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38:
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reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity – via the pitching motion – elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) rather than causing the spinning top to fall to its side.
1516:
27:
1817:
In the adjunct image, Earth's apsidal precession is illustrated. As the Earth travels around the Sun, its elliptical orbit rotates gradually over time. The eccentricity of its ellipse and the precession rate of its orbit are exaggerated for visualization. Most orbits in the Solar System have a much
1809:
do not really follow an identical ellipse each time, but actually trace out a flower-petal shape because the major axis of each planet's elliptical orbit also precesses within its orbital plane, partly in response to perturbations in the form of the changing gravitational forces exerted by other
1143:
Due to the way the torque vectors are defined, it is a vector that is perpendicular to the plane of the forces that create it. Thus it may be seen that the angular momentum vector will change perpendicular to those forces. Depending on how the forces are created, they will often rotate with the
987:
In the discussion above, the setup was kept unchanging by preventing pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over, the spinning top just pitches a little. This pitching motion
443:
181:
1153:
1679:(c. 190–120 BC) is generally accepted to be the earliest known astronomer to recognize and assess the precession of the equinoxes at about 1° per century (which is not far from the actual value for antiquity, 1.38°), although there is some minor dispute about whether he was. In
980:
The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some spinning is added to rotation around a vertical axis.
651:
894:
First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (around the wheelhub) is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a
575:
1363:
368:
799:{\displaystyle E\left(\exp \left(\left_{\times }\right){\boldsymbol {R}}\right)\approx E\left({\boldsymbol {R}}\right)+\left({\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\times {\boldsymbol {L}}\right)\cdot {\boldsymbol {v}}}
1766:
Precessional movement of the axis (left), precession of the equinox in relation to the distant stars (middle), and the path of the north celestial pole among the stars due to the precession. Vega is the bright star near the bottom
1094:
1607:
In astronomy, precession refers to any of several gravity-induced, slow and continuous changes in an astronomical body's rotational axis or orbital path. Precession of the equinoxes, perihelion precession, changes in the
589:
171:). If an object is asymmetric about its principal axis of rotation, the moment of inertia with respect to each coordinate direction will change with time, while preserving angular momentum. The result is that the
1276:{\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {\ mgr}{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}}={\frac {\tau }{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }\sin(\theta )}}}
96:. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called
472:
272:{\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {{\boldsymbol {I}}_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}{{\boldsymbol {I}}_{\mathrm {p} }\cos({\boldsymbol {\alpha }})}}}
865:
1641:
Axial precession is the movement of the rotational axis of an astronomical body, whereby the axis slowly traces out a cone. In the case of Earth, this type of precession is also known as the
117:
refers to any of several slow changes in an astronomical body's rotational or orbital parameters. An important example is the steady change in the orientation of the axis of rotation of the
1653:. Earth goes through one such complete precessional cycle in a period of approximately 26,000 years or 1° every 72 years, during which the positions of stars will slowly change in both
1116:
178:
The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows:
1048:
1138:
1511:{\displaystyle T_{\mathrm {p} }={\frac {4\pi ^{2}I_{\mathrm {s} }}{\ mgrT_{\mathrm {s} }}}={\frac {4\pi ^{2}I_{\mathrm {s} }\sin(\theta )}{\ \tau T_{\mathrm {s} }}}}
452:
861:(reaction) of the ground is pushing up on it at the point of contact with the support. These two opposite forces produce a torque which causes the top to precess.
2251:(The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)
2282:
2262:"An even larger value for a precession has been found, for a black hole in orbit around a much more massive black hole, amounting to 39 degrees each orbit"
1699:
had drifted roughly one degree over the course of fifty years relative to the position of the stars. The precession of Earth's axis was later explained by
438:{\displaystyle {\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)={\boldsymbol {R}}{\boldsymbol {I}}_{0}^{-1}{\boldsymbol {R}}^{T}{\boldsymbol {L}}}
336:, inelastic dissipation will tend to damp torque-free precession, and the rotation axis will align itself with one of the inertia axes of the body.
339:
For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix
959:
tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis. Section
2261:
2129:
2090:
1692:
1564:
give three types of corrections to the
Newtonian precession, of a gyroscope near a large mass such as Earth, described above. They are:
968:
of the wheel is moving away from the pivot axis, and so a force (again, a
Coriolis force) acts in the same direction as in the case of
646:{\displaystyle E\left({\boldsymbol {R}}\right)={\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\cdot {\frac {\boldsymbol {L}}{2}}}
1841:), which accurately predicted the anomalies. Deviating from Newton's law, Einstein's theory of gravitation predicts an extra term of
1583:, a general-relativistic correction accounting for the frame dragging by the Kerr metric of curved space near a large rotating mass.
