- Tidal force
The

**tidal force**is a secondary effect of theforce ofgravity and is responsible for thetide s. It arises because the gravitational acceleration experienced by a large body is not constant across itsdiameter . One side of the body has greateracceleration than itscenter of mass , and the other side of the body has lesser acceleration.Figure 1 shows

Comet Shoemaker-Levy 9 after it had broken up under the influence of Jupiter's tidal forces. The comet was falling into Jupiter, and the parts of the comet closest to Jupiter fell with a greater acceleration, due to the greater gravitational force. From the point of view of an observer riding on the comet, it would appear that the parts in front split off in the forward direction, while the parts in back split off in the backward direction. In reality, however, all parts of the comet were accelerating towardJupiter , but at different rates.**Explanation**When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. Figure 2 shows the differential force of gravity on a spherical body (body 1) exerted by another body (body 2). These so called "tidal forces" cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart.cite book |title=The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics |author=R Penrose |page=p. 264 |url=http://books.google.com/books?id=oI0grArWHUMC&pg=PA264&vq=tidal&dq=tidal+force&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U3N2jyScOSJZu-GxBiQrTnEiaL5QA

isbn=0192861980 |year=1999 |publisher=Oxford University Press ] Theis the distance from a planet at which tidal effects would cause an object to disintegrate because the differential force of gravity from the planet overcomes the attraction of the parts of the object for one another.cite book |title=The Solar System |page=p. 16 |url=http://books.google.com/books?id=Je61Y7UbqWgC&pg=PA16&vq=tide&dq=tidal+force&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U1a7gm97Sow1S17TnMtjsMYdSk1VQ#PPA16,M1Roche limit

author=Thérèse Encrenaz, J -P Bibring, M Blanc |isbn=3540002413 |year=2003 |publisher=Springer ] These strains would not occur if the gravitational field is uniform, since a uniform field only causes the entire body to accelerate together in the same direction and at the same rate.**Effects of tidal forces**In the case of an elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an

ellipsoid , with two bulges, pointing towards and away from the other body. An elliptical distortion is approximately what happens to the Earth's oceans under the action of the Moon. The Earth and Moon rotate about their common center of mass orbarycenter , and their gravitational attraction provides thecentripetal force necessary to maintain this motion. To an observer on the Earth, very close to this barycenter, the situation is one of the Earth as body 1 acted upon by the gravity of the Moon as body 2. All parts of the Earth are subject to the Moon's gravitational forces, causing the water in the oceans to redistribute, forming bulges on the sides near the Moon and far from the Moon.cite book |title=The Encyclopedia Americana: A Library of Universal Knowledge |author=Rollin A Harris |page=pp.611-617 |url=http://books.google.com/books?id=r8BPAAAAMAAJ&pg=PA612&dq=tidal+force&lr=&as_brr=0#PPA612,M1

publisher=Encyclopedia Americana Corp. |year=1920 |edition=Vol. 26 ]When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. If the body is close enough to its primary, this can result in a rotation which is tidally locked to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io.

Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that in addition to variations of

insolation associated withorbital forcing , harmonic beat variations in tidal forcing may contribute to climate changes. [*cite web*]

title =Millennial Climate Variability: Is There a Tidal Connection?

url =http://ams.allenpress.com/archive/1520-0442/15/4/pdf/i1520-0442-15-4-370.pdfTidal effects become particularly pronounced near small bodies of high mass, such as

neutron star s orblack hole s, where they are responsible for the "spaghettification " of infalling matter. Tidal forces create the oceanictide ofEarth 's oceans, where the attracting bodies are theMoon and theSun .Tidal forces are also responsible for

tidal locking andtidal acceleration .**Mathematical treatment**For a given (externally generated) gravitational field, the

**tidal acceleration**at a point with respect to a body is obtained by vectorially subtracting the gravitational acceleration at the center of the body from the actual gravitational acceleration at the point. Correspondingly, the term**"tidal force**" is used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant.Tidal acceleration does not require rotation or orbiting bodies; e.g. the body may be

freefall ing in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration.Newton's law of universal gravitation states that a particle of mass "m" a distance "r" from the center of a sphere of mass "M" feels a force of::$vec\; F\_g\; =\; -\; hat\; r\; ~\; G\; ~\; frac\{M\; m\}\{r^2\}$,

where $hat\; r$ is a

unit vector pointing from the body M to the particle m.Extending the description of "m" to a small body with spatial extent, suppose that "R" is the inter-object distance -- the distance from the center of "M" to the center of "m", and let "∆r" be the radius of "m" in the direction pointing towards "M". Hence the points on the surface of "m" are located at distance $r\; =\; R\; pm\; Delta\; r$ from the centre of "M". Using the above equation, and ignoring the small contribution due to "m"'s own mass, we have the gravitational force at these points as:

:$vec\; F\_g\; =\; -\; hat\; r\; ~\; G\; ~\; frac\{M\; m\}\{(R\; pm\; Delta\; r)^2\}$

Pulling out the "R"² term from the denominator gives:

:$vec\; F\_g\; =\; -\; hat\; r\; ~\; G\; ~\; frac\{M\; m\}\{R^2\}\; ~\; frac\{1\}\{(1\; pm\; Delta\; r\; /\; R)^2\}$

The

Maclaurin series of 1/(1 + "x")² is 1 - 2 "x" + 3 "x"² - ..., which gives a series expansion of::$vec\; F\_g\; =\; -\; hat\; r\; ~\; G\; ~\; frac\{M\; m\}\{R^2\}\; pm\; hat\; r\; ~\; G\; ~\; frac\{2\; M\; m\; \}\{R^2\}\; ~\; frac\{Delta\; r\}\{R\}\; mp\; ...$

The first term is the traditional gravitational force; all other terms are tidal force terms. Generally, the first is much more significant than the other terms, giving:

:$vec\; F\_t\; approx\; hat\; r\; ~\; G\; ~\; frac\{2\; M\; m\; \}\{R^2\}\; ~\; frac\{Delta\; r\}\{R\}$

The tidal forces can also be calculated away from the axis connecting the bodies, requiring a vector calculation of forces. In the plane perpendicular to the axis, the tidal force is directed inwards, and its magnitude is $F\_t/2$ in linear approximation as in Figure 5.

**ee also***

Tidal resonance

*Roche limit

*Tidal locking

*Tidal acceleration

*Galactic tide **References****External links***http://burro.astr.cwru.edu/Academics/Astr221/Gravity/tides.html

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