# gravitational force

**Isaac Newton's theory of universal gravitation**is a physical law describing the gravitational attraction between massive bodies. It is a part of classical mechanics and was first formulated in Newton's work

*Philosophiae Naturalis Principia Mathematica*, published in 1687. In modern language it states the following:

Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses:where:

Fis the magnitude of the gravitational force between the two point masses,Gis the gravitational constant,m_{1}is the mass of the first point mass,m_{2}is the mass of the second point mass,ris the distance between the two point masses.

Assuming SI units,

*F*is measured in newtons (N),

*m*

_{1}and

*m*

_{2}in kilograms (kg),

*r*in metres (m), and the constant

*G*is approximately equal to 6.67 × 10

^{−11}N m

^{2}kg

^{−2}. The value of the constant

*G*was first accurately determined from the results of Cavendish experiment conducted by the British scientist Henry Cavendish in 1798 (though Cavendish did not himself calculate a numerical value for

*G*

^{[1]}). This experiment was also the first test of Newton's theory of gravitation between masses in the laboratory. It took place 111 years after the publication of Newton's

*Principia*and 71 years after Newton's death, so none of Newton's calculations could use the value of

*G*; instead he could only calculate a force relative to another force.

Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of electrical force between two charged bodies. Both are inverse-square laws, in which force is inversely proportional to the square of the distance between the bodies. Coulomb's Law has the product of two charges in place of the product of the masses, and the electrostatic constant in place of the gravitational constant.

## Acceleration due to gravity

Let*a*

_{1}be the acceleration experienced by the first point mass due to the gravitational force exerted on it by the second point mass. Newton's second law states that

*F*=

*m*

_{1}

*a*

_{1}, meaning that

*a*

_{1}=

*F*/

*m*

_{1}. Substituting

*F*from the earlier equation gives:

- and similarly .

Assuming SI units, gravitational acceleration (as acceleration in general) is measured in metres per second squared (m/s

^{2}or m s

^{-2}). Non-SI units include gals, gees, and feet per second squared.

The force of gravity attracting a mass to another mass will also be accompanied by a force attracting to . Therefore the position of one mass from the second mass gravitationally accelerates according to

^{[2]}:

If

*m*

_{1}is negligible compared to

*m*

_{2}, small masses will have approximately the same acceleration. However, for appreciably large

*m*

_{1}, the combined acceleration should be considered. As an example, all small rocks dropped at the same position from an asteroid's surface will accelerate towards the asteroid, and crash into it, following roughly the same trajectory. However if a large object with mass comparable or larger to the asteroid's mass was released from this position, the gravitational acceleration on the asteroid itself should no longer be neglected for the two will collide sooner.

In the general case, the two masses can have an initial relative velocity such that their surfaces will not collide, in which case the mutual acceleration will lead to more complex trajectories. Some examples are elliptical orbits around their center of mass, or even gravity assisting "sling shots" flinging the masses apart. When only two masses are involved the trajectories can be solved symbolically,

^{[3]}but when three or more masses are considered the problem must in general be solved numerically.

If

*r*changes proportionally very little during an object's travel – as is the case when an object is falling near the surface of the earth – then the acceleration due to gravity appears very nearly constant (see also Earth's gravity). Across a large body, variations in

*r*, and the consequent variation in gravitational strength, can create a significant tidal force. For example, one side of the Earth is about 6,350 km closer to the Moon than the other. Although this is a small difference compared to the 385,000 km average separation, it is enough to cause a slight difference in the gravitational force exerted by the Moon on the Earth's oceans on each side relative to the average force exerted on the whole Earth. This difference is the cause of the tides.

## Bodies with spatial extent

If the bodies in question have spatial extent (rather than being theoretical point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies. In the limit, as the component point masses become "infinitely small", this entails integrating the force (in vector form, see below) over the extents of the two bodies.In this way it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its centre

^{[4]}. (This is not generally true for non-spherically-symmetrical bodies.)

For points

*inside*a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force. The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance r

_{0}from the center of the mass distribution

^{[5]}:

- The mass located at a radius
*r*<*r*_{0}causes the same force at*r*_{0}as if all of the mass enclosed within a sphere of radius r_{0}were concentrated at the center of the mass distribution (as noted above). - The mass located at a radius
*r*>*r*_{0}exerts no net gravitational force at*r*_{0}. I.e., the individual forces exerted by the elements of the sphere on the point at*r*_{0}cancel each other out.

## Vector form

Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in**bold**represent vectors.

- is the force applied on object 2 due to object 1

- is the gravitational constant

- and are respectively the masses of objects 1 and 2

- is the distance between objects 1 and 2

- is the unit vector from object 1 to 2

It can be seen that the vector form of the equation is the same as the scalar form given earlier, except that

**F**is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that

**F**

_{12}= −

**F**

_{21}.

## Gravitational field

The**gravitational field**is a vector field that describes the gravitational force which would be applied on an object in any given point in space, per unit mass. It is actually equal to the gravitational acceleration at that point.

It is a generalization of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the Earth and the Moon). For 2 objects (e.g. object 2 is a rocket, object 1 the Earth), we simply write instead of and instead of and define the gravitational field as:

so that we can write:

This formulation is dependent on the objects causing the field. The field has units of acceleration; in SI, this is m/s

^{2}.

Gravitational fields are also

**conservative**; that is, the work done by gravity from one position to another is

**path-independent**. This has the consequence that there exists a gravitational potential field

*V*(

**r**) such that

- .

