystyle {\hat {\varphi }}}
1827:{\displaystyle \mathbf {v} _{\|}}
4431:This section is an excerpt from
4402:
4385:
4368:
4334:
4275:
4198:
4092:
4065:
4038:
3813:
3773:
3746:
3708:
3691:
3670:
3647:
3630:
3609:
3586:
3569:
3548:
3500:
3485:
3470:
3176:
3154:
3109:
3101:
3047:
3012:
2984:
2916:
2886:
2810:is a perpendicular unit vector.
2677:
2658:
2292:
2176:
2007:
1979:
1843:
1814:
1785:
1770:
1761:
1644:
997:{\displaystyle {\vec {\omega }}}
916:
915:
902:
235:
5118:Linear/translational quantities
4993:. Addison-Wesley, Reading, MA.
3298:If we choose a reference point
5045:Angular Velocity of a particle
5041:A college text-book of physics
4972:
4947:
4901:
4871:
4691:
4680:
4648:
4609:
4406:
4389:
4372:
4338:
4327:
4288:
4279:
4268:
4211:
4202:
4191:
4134:
3440:
3415:
3293:instantaneous axis of rotation
3037:
3031:
2949:
2937:
2794:
2733:
2641:in cylindrical coordinates).
2621:
2591:
2567:
2564:
2558:
2546:
2540:
2528:
2507:
2504:
2498:
2486:
2480:
2471:
2436:
2433:
2427:
2400:
2394:
2370:
2364:
2337:
2331:
2310:
2225:
2222:
2216:
2201:
2195:
2183:
2143:
2137:
2096:
2090:
1720:
1708:
1586:
1580:
1545:
1539:
1420:
1414:
1282:
1267:
1092:
988:
16:Direction and rate of rotation
1:
5123:Angular/rotational quantities
4922:Hibbeler, Russell C. (2009).
4836:
4667:infinitesimal rotation matrix
1805:, with the radial component
1529:, and the linear velocity is
1148:revolves about a fixed origin
1004:, the lowercase Greek letter
538:Koopman–von Neumann mechanics
3967:Components from Euler angles
2930:). Taking polar coordinates
2923:{\displaystyle \mathbf {u} }
2893:{\displaystyle \mathbf {u} }
2826:Particle in three dimensions
2014:{\displaystyle \mathbf {v} }
1943:Here the cross-radial speed
1651:{\displaystyle \mathbf {r} }
1522:{\displaystyle \ell =r\phi }
1300:geocentric coordinate system
606:Non-inertial reference frame
7:
5043:By Arthur Lalanne Kimball (
4789:
4782:angular displacement tensor
1970:is the signed magnitude of
1368:with respect to the origin
1180:, with time in common. The
533:Appell's equation of motion
403:Inertial frame of reference
10:
6135:
5696:{\displaystyle {\dot {m}}}
5120:
4430:
2742:{\displaystyle {\hat {r}}}
2713:{\displaystyle {\dot {r}}}
1963:{\displaystyle v_{\perp }}
1356:Particle in two dimensions
1192:(°/s) is also common. The
1016:representation of how the
5394:specific angular momentum
5122:
5117:
5112:
5051:Pickering, Steve (2009).
5019:. Butterworth-Heinemann.
4878:Taylor, Barry N. (2009).
4425:
2955:{\displaystyle (r,\phi )}
2867:). However, as there are
2041:(linear speed) and angle
1726:{\displaystyle (r,\phi )}
1212:(also equivalent to s).
153:
127:
111:
100:
85:
74:
61:
57:rad ⋅ s
51:
37:
28:
23:
5065:University of Nottingham
3289:Euler's rotation theorem
2271:{\displaystyle \varphi }
1426:{\displaystyle \phi (t)}
1144:Orbital angular velocity
1010:angular frequency vector
696:Rotating reference frame
528:Hamilton–Jacobi equation
5518:rotational acceleration
4592:angular velocity vector
4441:angular velocity tensor
4433:Angular velocity tensor
3963:particles in the body.
2841:three-dimensional space
2054:{\displaystyle \theta }
1376:of the velocity vector
1374:perpendicular component
1184:of angular velocity is
1176:, with angle replacing
637:Newton's laws of motion
497:Newton's laws of motion
5697:
4953:
4767:
4742:
4698:
4655:
4581:
4416:
4348:
4106:
3976:
3953:
3921:
3854:
3798:
3725:
3514:
3447:
3352:
3323:
3278:
3184:
3133:
3058:
2956:
2924:
2894:
2836:
2804:
2775:
2743:
2714:
2685:
2631:
2446:
2272:
2252:
2232:
2159:
2103:
2055:
2035:
2015:
1993:
1964:
1934:
1857:
1828:
1799:
1747:
1727:
1692:
1672:
1652:
1626:
1593:
1523:
1490:
1470:
1427:
1398:
1381:
1339:through space is thus
1329:
1292:
1198:dimensionless quantity
1118:
1063:
998:
664:Simple harmonic motion
577:Euler's laws of motion
371:D'Alembert's principle
254:
186:
5698:
4989:Symon, Keith (1971).
4968:– via OpenStax.
4925:Engineering Mechanics
4853:Understanding physics
4768:
4743:
4699:
4656:
4582:
4445:skew-symmetric matrix
4417:
4349:
4107:
3974:
3954:
3922:
3860:due to the rotation.