868:
The response of a rotating system to an applied torque. When the device swivels, and some roll is added, the wheel tends to pitch.
876:
mounted. From inside to outside there are three axes of rotation: the hub of the wheel, the gimbal axis, and the vertical pivot.
1924:
2105:
2066:
2229:
2191:
2152:
2049:
1577:, a general-relativistic correction accounting for the Schwarzschild metric of curved space near a large non-rotating mass.
853:
that would intuitively result from the external torque. In the case of a toy top, its weight is acting downwards from its
1571:, a special-relativistic correction accounting for an object (such as a gyroscope) being accelerated along a curved path.
984:
It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous.
133:
Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the
2203:
1958:
1330:
is the distance between the center of mass and the pivot. The torque vector originates at the center of mass. Using
2335:
345:
that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal
1796:
1592:
1045:
produced by a torque. The general equation that relates the torque to the rate of change of angular momentum is:
850:
578:
570:{\displaystyle {\boldsymbol {R}}_{\text{new}}=\exp \left(\left_{\times }dt\right){\boldsymbol {R}}_{\text{old}}}
2450:
1580:
175:
of the angular velocities of the body about each axis will vary inversely with each axis' moment of inertia.
1818:
smaller eccentricity and precess at a much slower rate, making them nearly circular and nearly stationary.
1099:
93:
1695:
307–345 AD) made a similar discovery centuries later, noting that the position of the Sun during the
2305:
149:. The inertia matrix is composed of the moments of inertia of a body calculated with respect to separate
20:
146:
2326:
2221:
1684:
1616:
of its orbit over tens of thousands of years are all important parts of the astronomical theory of
946:
is forced closer to the pivot axis of the rotation (by the wheel spinning further), because of the
346:
1121:
1919:
1778:
1588:
842:
172:
1654:
150:
77:
2039:
2041:
Mechanical
Systems, Classical Models: Volume II: Mechanics of Discrete and Continuous Systems
879:
To distinguish between the two horizontal axes, rotation around the wheel hub will be called
2196:
Science and
Civilization in China: Mathematics and the Sciences of the Heavens and the Earth
653:
this unphysical tendency can be counteracted by repeatedly applying a small rotation vector
2010:
1834:
1613:
1561:
323:
is moment of inertia about either of the other two equal perpendicular principal axes, and
586:. The errors induced by finite time steps tend to increase the rotational kinetic energy:
8:
2440:
2412:
1826:
1700:
1623:
333:
2014:
2400:
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2184:
Our Place in the
Universe: Understanding Fundamental Astronomy from Ancient Discoveries
2123:
2084:
1838:
1811:
1790:
1782:
1658:
1089:{\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}}
85:
2311:
2347:
2276:
2225:
2199:
2187:
2148:
2045:
2023:
1998:
1954:
1904:
1829:
were prominent among the forms of experimental evidence leading to the acceptance of
1727:, but instead causing it to precess. The torque exerted by the planets, particularly
1707:, Earth has a non-spherical shape, bulging outward at the equator. The gravitational
1661:. Over this cycle, Earth's north axial pole moves from where it is now, within 1° of
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vector changes orientation with time. What makes this possible is a time-varying
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Discrepancies between the observed perihelion precession rate of the planet
329:
is the angle between the moment of inertia direction and the symmetry axis.
2376:
1914:
1881:
1147:
Under these circumstances the angular velocity of precession is given by:
858:
834:
89:
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996:
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2107:
11.4 Precession of a
Gyroscope - University Physics Volume 1 | OpenStax
2068:
11.4 Precession of a
Gyroscope - University Physics Volume 1 | OpenStax
1676:
26:
2419:
1591:(sometimes Schwarzschild precession) is used in the prediction of the
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1357:
921:
822:
66:
31:
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1909:
1739:
1724:
1007:
995:
performance at high speed. Precession is also the mechanism behind
934:
98:
81:
61:
46:
2198:, vol. 3, reprint edition. Cambridge: Cambridge University Press.
1552:. In general, the problem is more complicated than this, however.
1324:
is the angle between the spin axis and the axis of precession and
1144:
angular momentum vector, and then circular precession is created.
