WordNet

the 7th letter of the Roman alphabet (同)g
the act of accelerating; increasing the speed (同)quickening, speedup
an increase in rate of change; "modern science caused an acceleration of cultural change"
(physics) a rate of increase of velocity
a figurative movement toward some attraction; "the gravitation of the middle class to the suburbs"
movement downward resulting from gravitational attraction; "irrigation by gravitation rather than by pumps"
of or relating to or caused by gravitation (同)gravitative

PrepTutorEJDIC

加速 / 加速度
引力,引力の作用 / (…に)引き寄せられること《+『to(toward)』+『名』》
引力の,重力の

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/08/19 22:46:01」(JST)

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In physics, gravitational acceleration is the acceleration on an object caused by force of gravitation. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass.^[1] This equality is true regardless of the masses or compositions of the bodies.

At different points on Earth, objects fall with an acceleration between 9.78 and 9.82 m/s² depending on altitude, with a conventional standard value of exactly 9.80665 m/s² (approx. 32.174 ft/s²). Objects with low densities do not accelerate as rapidly due to buoyancy and air resistance.

For point masses

Newton's Law of Universal Gravitation states that there is a gravitational force between any two masses that is equal in magnitude for each mass, and is aligned to draw the two masses toward each other. The formula is:

where and are the two masses, is the gravitational constant, and is the distance between the two masses. The formula was derived for planetary motion where the distances between the planets and the Sun made it reasonable to consider the bodies to be point masses. (For a satellite in orbit, the 'distance' refers to the distance from the mass centers rather than, say, the altitude above a planet's surface.)

If one of the masses is much larger than the other, it is convenient to define a gravitational field around the larger mass as follows:^[2]

where is the mass of the larger body, and is a unit vector directed from the large mass to the smaller mass. The negative sign indicates that the force is an attractive force.

In that way, the force acting upon the smaller mass can be calculated as:

where is the force vector, is the smaller mass, and is a vector pointed toward the larger body. Note that has units of acceleration and is a vector function of location relative to the large body, independent of the magnitude (or even the presence) of the smaller mass.

This model represents the "far-field" gravitational acceleration associated with a massive body. When the dimensions of a body are not trivial compared to the distances of interest, the principle of superposition can be used for differential masses for an assumed density distribution throughout the body in order to get a more detailed model of the "near-field" gravitational acceleration. For satellites in orbit, the far-field model is sufficient for rough calculations of altitude versus period, but not for precision estimation of future location after multiple orbits.

The more detailed models include (among other things) the bulging at the equator for the Earth, and irregular mass concentrations (due to meteor impacts) for the Moon. The Gravity Recovery And Climate Experiment (GRACE) mission launched in 2002 consists of two probes, nicknamed "Tom" and "Jerry", in polar orbit around the Earth measuring differences in the distance between the two probes in order to more precisely determine the gravitational field around the Earth, and to track changes that occur over time. Similarly, the Gravity Recovery and Interior Laboratory (GRAIL) mission from 2011-2012 consisted of two probes ("Ebb" and "Flow") in polar orbit around the Moon to more precisely determine the gravitational field for future navigational purposes, and to infer information about the Moon's physical makeup.

Gravity model for Earth

The type of gravity model used for the Earth depends upon the degree of fidelity required for a given problem. For many problems such as aircraft simulation, it may be sufficient to consider gravity to be a constant, defined as:^[3]

9.80665 metres (32.1740 ft) per s²

based upon data from World Geodetic System 1984 (WGS-84), where is understood to be pointing 'down' in the local frame of reference.