3855:
3799:
3726:
3515:
3448:
3353:
3324:
3279:
3185:
3134:
3071:is the angle between
3059:
2957:
2925:
2895:
2878:Let the pseudovector
2833:
2805:
2776:
2744:
2715:
2686:
2632:
2447:
2273:
2253:
2233:
2160:
2104:
2056:
2036:
2016:
1994:
1965:
1935:
1867:The angular velocity
1858:
1829:
1800:
1748:
1728:
1693:
1673:
1653:
1627:
1594:
1524:
1491:
1489:{\displaystyle \phi }
1471:
1428:
1399:
1372:is determined by the
1363:
1330:
1293:
1252:Earth's rotation axis
1168:Angular velocity has
1158:Spin angular velocity
1128:to the instantaneous
1119:
1064:
999:
518:Hamiltonian mechanics
336:Statistical mechanics
255:
187:
6099:Kinematic properties
5678:
5528:angular acceleration
5300:angular displacement
4796:Angular acceleration
4755:
4711:
4677:
4598:
4454:
4361:
4123:
4013:
3941:
3874:
3808:
3738:
3527:
3465:
3365:
3333:
3302:
3222:
3149:
3086:
2969:
2934:
2912:
2882:
2785:
2753:
2724:
2695:
2648:
2456:
2282:
2262:
2242:
2172:
2116:
2065:
2045:
2025:
2003:
1974:
1947:
1878:
1838:
1809:
1757:
1737:
1705:
1682:
1662:
1640:
1603:
1533:
1504:
1480:
1437:
1408:
1388:
1306:
1258:
1134:angular displacement
1083:
1039:
979:
741:Angular acceleration
733:Rotational frequency
513:Lagrangian mechanics
506:Analytical mechanics
262:Second law of motion
215:
162:
117:coord transformation
6104:Rotational symmetry
5618:weighted position:
5435:rotational velocity
5383:kinematic viscosity
5103:Classical mechanics
4826:Rigid body dynamics
2452:which is equal to:
593:Harmonic oscillator
571:Equations of motion
206:Classical mechanics
200:Part of a series on
5693:
4930:Upper Saddle River
4897:Extract of page 27
4763:
4738:
4694:
4651:
4577:
4571:
4412:
4344:
4102:
3977:
3949:
3917:
3850:
3794:
3721:
3510:
3443:
3348:
3319:
3274:
3180:
3129:
3054:
2952:
2920:
2890:
2837:
2800:
2771:
2739:
2710:
2681:
2627:
2442:
2268:
2248:
2228:
2155:
2099:
2051:
2031:
2011:
1989:
1960:
1930:
1853:
1824:
1795:
1743:
1723:
1688:
1668:
1648:
1622:
1589:
1519:
1486:
1466:
1423:
1394:
1382:
1325:
1288:
1229:opposite direction
1202:reciprocal seconds
1190:degrees per second
1186:radians per second
1162:center of rotation
1114:
1059:
994:
909:Physics portal
523:Routhian mechanics
398:Frame of reference
250:
182:
65:SI base units
6081:
6080:
6076:
6075:
5690:
5643:moment of inertia
5026:978-0-7506-2896-9
5000:978-0-201-07392-8
4960:. Rice University
4939:978-0-13-607791-6
4891:978-1-4379-1558-7
4863:978-81-265-0882-2
4801:Angular frequency
4409:
4392:
4375:
4341:
4324:
4300:
4282:
4256:
4223:
4205:
4179:
4146:
4087:
4060:
4033:
3912:
3867:angular velocity
3792:
3753:
3677:
3616:
3555:
3401:
3377:
3345:
3124:
3044:
3009:
2797:
2768:
2736:
2707:
2671:
2624:
2612:
2594:
2582:
2525:
2468:
2418:
2385:
2355:
2322:
2305:
2251:{\displaystyle r}
2150:
2034:{\displaystyle v}
1925:
1905:
1746:{\displaystyle t}
1700:polar coordinates
1691:{\displaystyle P}
1671:{\displaystyle O}
1620:
1569:
1464:
1397:{\displaystyle r}
1285:
1270:
1206:rotation velocity
1130:plane of rotation
1095:
1077:angular frequency
1069:, represents the
1008:), also known as
991:
959:
958:
706:Centrifugal force
701:Centripetal force
657:Euler's equations
642:Relative velocity
418:Moment of inertia
248:
222:
195:
194:
6126:
6068:
6064:
6052:
6048:
6028:
6024:
6012:
6008:
5986:
5982:
5972:
5962:
5947:
5943:
5933:
5911:
5907:
5897:
5887:
5872:
5868:
5855:
5831:
5825:
5817:
5806:
5800:
5789:
5782:angular momentum
5767:
5761:
5753:
5738:
5732:
5722:
5709:
5703:
5702:
5700:
5699:
5694:
5692:
5691:
5683:
5656:
5650:
5634:
5611:
5592:
5587:
5565:
5543:
5537:
5520:
5504:
5482:
5476:
5467:angular velocity
5464:
5445:
5444:
5431:
5430:
5423:rotational speed
5419:
5418:
5401:
5390:
5375:
5365:
5346:
5345:
5328:
5323:
5309:
5297:
5277:
5258:
5257:
5251:
5245:
5234:
5208:
5188:
5173:
5157:
5150:
5115:
5114:
5096:
5089:
5082:
5073:
5072:
5068:
5030:
5004:
4981:
4976:
4970:
4969:
4967:
4965:
4957:Angular Velocity
4954:Singh, Sunil K.
4951:
4945:
4943:
4919:
4913:
4912:
4905:
4899:
4895:
4875:
4869:
4867:
4847:
4821:Orthogonal group
4806:Angular momentum
4772:
4770:
4769:
4764:
4762:
4747:
4745:
4744:
4739:
4737:
4726:
4718:
4703:
4701:
4700:
4695:
4687:
4660:
4658:
4657:
4652:
4647:
4646:
4634:
4633:
4621:
4620:
4605:
4586:
4584:
4583:
4578:
4576:
4575:
4563:
4562:
4551:
4550:
4534:
4533:
4514:
4513:
4500:
4499:
4488:
4487:
4421:
4419:
4418:
4413:
4411:
4410:
4405:
4400:
4394:
4393:
4388:
4383:
4377:
4376:
4371:
4366:
4353:
4351:
4350:
4345:
4343:
4342:
4337:
4332:
4326:
4325:
4317:
4302:
4301:
4293:
4284:
4283:
4278:
4273:
4258:
4257:
4249:
4225:
4224:
4216:
4207:
4206:
4201:
4196:
4181:
4180:
4172:
4148:
4147:
4139:
4130:
4111:
4109:
4108:
4103:
4101:
4100:
4095:
4089:
4088:
4080:
4074:
4073:
4068:
4062:
4061:
4053:
4047:
4046:
4041:
4035:
4034:
4026:
4020:
3958:
3956:
3955:
3950:
3948:
3926:
3924:
3923:
3918:
3913:
3911:
3910:
3901:
3900:
3892:
3886:
3881:
3859:
3857:
3856:
3851:
3822:
3821:
3816:
3803:
3801:
3800:
3795:
3793:
3791:
3783:
3782:
3781:
3776:
3766:
3761:
3760:
3755:
3754:
3749:
3744:
3730:
3728:
3727:
3722:
3717:
3716:
3711:
3705:
3701:
3700:
3699:
3694:
3685:
3684:
3679:
3678:
3673:
3668:
3656:
3655:
3650:
3644:
3640:
3639:
3638:
3633:
3624:
3623:
3618:
3617:
3612:
3607:
3595:
3594:
3589:
3583:
3579:
3578:
3577:
3572:
3563:
3562:
3557:
3556:
3551:
3546:
3534:
3519:
3517:
3516:
3511:
3509:
3508:
3503:
3494:
3493:
3488:
3479:
3478:
3473:
3452:
3450:
3449:
3444:
3439:
3438:
3437:
3432:
3422:
3411:
3403:
3402:
3397:
3396:
3391:
3385:
3379:
3378:
3370:
3357:
3355:
3354:
3349:
3347:
3346:
3338:
3328:
3326:
3325:
3320:
3318:
3317:
3316:
3311:
3283:
3281:
3280:
3275:
3273:
3272:
3260:
3259:
3247:
3246:
3234:
3233:
3189:
3187:
3186:
3181:
3179:
3171:
3163:
3162:
3157:
3138:
3136:
3135:
3130:
3125:
3123:
3122:
3113:
3112:
3104:
3098:
3093:
3063:
3061:
3060:
3055:
3050:
3045:
3040:
3020:
3015:
3010:
3008:
3000:
2992:
2987:
2976:
2961:
2959:
2958:
2953:
2929:
2927:
2926:
2921:
2919:
2899:
2897:
2896:
2891:
2889:
2820:parity inversion
2809:
2807:
2806:
2801:
2799:
2798:
2790:
2780:
2778:
2777:
2772:
2770:
2769:
2761:
2748:
2746:
2745:
2740:
2738:
2737:
2729:
2719:
2717:
2716:
2711:
2709:
2708:
2700:
2690:
2688:
2687:
2682:
2680:
2672:
2670:
2662:
2661:
2652:
2636:
2634:
2633:
2628:
2626:
2625:
2617:
2614:
2613:
2605:
2596:
2595:
2587:
2584:
2583:
2575:
2527:
2526:
2518:
2470:
2469:
2461:
2451:
2449:
2448:
2443:
2420:
2419:
2411:
2387:
2386:
2378:
2357:
2356:
2348:
2324:
2323:
2315:
2306:
2304:
2296:
2295:
2286:
2277:
2275:
2274:
2269:
2257:
2255:
2254:
2249:
2237:
2235:
2234:
2229:
2179:
2165:
2162:
2161:
2156:
2151:
2146:
2126:
2108:
2106:
2105:
2100:
2077:
2076:
2060:
2058:
2057:
2052:
2040:
2038:
2037:
2032:
2021:gives magnitude
2020:
2018:
2017:
2012:
2010:
1998:
1996:
1995:
1990:
1988:
1987:
1982:
1969:
1967:
1966:
1961:
1959:
1958:
1940:
1937:
1936:
1931:
1926:
1921:
1920:
1911:
1906:
1904:
1896:
1888:
1862:
1860:
1859:
1854:
1852:
1851:
1846:
1833:
1831:
1830:
1825:
1823:
1822:
1817:
1804:
1802:
1801:
1796:
1794:
1793:
1788:
1779:
1778:
1773:
1764:
1752:
1750:
1749:
1744:
1732:
1730:
1729:
1724:
1697:
1695:
1694:
1689:
1677:
1675:
1674:
1669:
1658:from the origin
1657:
1655:
1654:
1649:
1647:
1631:
1629:
1628:
1623:
1621:
1613:
1598:
1596:
1595:
1590:
1570:
1568:
1560:
1552:
1528:
1526:
1525:
1520:
1495:
1493:
1492:
1487:
1475:
1473:
1472:
1467:
1465:
1463:
1455:
1447:
1432:
1430:
1429:
1424:
1403:
1401:
1400:
1395:
1337:tangential speed
1334:
1332:
1331:
1326:
1324:
1313:
1297:
1295:
1294:
1289:
1287:
1286:
1278:
1272:
1271:
1263:
1123:
1121:
1120:
1115:
1110:
1105:
1097:
1096:
1088:
1068:
1066:
1065:
1060:
1055:
1018:angular position
1003:
1001:
1000:
995:
993:
992:
984:
973:
967:angular velocity
951:
944:
937:
924:
919:
918:
911:
907:
906:
812:Johann Bernoulli
807:Daniel Bernoulli
728:Tangential speed
632:
608:
583:Fictitious force
578:
430:Mechanical power
420:
361:Angular momentum
259:
257:
256:
251:
249:
247:
239:
238:
229:
224:
223:
197:
196:
191:
189:
188:
183:
181:
180:
172:
171:
149:
131:other quantities
129:Derivations from
119:
66:
46:
33:
24:Angular velocity
21:
20:
6134:
6133:
6129:
6128:
6127:
6125:
6124:
6123:
6084:
6083:
6082:
6077:
6066:
6065:
6060:
6050:
6049:
6044:
6026:
6025:
6020:
6010:
6009:
6004:
5984:
5983:
5978:
5968:
5958:
5945:
5944:
5939:
5927:
5909:
5908:
5903:
5893:
5883:
5870:
5869:
5867:
5861:
5851:
5829:
5826:
5823:
5815:
5804:
5801:
5795:
5792:angular impulse
5785:
5765:
5762:
5759:
5751:
5736:
5733:
5728:
5718:
5707:
5704:
5682:
5681:
5679:
5676:
5675:
5674:
5654:
5651:
5646:
5619:
5612:
5607:
5590:
5588:
5581:
5566:
5561:
5541:
5538:
5531:
5521:
5516:
5505:
5500:
5480:
5477:
5470:
5460:
5446:
5438:
5433:
5426:
5421:
5414:
5409:
5402:
5397:
5392:
5386:
5376:
5371:
5361:
5347:
5341:
5336:
5324:
5321:
5316:
5310:
5303:
5293:
5278:
5273:
5263:
5253:
5247:
5241:
5236:
5230:
5209:
5204:
5189:
5184:
5174:
5169:
5153:
5146:
5108:
5100:
5037:
5027:
5001:
4985:
4984:
4977:
4973:
4963:
4961:
4952:
4948:
4940:
4920:
4916:
4907:
4906:
4902:
4892:
4876:
4872:
4864:
4848:
4844:
4839:
4792:
4787:
4786:
4775:position vector
4758:
4756:
4753:
4752:
4733:
4722:
4714:
4712:
4709:
4708:
4683:
4678:
4675:
4674:
4642:
4638:
4629:
4625:
4616:
4612:
4601:
4599:
4596:
4595:
4570:
4569:
4564:
4558:
4554:
4552:
4546:
4542:
4536:
4535:
4529:
4525:
4520:
4515:
4509:
4505:
4502:
4501:
4495:
4491:
4489:
4483:
4479:
4474:
4464:
4463:
4455:
4452:
4451:
4436:
4428:
4401:
4399:
4398:
4384:
4382:
4381:
4367:
4365:
4364:
4362:
4359:
4358:
4333:
4331:
4330:
4316:
4315:
4292:
4291:
4274:
4272:
4271:
4248:
4247:
4215:
4214:
4197:
4195:
4194:
4171:
4170:
4138:
4137:
4126:
4124:
4121:
4120:
4096:
4091:
4090:
4079:
4078:
4069:
4064:
4063:
4052:
4051:
4042:
4037:
4036:
4025:
4024:
4016:
4014:
4011:
4010:
4004:Euler rotations
3969:
3944:
3942:
3939:
3938:
3906:
3902:
3896:
3888:
3887:
3885:
3877:
3875:
3872:
3871:
3817:
3812:
3811:
3809:
3806:
3805:
3784:
3777:
3772:
3771:
3767:
3765:
3756:
3745:
3743:
3742:
3741:
3739:
3736:
3735:
3712:
3707:
3706:
3695:
3690:
3689:
3680:
3669:
3667:
3666:
3665:
3664:
3660:
3651:
3646:
3645:
3634:
3629:
3628:
3619:
3608:
3606:
3605:
3604:
3603:
3599:
3590:
3585:
3584:
3573:
3568:
3567:
3558:
3547:
3545:
3544:
3543:
3542:
3538:
3530:
3528:
3525:
3524:
3504:
3499:
3498:
3489:
3484:
3483:
3474:
3469:
3468:
3466:
3463:
3462:
3459:
3433:
3428:
3427:
3426:
3418:
3407:
3392:
3387:
3386:
3384:
3383:
3369:
3368:
3366:
3363:
3362:
3337:
3336:
3334:
3331:
3330:
3312:
3307:
3306:
3305:
3303:
3300:
3299:
3268:
3264:
3255:
3251:
3242:
3238:
3229:
3225:
3223:
3220:
3219:
3196:
3175:
3167:
3158:
3153:
3152:
3150:
3147:
3146:
3118:
3114:
3108:
3100:
3099:
3097:
3089:
3087:
3084:
3083:
3046:
3021:
3019:
3011:
3001:
2993:
2991:
2983:
2972:
2970:
2967:
2966:
2935:
2932:
2931:
2915:
2913:
2910:
2909:
2885:
2883:
2880:
2879:
2873:right-hand rule
2828:
2789:
2788:
2786:
2783:
2782:
2760:
2759:
2754:
2751:
2750:
2728:
2727:
2725:
2722:
2721:
2699:
2698:
2696:
2693:
2692:
2676:
2663:
2657:
2653:
2651:
2649:
2646:
2645:
2616:
2615:
2604:
2603:
2586:
2585:
2574:
2573:
2517:
2516:
2460:
2459:
2457:
2454:
2453:
2410:
2409:
2377:
2376:
2347:
2346:
2314:
2313:
2297:
2291:
2287:
2285:
2283:
2280:
2279:
2263:
2260:
2259:
2243:
2240:
2239:
2175:
2173:
2170:
2169:
2127:
2125:
2117:
2114:
2072:
2068:
2066:
2063:
2062:
2046:
2043:
2042:
2026:
2023:
2022:
2006:
2004:
2001:
2000:
1983:
1978:
1977:
1975:
1972:
1971:
1954:
1950:
1948:
1945:
1944:
1916:
1912:
1910:
1897:
1889:
1887:
1879:
1876:
1847:
1842:
1841:
1839:
1836:
1835:
1818:
1813:
1812:
1810:
1807:
1806:
1789:
1784:
1783:
1774:
1769:
1768:
1760:
1758:
1755:
1754:
1738:
1735:
1734:
1706:
1703:
1702:
1683:
1680:
1679:
1663:
1660:
1659:
1643:
1641:
1638:
1637:
1612:
1604:
1601:
1600:
1561:
1553:
1551:
1534:
1531:
1530:
1505:
1502:
1501:
1496:is measured in
1481:
1478:
1477:
1456:
1448:
1446:
1438:
1435:
1434:
1409:
1406:
1405:
1389:
1386:
1385:
1358:
1353:
1320:
1309:
1307:
1304:
1303:
1277:
1276:
1262:
1261:
1259:
1256:
1255:
1234:For example, a
1217:right-hand rule
1106:
1101:
1087:
1086:
1084:
1081:
1080:
1051:
1040:
1037:
1036:
983:
982:
980:
977:
976:
971:
955:
914:
901:
900:
893:
892:
891:
766:
758:
757:
737:
691:Circular motion
685:
675:
674:
673:
630:
600:
597:
576:
555:
547:
546:
543:
542:
500:
490:
482:
481:
480:
439:
435:Mechanical work
428:
412:
350:
342:
341:
340:
295:
287:
264:
240:
234:
230:
228:
219:
218:
216:
213:
212:
173:
167:
166:
165:
163:
160:
159:
135:
132:
130:
120:
115:
114:
113:Behaviour under
64:
44:
40:
17:
12:
11:
5:
6132:
6122:
6121:
6116:
6111:
6109:Temporal rates
6106:
6101:
6096:
6079:
6078:
6074:
6073:
6054:
6038:
6036:
6033:
6014:
5998:
5996:
5992:
5991:
5952:
5921:
5919:
5916:
5877:
5865:
5845:
5843:
5839:
5838:
5809:
5779:
5777:
5774:
5745:
5712:
5689:
5686:
5671:Mass flow rate
5668:
5664:
5663:
5661:
5659:
5640:
5637:
5635:
5616:
5601:
5597:
5596:
5594:
5575:
5573:
5570:
5568:
5555:
5553:
5549:
5548:
5546:
5525:
5514:
5511:
5509:
5494:
5492:
5488:
5487:
5485:
5454:
5407:
5404:
5380:
5355:
5334:
5330:
5329:
5314:
5287:
5285:
5282:
5267:
5224:
5222:
5218:
5217:
5215:
5213:
5198:
5195:
5193:
5178:
5163:
5159:
5158:
5151:
5144:
5141:
5138:
5135:
5132:
5129:
5125:
5124:
5121:
5119:
5113:
5110:
5109:
5099:
5098:
5091:
5084:
5076:
5070:
5069:
5048:
5036:
5035:External links
5033:
5032:
5031:
5025:
5013:Lifshitz, E.M.