1728:
1662:
1630:
1617:
992:
924:
of the rotating motion around the (vertical) pivot axis. Section
103:
1863:, which accurately gives the observed excess turning rate of 43
1595:
of the planets, most notably for the accurate prediction of the
1033:
in the direction of that torque. This causes the top to precess.
845:
of the external torque are constant, the spin axis will move at
2300:
1757:
1748:
1549:
1012:
896:
873:
830:
107:
37:
2312:
Explanation and derivation of formula for precession of a top
2104:
Moebs, William; Ling, Samuel J.; Sanny, Jeff (Sep 19, 2016).
2065:
Moebs, William; Ling, Samuel J.; Sanny, Jeff (Sep 19, 2016).
1802:
1688:
838:
118:
1712:
937:
with respect to the rotation around the pivot axis, and as
2371:
1140:
are the torque and angular momentum vectors respectively.
991:
Precession or gyroscopic considerations have an effect on
837:, but all rotating objects can undergo precession. If the
2264:. 18 March 2008. Archived from the original on 2018-08-07
2194:, p. 120; see also Needham, Joseph; Wang, Ling. (1995) .
1806:
1716:
833:
is applied to it. The phenomenon is commonly seen in a
2044:. Springer Science & Business Media. p. 420.
902:
In the picture, a section of the wheel has been named
2324:
2172:. Classical Philology, 86, (1991), 48–63. p. 59.
1975:"Lecture 26 – Torque-free rotation – body-fixed axes"
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1310:
is the angular velocity of spin about the spin axis,
1156:
1124:
1102:
1051:
977:. Note that both arrows point in the same direction.
887:. Rotation around the vertical pivot axis is called
883:, and rotation around the gimbal axis will be called
677:
592:
475:
371:
312:
is the moment of inertia about the axis of symmetry,
184:
1997:
Sharma, Ishan; Burns, Joseph A.; Hui, C.-H. (2005).
1719:
apply torque to the equator, attempting to pull the
16:
Periodic change in the direction of a rotation axis
2147:. New York: Taylor and Francis Group. p. 71.
1999:"Nutational damping times in solids of revolution"
1510:
1275:
1132:
1110:
1088:
798:
645:
569:
437:
271:
128:
2003:Monthly Notices of the Royal Astronomical Society
1810:planets. This is called perihelion precession or
1024:and the weight of the top causes a change in the
2432:
2281:: CS1 maint: bot: original URL status unknown (
2186:, second edition. Cham, Switzerland: Springer.
1542:is the period of spin about the spin axis, and
2103:
2064:
1996:
1631:Axial precession (precession of the equinoxes)
445:Precession occurs by repeatedly recalculating
1555:
301:is the spin rate about the axis of symmetry,
1891:For the precession of the Moon's orbit, see
92:, whereas the third Euler angle defines the
950:, with respect to the vertical pivot axis,
933:, therefore, has a lot of angular rotating
88:it can be defined as a change in the first
2128:: CS1 maint: location missing publisher (
2089:: CS1 maint: location missing publisher (
2037:
1925:Precession as a form of parallel transport
911:. At the depicted moment in time, section
2031:
2022:
1669:, with an angular radius of about 23.5°.
2167:
2142:
1777:
1006:
1002:
863:
365:, the instantaneous angular velocity is
36:
25:
1785:—the orbit rotates gradually over time.
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1943:
2317:Precession and the Milankovich theory
2215:
2209:
1950:Analytical Mechanics of Space Systems
1773:
1111:{\displaystyle {\boldsymbol {\tau }}}
106:, there are two types of precession:
2170:On the cosmical mysteries of Mithras
1972:
1560:The special and general theories of
1320:is the acceleration due to gravity,
872:The device depicted on the right is
359:and fixed external angular momentum
145:, or more precisely, a time-varying
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817:) is the phenomenon in which the
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2024:10.1111/j.1365-2966.2005.08864.x
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1738:
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1069:
332:When an object is not perfectly
2254:
2249:Einstein's Theory of Relativity
2238:
1797:Anomalous perihelion precession
1593:anomalous perihelion precession
129:Torque-free or Torque neglected
2176:
2161:
2136:
2097:
2058:
1990:
1966:
1937:
1482:
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821:of a spinning object (e.g., a
263:
255:
1:
1930:
1015:caused by the normal force –
2320:From Stargazers to Starships
2038:Teodorescu, Petre P (2002).