If it is desirable to model an object's weight on Earth as a function of latitude, one could use the following (^[3] p. 41):

where

= 9.832 metres (32.26 ft) per s²
= 9.806 metres (32.17 ft) per s²
= 9.780 metres (32.09 ft) per s²
lat = latitude, between −90 and 90 degrees

Both these models take into account the centrifugal relief that is produced by the rotation of the Earth, and neither accounts for changes in gravity with changes in altitude. It is worth noting that for the mass attraction effect by itself, the gravitational acceleration at the equator is about 0.18% less than that at the poles due to being located farther from the mass center. When the rotational component is included (as above), the gravity at the equator is about 0.53% less than that at the poles, with gravity at the poles being unaffected by the rotation. So the rotational component change due to latitude (0.35%) is about twice as significant as the mass attraction change due to latitude (0.18%), but both reduce strength of gravity at the equator as compared to gravity at the poles.

Note that for satellites, orbits are decoupled from the rotation of the Earth so the orbital period is not necessarily one day, but also that errors can accumulate over multiple orbits so that accuracy is important. For such problems, the rotation of the Earth would be immaterial unless variations with longitude are modeled. Also, the variation in gravity with altitude becomes important, especially for highly elliptical orbits.

The Earth Gravitational Model 1996 (EGM96) contains 130,676 coefficients that refine the model of the Earth's gravitational field (^[3] p. 40). The most significant correction term is about two orders of magnitude more significant than the next largest term (^[3] p. 40). That coefficient is referred to as the term, and accounts for the flattening of the poles, or the oblateness, of the Earth. (A shape elongated on its axis-of-symmetry, like an American football, would be called prolate.) A gravitational potential function can be written for the change in potential energy for a unit mass that is brought from infinity into proximity to the Earth. Taking partial derivatives of that function with respect to a coordinate system will then resolve the directional components of the gravitational acceleration vector, as a function of location. The component due to the Earth's rotation can then be included, if appropriate, based on a sidereal day relative to the stars (≈366.24 days/year) rather than on a solar day (≈365.24 days/year). That component is perpendicular to the axis of rotation rather than to the surface of the Earth.

A similar model adjusted for the geometry and gravitational field for Mars can be found in publication NASA SP-8010.^[4]

The barycentric gravitational acceleration at a point in space is given by:

where:

M is the mass of the attracting object, is the unit vector from center-of-mass of the attracting object to the center-of-mass of the object being accelerated, r is the distance between the two objects, and G is the gravitational constant.

When this calculation is done for objects on the surface of the Earth, or aircraft that rotate with the Earth, one has to account that the Earth is rotating and the centrifugal acceleration has to be subtracted from this. For example, the equation above gives the acceleration at 9.820 m/s², when GM = 3.986×10¹⁴ m³/s², and R=6.371×10⁶ m. The centripetal radius is r = R cos(latitude), and the centripetal time unit is approximately (day / 2π), reduces this, for r = 5×10⁶ metres, to 9.79379 m/s², which is closer to the observed value.

Disregarding air resistance, all small masses dropped simultaneously from the same height will hit the ground at the same time; for example,^{[citation needed]} during Apollo 15 an astronaut on the Moon simultaneously dropped a feather and a hammer and they reached the ground at the same time.

General relativity

In Einstein's theory of general relativity, gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. In Einstein's theory, masses distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. The gravitational force is a fictitious force. There is no gravitational acceleration, in that the proper acceleration and hence four-acceleration of objects in free fall are zero. Rather than undergoing an acceleration, objects in free fall travel along straight lines (geodesics) on the curved spacetime.

References

^ Gerald James Holton and Stephen G. Brush (2001). Physics, the human adventure: from Copernicus to Einstein and beyond (3rd ed.). Rutgers University Press. p. 113. ISBN 978-0-8135-2908-0.
^ Fredrick J. Bueche (1975). Introduction to Physics for Scientists and Engineers, 2nd Ed. USA: Von Hoffmann Press. ISBN 0-07-008836-5.
^ ^a ^b ^c ^d Brian L. Stevens; Frank L. Lewis (2003). Aircraft Control And Simulation, 2nd Ed. Hoboken, New Jersey: John Wiley & Sons, Inc. ISBN 0-471-37145-9.
^ Richard B. Noll; Michael B. McElroy (1974), Models of Mars' Atmosphere [1974], Greenbelt, Maryland: NASA Goddard Space Flight Center, SP-8010.