5005:
4999:
4983:
4982:
4971:
4946:
4938:
4914:
4900:
4890:
4870:
4862:
4841:
4840:
4838:
4835:
4834:
4833:
4828:
4823:
4818:
4813:
4811:Areal velocity
4808:
4803:
4798:
4791:
4788:
4761:
4749:
4748:
4736:
4732:
4729:
4725:
4721:
4717:
4693:
4690:
4686:
4682:
4650:
4645:
4641:
4637:
4632:
4628:
4624:
4619:
4615:
4611:
4608:
4604:
4588:
4587:
4574:
4568:
4565:
4561:
4557:
4553:
4549:
4545:
4541:
4538:
4537:
4532:
4528:
4524:
4521:
4519:
4516:
4512:
4508:
4504:
4503:
4498:
4494:
4490:
4486:
4482:
4478:
4475:
4473:
4470:
4469:
4467:
4462:
4459:
4437:
4429:
4427:
4424:
4408:
4404:
4397:
4391:
4387:
4380:
4374:
4370:
4355:
4354:
4340:
4336:
4329:
4323:
4320:
4314:
4311:
4308:
4305:
4299:
4296:
4290:
4287:
4281:
4277:
4270:
4267:
4264:
4261:
4255:
4252:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4222:
4219:
4213:
4210:
4204:
4200:
4193:
4190:
4187:
4184:
4178:
4175:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4145:
4142:
4136:
4133:
4129:
4113:
4112:
4099:
4094:
4086:
4083:
4077:
4072:
4067:
4059:
4056:
4050:
4045:
4040:
4032:
4029:
4023:
4019:
4006:). Therefore:
3999:
3998:
3995:
3992:
3981:Leonhard Euler
3968:
3965:
3947:
3928:
3927:
3916:
3909:
3905:
3899:
3895:
3891:
3884:
3880:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3820:
3815:
3790:
3787:
3780:
3775:
3770:
3764:
3759:
3752:
3748:
3732:
3731:
3720:
3715:
3710:
3704:
3698:
3693:
3688:
3683:
3676:
3672:
3663:
3659:
3654:
3649:
3643:
3637:
3632:
3627:
3622:
3615:
3611:
3602:
3598:
3593:
3588:
3582:
3576:
3571:
3566:
3561:
3554:
3550:
3541:
3537:
3533:
3507:
3502:
3497:
3492:
3487:
3482:
3477:
3472:
3458:
3455:
3454:
3453:
3442:
3436:
3431:
3425:
3421:
3417:
3414:
3410:
3406:
3400:
3395:
3390:
3382:
3376:
3373:
3344:
3341:
3315:
3310:
3271:
3267:
3263:
3258:
3254:
3250:
3245:
3241:
3237:
3232:
3228:
3195:
3192:
3191:
3190:
3178:
3174:
3170:
3166:
3161:
3156:
3140:
3139:
3128:
3121:
3117:
3111:
3107:
3103:
3096:
3092:
3065:
3064:
3053:
3049:
3043:
3039:
3036:
3033:
3030:
3027:
3024:
3018:
3014:
3007:
3004:
2999:
2996:
2990:
2986:
2982:
2979:
2975:
2951:
2948:
2945:
2942:
2939:
2918:
2888:
2827:
2824:
2796:
2793:
2767:
2764:
2758:
2735:
2732:
2706:
2703:
2679:
2675:
2669:
2666:
2660:
2656:
2623:
2620:
2611:
2608:
2602:
2599:
2593:
2590:
2581:
2578:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2524:
2521:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2467:
2464:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2417:
2414:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2384:
2381:
2375:
2372:
2369:
2366:
2363:
2360:
2354:
2351:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2321:
2318:
2312:
2309:
2303:
2300:
2294:
2290:
2267:
2247:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2178:
2154:
2149:
2145:
2142:
2139:
2136:
2133:
2130:
2124:
2121:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2075:
2071:
2050:
2030:
2009:
1986:
1981:
1957:
1953:
1929:
1924:
1919:
1915:
1909:
1903:
1900:
1895:
1892:
1886:
1883:
1850:
1845:
1821:
1816:
1792:
1787:
1782:
1777:
1772:
1767:
1763:
1742:
1722:
1719:
1716:
1713:
1710:
1687:
1678:to a particle
1667:
1646:
1619:
1616:
1611:
1608:
1588:
1585:
1582:
1579:
1576:
1573:
1567:
1564:
1559:
1556:
1550:
1547:
1544:
1541:
1538:
1518:
1515:
1512:
1509:
1485:
1462:
1459:
1454:
1451:
1445:
1442:
1422:
1419:
1416:
1413:
1393:
1357:
1354:
1352:
1349:
1323:
1319:
1316:
1312:
1284:
1281:
1275:
1269:
1266:
1250:) parallel to
1166:
1165:
1155:
1113:
1109:
1104:
1100:
1094:
1091:
1058:
1054:
1050:
1047:
1044:
990:
987:
957:
956:
954:
953:
946:
939:
931:
928:
927:
926:
925:
912:
895:
894:
890:
889:
884:
879:
874:
869:
864:
859:
854:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
768:
767:
764:
763:
760:
759:
756:
755:
736:
735:
730:
725:
720:
718:Coriolis force
715:
714:
713:
703:
698:
693:
687:
686:
681:
680:
677:
676:
672:
671:
666:
661:
660:
659:
654:
644:
639:
634:
627:
616:
615:
614:
609:
596:
595:
590:
585:
580:
573:
568:
563:
557:
556:
553:
552:
549:
548:
545:
544:
541:
540:
535:
530:
525:
520:
515:
509:
503:
501:
494:
491:
488:
487:
484:
483:
479:
478:
473:
468:
463:
458:
453:
448:
443:
437:
432:
426:
421:
410:
405:
400:
395:
390:
389:
388:
383:
373:
368:
363:
358:
352:
351:
348:
347:
344:
343:
339:
338:
333:
328:
323:
318:
313:
308:
303:
297:
296:
293:
292:
289:
288:
286:
285:
280:
275:
269:
266:
265:
260:
246:
243:
237:
233:
227:
209:
208:
202:
201:
193:
192:
179:
176:
170:
157:
151:
150:
133:
128:
125:
124:
121:
112:
109:
108:
105:
98:
97:
90:
83:
82:
79:
72:
71:
68:
59:
58:
55:
49:
48:
41:
39:Common symbols
38:
35:
34:
26:
25:
15:
9:
6:
4:
3:
2:
6131:
6120:
6117:
6115:
6112:
6110:
6107:
6105:
6102:
6100:
6097:
6095:
6092:
6091:
6089:
6072:
6063:
6058:
6055:
6047:
6042:
6039:
6037:
6034:
6032:
6023:
6018:
6015:
6007:
6002:
5999:
5997:
5994:
5993:
5990:
5981:
5976:
5971:
5966:
5961:
5956:
5953:
5951:
5942:
5937:
5932:
5931:
5925:
5922:
5920:
5917:
5915:
5906:
5901:
5896:
5891:
5886:
5881:
5878:
5876:
5864:
5859:
5854:
5849:
5846:
5844:
5841:
5840:
5837:
5833:
5821:
5813:
5810:
5808:
5799:
5793:
5788:
5783:
5780:
5778:
5775:
5773:
5769:
5757:
5749:
5746:
5744:
5740:
5731:
5726:
5721:
5716:
5713:
5711:
5687:
5684:
5672:
5669:
5666:
5665:
5662:
5660:
5658:
5649:
5644:
5641:
5638:
5636:
5633:
5630:
5626:
5622:
5617:
5615:
5610:
5605:
5602:
5599:
5598:
5595:
5586:
5585:
5579:
5576:
5574:
5571:
5569:
5564:
5559:
5556:
5554:
5551:
5550:
5547:
5545:
5536:
5535:
5529:
5526:
5524:
5519:
5515:
5512:
5510:
5508:
5503:
5498:
5495:
5493:
5490:
5489:
5486:
5484:
5475:
5474:
5468:
5463:
5458:
5457:angular speed
5455:
5453:
5449:
5443:
5442:
5436:
5429:
5424:
5417:
5412:
5408:
5405:
5400:
5395:
5389:
5384:
5381:
5379:
5374:
5369:
5364:
5359:
5356:
5354:
5350:
5344:
5339:
5335:
5332:
5331:
5327:
5319:
5315:
5313:
5308:
5307:
5301:
5296:
5291:
5288:
5286:
5283:
5281:
5276:
5271:
5268:
5266:
5262:
5256:
5250:
5244:
5239:
5233:
5228:
5225:
5223:
5220:
5219:
5216:
5214:
5212:
5207:
5202:
5199:
5196:
5194:
5192:
5187:
5182:
5179:
5177:
5172:
5167:
5164:
5161:
5160:
5156:
5152:
5149:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5126:
5116:
5111:
5107:
5104:
5097:
5092:
5090:
5085:
5083:
5078:
5077:
5074:
5066:
5062:
5058:
5057:Sixty Symbols
5054:
5049:
5046:
5042:
5039:
5038:
5028:
5022:
5018:
5014:
5010:
5006:
5002:
4996:
4992:
4987:
4986:
4980:
4975:
4959:
4958:
4950:
4941:
4935:
4931:
4927:
4926:
4918:
4910:
4904:
4898:
4893:
4887:
4883:
4882:
4874:
4865:
4859:
4855:
4854:
4846:
4842:
4832:
4829:
4827:
4824:
4822:
4819:
4817:
4814:
4812:
4809:
4807:
4804:
4802:
4799:
4797:
4794:
4793:
4784:
4783:
4778:
4776:
4727:
4719:
4707:
4706:
4705:
4688:
4673:
4672:cross product
4669:
4668:
4662:
4643:
4639:
4635:
4630:
4626:
4622:
4617:
4613:
4606:
4593:
4572:
4566:
4559:
4555:
4547:
4543:
4539:
4530:
4526:
4522:
4517:
4510:
4506:
4496:
4492:
4484:
4480:
4476:
4471:
4465:
4460:
4450:
4449:
4448:
4446:
4442:
4434:
4423:
4395:
4378:
4321:
4318:
4312:
4309:
4306:
4303:
4297:
4294:
4285:
4265:
4262:
4259:
4253:
4250:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4220:
4217:
4208:
4188:
4185:
4182:
4176:
4173:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4143:
4140:
4131:
4119:
4118:
4117:
4097:
4084:
4081:
4075:
4070:
4057:
4054:
4048:
4043:
4030:
4027:
4021:
4009:
4008:
4007:
4005:
3996:
3993:
3990:
3989:
3988:
3986:
3982:
3973:
3964:
3962:
3937:
3933:
3914:
3907:
3903:
3893:
3882:
3870:
3869:
3868:
3866:
3861:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
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1955:
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1241:
1237:
1236:geostationary
1232:
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1211:
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1203:
1199:
1195:
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1137:
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1131:
1127:
1111:
1107:
1098:
1078:
1074:
1073:
1072:angular speed
1045:
1042:
1033:
1031:
1027:
1023:
1019:
1015:
1011:
1007:
985:
974:
968:
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754:
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244:
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211:
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56:
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50:
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42:
36:
32:
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22:
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5786:
5729:
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5647:
5631:
5628:
5624:
5620:
5608:
5583:
5582:
5578:angular jerk
5562:
5533:
5532:
5501:
5497:acceleration
5472:
5471:
5466:
5461:
5440:
5439:
5427:
5415:
5398:
5387:
5372:
5362:
5342:
5305:
5304:
5294:
5274:
5261:displacement
5254:
5248:
5242:
5231:
5205:
5185:
5170:
5154:
5147:
5056:
5044:
5016:
5009:Landau, L.D.
4990:
4974:
4962:. Retrieved
4956:
4949:
4924:
4917:
4903:
4880:
4873:
4852:
4845:
4780:
4750:
4665:
4663:
4589:
4447:defined by:
4438:
4356:
4114:
4000:
3985:Euler angles
3978:
3960:
3935:
3932:single point
3931:
3929:
3864:
3862:
3733:
3460:
3297:
3286:
3212:
3204:rigid bodies
3201:
3197:
3141:
3076:
3072:
3068:
3066:
2905:
2901:
2877:
2868:
2864:
2860:
2856:
2852:
2849:pseudovector
2844:
2838:
2816:pseudoscalar
2812:
2643:
2167:
2111:
1942:
1873:
1868:
1866:
1634:
1383:
1377:
1369:
1365:
1340:
1243:
1233:
1214:
1208:in units of
1205:
1167:
1157:
1143:
1138:
1076:
1070:
1034:
1014:pseudovector
1009:
970:
966:
960:
752:
751: /
747: /
745:displacement
743: /
604: /
566:Displacement
504:
495:
489:Formulations
476:Virtual work
416: /
356:Acceleration
349:Fundamentals
146:
142:
137:
136:
123:pseudovector
53:SI unit
43:
18:
5318:solid angle
5061:Brady Haran
4664:This is an
4594:components
2639:Unit vector
1698:, with its
1248:unit vector
1219:, implying
1188:, although
1022:orientation
887:von Neumann
554:Core topics
6088:Categories
6053:m s, N m s
6029:m s,
5975:Lagrangian
5900:Lagrangian
5140:Dimensions
5128:Dimensions
4837:References
3983:using his
2720:, because
2109:, so that
1599:, so that
822:d'Alembert
802:Maupertuis
765:Scientists
647:Rigid body
321:Kinematics
94:rigid body
6069:ms,
5688:˙
5411:frequency
5338:frequency
5017:Mechanics
4991:Mechanics
4831:Vorticity
4731:Ω
4720:×
4716:ω
4689:×
4685:ω
4640:ω
4627:ω
4614:ω
4603:ω
4556:ω
4544:ω
4540:−
4527:ω
4523:−
4507:ω
4493:ω
4481:ω
4477:−
4458:Ω
4407:^
4390:^
4373:^
4339:^
4322:˙
4319:γ
4310:β
4307:
4298:˙
4295:α
4280:^
4266:γ
4263:
4254:˙
4251:β
4245:−
4242:γ
4239:
4233:β
4230:
4221:˙
4218:α
4203:^
4189:γ
4186:
4177:˙
4174:β
4165:γ
4162:
4156:β
4153:
4144:˙
4141:α
4128:ω
4085:˙
4082:γ
4058:˙
4055:β
4031:˙
4028:α
4018:ω
3946:ω
3894:×
3879:ω
3751:˙
3687:⋅
3675:˙
3626:⋅
3614:˙
3565:⋅
3553:˙
3532:ω
3424:−
3413:×
3409:ω
3399:˙
3375:˙
3343:˙
3266:ω
3253:ω
3240:ω
3227:ω
3173:×
3169:ω
3160:⊥
3106:×
3091:ω
3035:θ
3029:
2998:ϕ
2981:ω
2974:ω
2947:ϕ
2875:is used.