1839:General Theory of Relativity
1602:
1133:{\displaystyle \mathbf {L} }
1037:Precession is the change of
80:of the rotational axis of a
7:
1898:
1643:precession of the equinoxes
1360:of precession is given by:
813:Torque-induced precession (
123:precession of the equinoxes
21:Precession (disambiguation)
10:
2467:
2222:Cambridge University Press
1953:, AIAA, pp. 149–150,
1890:
1874:
1794:
1788:
1634:
1556:Relativistic (Einsteinian)
829:in space when an external
110:-free and torque-induced.
18:
2145:Fundamentals of Astronomy
2143:Barbieri, Cesare (2007).
1665:, in a circle around the
1651:precession of the equator
1581:Lense–Thirring precession
1674:ancient Greek astronomer
899:around the gimbal axis.
841:of the rotation and the
347:moment of inertia tensor
290:is the precession rate,
84:body. In an appropriate
2168:Swerdlow, Noel (1991).
1920:Precession (mechanical)
1589:Schwarzschild geodesics
137:is a constant, but the
1825:and that predicted by
1805:of planets around the
1786:
1723:into the plane of the
1655:equatorial coordinates
1612:to its orbit, and the
1512:
1277:
1134:
1112:
1090:
1034:
869:
800:
659:perpendicular to both
647:
571:
439:
273:
70:
34:
1781:
1731:, also plays a role.
1513:
1278:
1135:
1113:
1091:
1010:
1003:Classical (Newtonian)
867:
815:gyroscopic precession
801:
648:
579:skew-symmetric matrix
572:
451:and applying a small
440:
274:
40:
29:
2451:Dynamics (mechanics)
2308:at Wikimedia Commons
2216:Bradt, Hale (2007).
1973:Boal, David (2001).
1837:(in particular, his
1835:Theory of Relativity
1647:lunisolar precession
1610:tilt of Earth's axis
1575:de Sitter precession
1364:
1154:
1122:
1100:
1049:
675:
590:
473:
369:
182:
19:For other uses, see
2182:Sun, Kwok. (2017).
2015:2005MNRAS.359...79S
1827:classical mechanics
1624:Milankovitch cycles
1356:, we find that the
463:for the short time
417:
76:is a change in the
1812:apsidal precession
1791:Apsidal precession
1787:
1783:Apsidal precession
1774:Apsidal precession
1659:ecliptic longitude
1597:apsidal precession
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2304:Media related to
2231:978-0-521-53551-9
2218:Astronomy Methods
2192:978-3-319-54171-6
2154:978-0-7503-0886-1
2051:978-1-4020-8988-6
1945:Schaub, Hanspeter
1905:Larmor precession
1867:every 100 years.
1701:Newtonian physics
1687:scholar-official
1569:Thomas precession
1531:moment of inertia
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1705:oblate spheroid
1697:winter solstice
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1005:
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966:
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948:Coriolis effect
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453:rotation vector
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166:
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154:
151:coordinate axes
131:
121:, known as the
94:rotation itself
86:reference frame
64:
57:
55:
51:
49:
42:
24:
17:
12:
11:
5:
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2322:
2314:
2309:
2295:
2294:External links
2292:
2289:
2288:
2253:
2237:
2230:
2224:. p. 66.