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English Journal

Effects of gravitational acceleration on cardiovascular autonomic control in resting humans.

Fontolliet T1, Pichot V, Antonutto G, Bonjour J, Capelli C, Tam E, Barthélémy JC, Ferretti G.
European journal of applied physiology.Eur J Appl Physiol.2015 Jul;115(7):1417-27. doi: 10.1007/s00421-015-3117-9. Epub 2015 Feb 14.
PURPOSE: Previous studies of cardiovascular responses in hypergravity suggest increased sympathetic regulation. The analysis of spontaneous heart rate variability (HRV) parameters and spontaneous baroreflex sensitivity (BRS) informs on the reciprocal balance of parasympathetic and sympathetic regula
PMID 25681109

Effect of soccer heading ball speed on S100B, sideline concussion assessments and head impact kinematics.

Dorminy M1, Hoogeveen A, Tierney RT, Higgins M, McDevitt JK, Kretzschmar J.
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OBJECTIVE: To examine the effect of soccer heading ball speed on S-100B serum concentration, concussion sideline assessments and linear head impact acceleration.METHODS: Sixteen division I soccer players participated in this pre-test post-test design study. Athletes performed five standing headers o
PMID 26004753

Experimental and computational investigation of the trajectories of blood drops ejected from the nose.

Geoghegan PH1, Spence CJ, Wilhelm J, Kabaliuk N, Taylor MC, Jermy MC.
International journal of legal medicine.Int J Legal Med.2015 Mar 15. [Epub ahead of print]
Blood expirated from the nose may leave a characteristic bloodstain at a crime scene which can provide important clues for reconstructing events during a violent assault. Little research has been done on the typical velocities, trajectories and size distribution that can be expected from expirated b
PMID 25773915

Japanese Journal

遠心鋳造における等軸晶挙動に及ぼす回転軸の傾きの影響 (鉄と鋼) -- (特集号「凝固時の偏析の生成および制御」)

江阪久雄,徳差圭,篠塚計
ふぇらむ = Bulletin of the Iron and Steel Institute of Japan : (一社)日本鉄鋼協会会報 99(2), 149-155, 2013
… Since the contribution of gravitational acceleration to the movement of solid/liquid mixture decreases with increasing η, the turbulence of the surface of liquid phase decreases with increasing η. … The liquid phase is completely stabilized by centrifugal force, because the effect of acceleration due to gravity vanishes in this condition. …
NAID 130003372504

On the Effect of Forced-Oscillation Amplitude upon the Flow in a Cubic Cavity Heated Below

HIRATA Katsuya,TANIGAWA Hirochika,NAKAMURA Noriyuki,FUJITA Satoshi,FUNAKI Jiro
Journal of Fluid Science and Technology 8(1), 106-119, 2013
… The authors numerically investigate the forced-oscillation-frequency responses on flow and thermal characteristics of the thermal convection in a cubic cavity heated from below in the gravitational field, concerning global quantities such as spatially-averaged kinetic energy and so on. … Even at large acceleration amplitudes, the optimum frequency exists. … At the optimum frequency, periodicity is exact, while the wave form is far from sinusoidal at a non-dimensional acceleration amplitude η ≥ 1. …
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Construction of Two-Axis Acceleration Sensor Using a Cross-Coupled Vibrator

Terada Jiro,Uetsuji Yasutomo,Sugawara Sumio
Jpn J Appl Phys 51(10), 106602-106602-6, 2012-10-25
… We describe an acceleration sensor composed of four vibration bars, with a detection mechanism in which the resonant frequencies of the four bars are brought close together. … Using this structure, acceleration can be easily calculated by differential detection of the oscillation amplitude signals of each of the four vibration bars. …
NAID 150000105040