2795:^
2792:φ
2766:˙
2763:φ
2734:^
2705:˙
2622:^
2619:φ
2610:˙
2607:φ
2592:^
2580:˙
2562:φ
2556:
2544:φ
2538:
2532:−
2523:˙
2520:φ
2502:φ
2496:
2484:φ
2478:
2466:˙
2431:φ
2425:
2416:˙
2413:φ
2398:φ
2392:
2383:˙
2368:φ
2362:
2353:˙
2350:φ
2341:−
2335:φ
2329:
2320:˙
2266:φ
2220:φ
2214:
2199:φ
2193:
2141:θ
2135:
2120:ω
2094:θ
2088:
2074:⊥
2049:θ
1985:⊥
1956:⊥
1918:⊥
1894:ϕ
1882:ω
1849:⊥
1820:‖
1791:⊥
1776:‖
1718:ϕ
1607:ω
1578:ω
1558:ℓ
1517:ϕ
1508:ℓ
1484:ϕ
1453:ϕ
1441:ω
1412:ϕ
1322:ω
1298:, in the
1283:^
1268:^
1265:ω
1221:clockwise
1170:dimension
1112:ω
1103:ω
1093:^
1090:ω
1057:‖
1053:ω
1049:‖
1043:ω
1030:direction
989:→
986:ω
867:Liouville
749:frequency
669:Vibration
386:potential
311:Continuum
306:Celestial
283:Textbooks
175:−
155:Dimension
102:Conserved
92:yes (for
87:Intensive
76:Extensive
6119:Velocity
6013:m s, N s
5836:J s
5832:m s
5807:m s
5768:m s
5739:m s
5715:momentum
5368:velocity
5238:position
5227:distance
5181:absement
5106:SI units
5063:for the
5015:(1997).
4816:Isometry
4790:See also
2781:because
2644:Knowing
2238:, being
1345:prograde
1225:negation
1178:distance
1174:velocity
969:(symbol
922:Category
847:Hamilton
832:Lagrange
827:Clairaut
792:Horrocks
753:velocity
723:Pendulum
711:reactive
683:Rotation
652:dynamics
602:Inertial
588:Friction
471:Velocity
446:Momentum
326:Kinetics
316:Dynamics
294:Branches
278:Timeline
6114:Tensors
6041:rotatum
5820:actergy
5756:actergy
5725:impulse
5645::
5396::
5326:rad, sr
3865:orbital
1498:radians
1240:equator
1182:SI unit
1026:rotates
1012:, is a
963:physics
882:Koopman
842:Poisson
837:Laplace
782:Huygens
777:Galileo
622: (
561:Damping
414:Inertia
408:Impulse
381:kinetic
331:Statics
301:Applied
273:History
5955:energy
5936:moment
5924:torque
5880:energy
5858:weight
5830:
5812:action
5805:
5766:
5748:action
5737:
5708:
5655:
5627:⟩ = ∑
5591:
5542:
5481:
5023:
4997:
4964:21 May
4936:
4888:
4860:
4751:where
4426:Tensor
4357:where
3936:single
3734:where
3216:gimbal
3208:tensor
3067:where
1194:radian
1152:origin
1126:normal
920:
872:Appell
857:Cauchy
852:Jacobi
797:Halley
787:Newton
772:Kepler
624:linear
620:Motion
466:Torque
441:Moment
376:Energy
366:Couple
6094:Angle
6057:power
6017:power
5987:m s,
5948:m s,
5912:m s,
5873:m s,
5848:force
5358:speed
5290:angle
4944:(EM1)
4868:(UP1)
4773:is a
4443:is a
2637:(see
1476:. If
1210:hertz
1196:is a
1006:omega
877:Gibbs
862:Routh
817:Euler
456:Speed
451:Space
393:Force
96:only)
6001:yank
5965:work
5890:work
5604:mass
5558:jerk
5270:area
5201:time
5166:time
5021:ISBN
4995:ISBN
4966:2021
4934:ISBN
4886:ISBN
4858:ISBN
4439:The
3075:and
2904:and
2863:and
2839:In
1075:(or
461:Time
424:Mass
6035:MLT
5950:N m
5918:MLT
5871:kg
5776:MLT
5772:J s
5743:N s
5589:rad
5567:m s
5540:rad
5507:m s
5479:rad
5403:m s
5378:m s
5312:rad
5191:m s
4304:cos
4260:sin
4236:cos
4227:sin
4183:cos
4159:sin
4150:sin
3961:all
3287:By
3026:sin
2869:two
2835:up.
2553:cos
2535:sin
2493:sin
2475:cos
2422:cos
2389:sin
2359:sin
2326:cos
2211:sin
2190:cos
2132:sin
2085:sin
1136:.
1132:or
1124:is
1032:.