2208:
2175:
2160:
2153:
2135:
2096:
2057:
2050:
2030:
1989:
1965:
1959:
1935:
1934:
1932:
1929:
1928:
1927:
1922:
1917:
1912:
1907:
1900:
1897:
1875:Main article:
1872:
1869:
1789:Main article:
1775:
1772:
1765:
1764:
1755:
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1734:
1733:
1635:Main article:
1632:
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855:center of mass
825:) describes a
810:
809:Torque-induced
807:
794:
790:
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781:
777:
773:
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760:
755:
751:
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735:
732:
728:
723:
718:
713:
708:
704:
700:
695:
691:
688:
684:
680:
671:, noting that
640:
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611:
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147:inertia matrix
130:
127:
113:In astronomy,
56:
50:
41:
15:
9:
6:
4:
3:
2:
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2204:0-521-05801-5
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2020:
2016:
2012:
2008:
2004:
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1993:
1976:
1969:
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1960:9781600860270
1956:
1952:
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1936:
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1923:
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1918:
1916:
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1882:Orbital nodes
1878:
1868:
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1681:ancient China
1678:
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1667:ecliptic pole
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1379:
1368:
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1318:
1314:is the mass,
1305:
1304:
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1290:
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1230:
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1192:
1186:
1183:
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1080:
1057:
1044:
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1000:
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997:gyrocompasses
994:
989:
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978:
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378:
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100:
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63:
48:
39:
33:
28:
22:
2413:Solar System
2266:. Retrieved
2256:
2248:
2240:
2217:
2211:
2195:
2183:
2178:
2169:
2163:
2144:
2138:
2112:. Retrieved
2106:
2099:
2073:. Retrieved
2067:
2060:
2040:
2033:
2006:
2002:
1992:
1981:. Retrieved
1968:
1949:
1939:
1915:Polar motion
1880:
1855:
1847:
1820:
1816:
1800:
1709:tidal forces
1671:
1650:
1646:
1642:
1640:
1621:
1614:eccentricity
1606:
1586:
1559:
1544:
1535:
1522:
1519:
1348:
1333:
1332:
1326:
1316:
1302:
1301:
1288:
1285:
1146:
1142:
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1029:
1017:
990:
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983:
979:
970:
961:
952:
939:
926:
913:
904:
901:
893:
888:
884:
880:
878:
871:
859:normal force
847:right angles
814:
812:
667:
661:
655:
465:
457:
456:
447:
361:
351:
350:
341:
338:
331:
325:
315:
314:
304:
303:
293:
292:
282:
281:
177:
167:
161:
155:
132:
114:
112:
97:
73:
72:
2401:Outer space
2389:Spaceflight
2353:Mathematics
1888:over time.
1703:. Being an
1685:Jin-dynasty
1599:of Mercury
90:Euler angle
78:orientation
69:of a planet
2441:Precession
2435:Categories
2306:Precession
2268:2023-11-15
2114:23 October
2075:23 October
1983:2008-09-17
1931:References
1865:arcseconds
1795:See also:
1677:Hipparchus
1562:relativity
920:is at the
115:precession
74:Precession
2365:Astronomy
2206:, p. 220.
2124:cite book
2085:cite book
2009:(1): 79.
1603:Astronomy
1491:τ
1480:θ
1474:
1450:π
1390:π
1265:θ
1259:
1244:ω
1226:τ
1206:ω
1160:ω
1105:τ
1054:τ
922:perimeter
851:direction
843:magnitude
823:gyroscope
789:⋅
776:×
759:ω
731:≈
712:×
690:
631:⋅
614:ω
540:×
510:ω
495:
411:−
374:ω
260:α
253:
222:ω
188:ω
173:component
67:obliquity
32:gyroscope
2277:cite web
2247:(1924),
2245:Max Born
1947:(2003),
1910:Nutation
1899:See also
1831:Einstein
1767:(right).
1725:ecliptic
1618:ice ages
935:velocity
889:rotation
885:pitching
881:spinning
857:and the
577:for the
469:; e.g.:
99:nutation
82:rotating
62:Nutation
47:Rotation
2425:Science
2341:Physics
2327:Portals
2011:Bibcode
1886:precess
1860:
1844:
1823:Mercury
1729:Jupiter
1711:of the
1663:Polaris
1548:is the
1529:is the
1353:
1339:
1295:is the
993:bicycle
849:to the
104:physics
2228:
2202:
2190:
2151:
2048:
1957:
1803:orbits
1683:, the
1550:torque
1520:Where
1488:
1413:
1358:period
1286:where
1178:
1096:where
1013:torque
897:torque
874:gimbal
831:torque
279:where
153:(e.g.
108:torque
60:
58:
52:
45:
43:
2446:Earth
2377:Stars
1978:(PDF)
1884:also
1689:Yu Xi
1649:, or
1622:(See
839:speed
334:rigid
119:Earth
102:. In
2283:link
2226:ISBN
2200:ISBN
2188:ISBN
2149:ISBN
2130:link
2116:2020
2091:link
2077:2020
2046:ISBN
1955:ISBN
1801:The
1715:and
1713:Moon
1672:The
1657:and
1587:The
1118:and
1041:and
1011:The
827:cone
819:axis
665:and
2019:doi
2007:359
1833:'s
1807:Sun
1717:Sun
1693:fl.
1471:sin
1256:sin
687:exp
563:old
525:old
492:exp
484:new
250:cos
65:in
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2279:}}
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256:(
244:p
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214:s
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194:p
168:z
162:y
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