Nutation
Nutation (from
In a rigid body
If a top is set at a tilt on a horizontal surface and spun rapidly, its rotational axis starts precessing about the vertical. After a short interval, the top settles into a motion in which each point on its rotation axis follows a circular path. The vertical force of gravity produces a horizontal torque τ about the point of contact with the surface; the top rotates in the direction of this torque with an angular velocity Ω such that at any moment
- (vector cross product)
where L is the instantaneous angular momentum of the top.[3]
Initially, however, there is no precession, and the upper part of the top falls sideways and downward, thereby tilting. This gives rise to an imbalance in torques that starts the precession. In falling, the top overshoots the amount of tilt at which it would precess steadily and then oscillates about this level. This oscillation is called nutation. If the motion is damped, the oscillations will die down until the motion is a steady precession.[3][4]
The physics of nutation in tops and
If the top has mass M and its center of mass is at a distance l from the pivot point, its gravitational potential relative to the plane of the support is
In a coordinate system where the z axis is the axis of symmetry, the top has
In terms of the Euler angles, this is
If the Euler–Lagrange equations are solved for this system, it is found that the motion depends on two constants a and b (each related to a constant of motion). The rate of precession is related to the tilt by
The tilt is determined by a differential equation for u = cos(θ) of the form
where f is a
Astronomy
The nutation of a planet occurs because the gravitational effects of other bodies cause the speed of its axial precession to vary over time, so that the speed is not constant. English astronomer James Bradley discovered the nutation of Earth's axis in 1728.
Earth
It has been suggested that this section be Earth's nutation. (Discuss ) (October 2020) |
Nutation subtly changes the
In the case of Earth, the principal sources of tidal force are the Sun and Moon, which continuously change location relative to each other and thus cause nutation in Earth's axis. The largest component of Earth's nutation has a period of 18.6 years, the same as that of the precession of the Moon's orbital nodes.[1] However, there are other significant periodic terms that must be accounted for depending upon the desired accuracy of the result. A mathematical description (set of equations) that represents nutation is called[by whom?] a "theory of nutation".[citation needed] In the theory, parameters are adjusted in a more or less ad hoc method to obtain the best fit to data. Simple rigid body dynamics do not give the best theory; one has to account for deformations of the Earth, including mantle inelasticity and changes in the core–mantle boundary.[7]
The principal term of nutation is due to the regression of the Moon's
In popular culture
In the 1961 disaster film
In Star Trek: The Next Generation, rapidly 'cycling' or 'changing' the 'shield nutation' is frequently mentioned as a means by which to delay the antagonist in their efforts to break through the defences and pillage the Enterprise or other spacecraft.
See also
Notes
- ^ ISBN 9780521675963.
- ISBN 9780691135373.
- ^ a b Feynman, Leighton & Sands 2011, pp. 20–7[, clarification needed],
- ^ Goldstein 1980, p. 220
- ^ Goldstein 1980, p. 217
- ^ Goldstein 1980, pp. 213–217
- ^ "Resolution 83 on non-rigid Earth nutation theory". International Earth Rotation and Reference Systems Service. Federal Agency for Cartography and Geodesy. 2 April 2009. Retrieved 2012-08-06.
- ^ "Basics of Space Flight, Chapter 2". Jet Propulsion Laboratory/NASA. 28 August 2013. Retrieved 2015-03-26.
- ^ "NeoProgrammics - Science Computations".
References
- The Feynman Lectures on Physics Vol. I Ch. 20: Rotation in space
- Goldstein, Herbert (1980). Classical mechanics (2d ed.). Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0201029189.
- Lambeck, Kurt (2005). The earth's variable rotation : geophysical causes and consequences (Digitally printed 1st pbk. ed.). Cambridge: ISBN 9780521673303.
- Munk, Walter H.; MacDonald, Gordon J.F. (1975). The rotation of the earth : a geophysical discussion. Reprint. with corr. Cambridge, Eng.: Cambridge University Press. ISBN 9780521207782.