1020:or
975:or
961:In
145:/ d
141:= d
81:yes
62:In
6090::
6067:kg
6059::
6051:kg
6043::
6027:kg
6019::
6011:kg
6003::
5995:MT
5985:kg
5977::
5973:,
5967::
5963:,
5957::
5946:kg
5938::
5934:,
5926::
5910:kg
5902::
5898:,
5892::
5888:,
5882::
5860::
5856:,
5850::
5842:MT
5834:,
5828:kg
5822::
5818:,
5816:𝒮
5814::
5803:kg
5794::
5790:,
5784::
5770:,
5764:kg
5758::
5754:,
5752:𝒮
5750::
5741:,
5735:kg
5727::
5723:,
5717::
5706:kg
5673::
5667:MT
5653:kg
5639:ML
5614:kg
5606::
5580::
5560::
5530::
5499::
5469::
5465:,
5459::
5452:Hz
5450:,
5437::
5432:,
5425::
5420:,
5413::
5385::
5370::
5366:,
5360::
5353:Hz
5351:,
5340::
5320::
5302::
5298:,
5292::
5272::
5259:,
5252:,
5246:,
5240::
5235:,
5229::
5203::
5183::
5168::
5059:.
5055:.
5011:;
4928:.
4777:.
4704::
4661:.
3284:.
3210:.
1632:.
1231:.
1154:.
965:,
107:no
6071:W
6062:P
6046:P
6031:W
6022:P
6006:Y
5989:J
5980:L
5970:W
5960:E
5941:M
5930:τ
5914:J
5905:L
5895:W
5885:E
5875:N
5866:g
5863:F
5853:F
5824:ℵ
5798:L
5796:Δ
5787:L
5760:ℵ
5730:J
5720:p
5710:s
5685:m
5657:m
5648:I
5632:x
5629:m
5625:x
5623:⟨
5621:M
5609:m
5600:M
5593:s
5584:ζ
5572:T
5563:j
5552:T
5544:s
5534:α
5523:s
5513:T
5502:a
5491:T
5483:s
5473:ω
5462:ω
5448:s
5441:n
5428:n
5416:f
5406:T
5399:h
5391:,
5388:ν
5373:v
5363:v
5349:s
5343:f
5333:T
5322:Ω
5306:θ
5295:θ
5284:1
5280:m
5275:A
5265:m
5255:x
5249:s
5243:r
5232:d
5221:1
5211:s
5206:t
5197:T
5186:A
5176:s
5171:t
5162:T
5155:θ
5148:θ
5143:1
5137:L
5134:L
5131:1
5095:e
5088:t
5081:v
5067:.
5047:)
5029:.
5003:.
4942:.
4911:.
4894:.
4866:.
4785:.
4760:r
4735:r
4728:=
4724:r
4692:)
4681:(
4649:)
4644:z
4636:,
4631:y
4623:,
4618:x
4610:(
4607:=
4573:)
4567:0
4560:x
4548:y
4531:x
4518:0
4511:z
4497:y
4485:z
4472:0
4466:(
4461:=
4435:.
4403:k
4396:,
4386:j
4379:,
4369:i
4335:k
4328:)
4313:+
4289:(
4286:+
4276:j
4269:)
4212:(
4209:+
4199:i
4192:)
4168:+
4135:(
4132:=
4098:3
4093:u
4076:+
4071:2
4066:u
4049:+
4044:1
4039:u
4022:=
3915:,
3908:2
3904:r
3898:v
3890:r
3883:=
3848:,
3845:3
3842:,
3839:2
3836:,
3833:1
3830:=
3827:i
3824:,
3819:i
3814:e
3789:t
3786:d
3779:i
3774:e
3769:d
3763:=
3758:i
3747:e
3719:,
3714:2
3709:e
3703:)
3697:1
3692:e
3682:3
3671:e
3662:(
3658:+
3653:1
3648:e
3642:)
3636:3
3631:e
3621:2
3610:e
3601:(
3597:+
3592:3
3587:e
3581:)
3575:2
3570:e
3560:1
3549:e
3540:(
3536:=
3506:3
3501:e
3496:,
3491:2
3486:e
3481:,
3476:1
3471:e
3441:)
3435:0
3430:r
3420:r
3416:(
3405:+
3394:0
3389:r
3381:=
3372:r
3340:r
3314:0
3309:r
3270:1
3262:+
3257:2
3249:=
3244:2
3236:+
3231:1
3177:r
3165:=
3155:v
3127:.
3120:2
3116:r
3110:v
3102:r
3095:=
3077:v
3073:r
3069:θ
3052:,
3048:u
3042:r
3038:)
3032:(
3023:v
3017:=
3013:u
3006:t
3003:d
2995:d
2989:=
2985:u
2978:=
2950:)
2944:,
2941:r
2938:(
2917:u
2906:v
2902:r
2887:u
2865:v
2861:r
2857:r
2853:r
2845:r
2757:r
2731:r
2702:r
2678:v
2674:=
2668:t
2665:d
2659:r
2655:d
2601:r
2598:+
2589:r
2577:r
2571:=
2568:)
2565:)
2559:(
2550:,
2547:)
2541:(
2529:(
2514:r
2511:+
2508:)
2505:)
2499:(
2490:,
2487:)
2481:(
2472:(
2463:r
2440:,
2437:)
2434:)
2428:(
2407:r
2404:+
2401:)
2395:(
2380:r
2374:,
2371:)
2365:(
2344:r
2338:)
2332:(
2317:r
2311:(
2308:=
2302:t
2299:d
2293:r
2289:d
2246:r
2226:)
2223:)
2217:(
2208:r
2205:,
2202:)
2196:(
2187:r
2184:(
2181:=
2177:r
2153:.
2148:r
2144:)
2138:(
2129:v
2123:=
2097:)
2091:(
2082:v
2079:=
2070:v
2029:v
2008:v
1980:v
1952:v
1928:.
1923:r
1914:v
1908:=
1902:t
1899:d
1891:d
1885:=
1869:ω
1844:v
1815:v
1786:v
1781:+
1771:v
1766:=
1762:v
1741:t
1721:)
1715:,
1712:r
1709:(
1686:P
1666:O
1645:r
1618:r
1615:v
1610:=
1587:)
1584:t
1581:(
1575:r
1572:=
1566:t
1563:d
1555:d
1549:=
1546:)
1543:t
1540:(
1537:v
1514:r
1511:=
1461:t
1458:d
1450:d
1444:=
1421:)
1418:t
1415:(
1392:r
1380:.
1378:v
1370:O
1366:P
1341:v
1318:r
1315:=
1311:v
1280:Z
1274:=
1254:(
1244:ω
1108:/
1099:=
1046:=
972:ω
950:e
943:t
936:v
626:)
245:t
242:d
236:p
232:d
226:=
221:F
178:1
169:T
147:t
143:θ
138:ω
104:?
89:?
78:?
70:s
45:ω
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