*m*

_{1}is a point mass or the mass of a sphere with homogeneous mass distribution, the force field

**g**(

**r**) outside the sphere is isotropic, i.e., depends only on the distance

*r*from the center of the sphere. In that case

## Problems with Newton's theory

Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used. Deviations from it are small when the dimensionless quantities*φ*/

*c*

^{2}and

*(v/c)*are both much less than one, where

^{2}*φ*is the gravitational potential,

*v*is the velocity of the objects being studied, and

*c*is the speed of light.

^{[6]}For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since

where

*r*

_{orbit}is the radius of the Earth's orbit around the Sun.

In situations where either dimensionless parameter is large, then general relativity must be used to describe the system. General relativity reduces to Newtonian gravity in the limit of small potential and low velocities, so Newton's law of gravitation is often said to be the low-gravity limit of general relativity.

### Theoretical concerns

- There is no immediate prospect of identifying the mediator of gravity. Attempts by theorists to identify the relationship between the gravitational force and other known fundamental forces are not yet resolved, although considerable headway has been made over the last 50 years (See: Theory of everything and Standard Model). Newton himself felt the inexplicable
*action at a distance*to be unsatisfactory (see "Newton's reservations" below). - Newton's theory requires that gravitational force is transmitted instantaneously. Given classical assumptions of the nature of space and time before the development of general relativity, a propagation delay leads to unstable orbits.

### Disagreement with observation

- Newton's theory does not fully explain the precession of the perihelion of the orbit of the planets, especially of planet Mercury
^{[7]}. There is a 43 arcsecond per century discrepancy between the Newtonian prediction, which arises only from the gravitational tugs of the other planets, and the observed precession. - The predicted deflection of light by gravity using Newton's theory is only half the deflection actually observed. General relativity is in closer agreement with the observations.

### Newton's reservations

While Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" which his equations implied. He never, in his words, "assigned the cause of this power". In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity. Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science.He lamented that "philosophers have hitherto attempted the search of nature in vain" for the source of the gravitational force, as he was convinced "by many reasons" that there were "causes hitherto unknown" that were fundamental to all the "phenomena of nature". These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer is yet to be found. In Newton's 1713

*General Scholium*in the second edition of

*Principia*:

*I have not yet been able to discover the cause of these properties of gravity from phenomena and I feign no hypotheses... It is enough that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies. That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it.*^{[8]}

### Einstein's solution

These objections were mooted by Einstein's theory of general relativity, in which 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. This allowed a description of the motions of light and mass that was consistent with all available observations.Newton's theory continues to be used as an excellent approximation of the effects of gravity. Relativity is only required when there is a need for extreme accuracy, or when dealing with gravitation for very massive objects.

## See also

- Newton's cannonball
- Newton's laws of motion
- Orbital mechanics - the analysis of Newton's laws as it applies to orbits

## Notes

1. ^ [1]The Michell-Cavendish Experiment, Laurent Hodges

2. ^ Equations 376 and 377, [2]

3. ^ Equations 378 to 381, [3]

4. ^ - Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: Isaac Newton,

5. ^ [4]

6. ^ Misner, Charles W.; Kip S. Thorne & John Archibald Wheeler (1973),

7. ^ - Max Born (1924),

8. ^ -

2. ^ Equations 376 and 377, [2]

3. ^ Equations 378 to 381, [3]

4. ^ - Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: Isaac Newton,

*The Principia*: Mathematical Principles of Natural Philosophy. Preceded by*A Guide to Newton's Principia*, by I.Bernard Cohen. University of California Press 1999 ISBN 0-520-08816-6 ISBN 0-520-08817-45. ^ [4]

6. ^ Misner, Charles W.; Kip S. Thorne & John Archibald Wheeler (1973),

*Gravitation*, New York: W. H.Freeman and Company, ISBN 0-7167-0344-0 Page 1049.7. ^ - Max Born (1924),

*Einstein's Theory of Relativity*(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.)8. ^ -

*The Construction of Modern Science: Mechanisms and Mechanics*, by Richard S. Westfall. Cambridge University Press 1978**Physical cosmology**, as a branch of astronomy, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. Cosmology involves itself with studying the motions of the celestial bodies and the first cause.

**.....**Click the link for more information.

The

**age of the universe**, in physics, is the time elapsed between the Big Bang and the present day. Current observations suggest that this is about 13.7 billion years, with an uncertainty of about +/-200 million years.**.....**Click the link for more information.**Big Bang**is the cosmological model of the universe whose primary assertion is that the universe has expanded into its current state from a primordial condition of enormous density and temperature.

**.....**Click the link for more information.

**Blue Shift**refers to a shortening of a transmitted signal's wavelength, and/or an increase in its frequency. The name comes from the fact that the shorter-wavelength end of the optical spectrum is the blue (or violet) end, hence, when visible light is compacted in

**.....**Click the link for more information.

In standard cosmology,

**'comoving**' distance or**'proper distance**' is one of several distance measures used by cosmologists to define distances between objects.## Comoving coordinates

**.....**Click the link for more information.**cosmic microwave background radiation**(most often abbreviated

**CMB**but occasionally

**CMBR**,

**CBR**or

**MBR**, also referred to as

**relic radiation**) is a form of electromagnetic radiation discovered in 1965 that fills the entire universe

^{[1]}.

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In physical cosmology,

**dark energy**is a hypothetical form of energy that permeates all of space and tends to increase the rate of expansion of the universe.^{[1]}**.....**Click the link for more information. In astrophysics and cosmology,

**dark matter**is hypothetical matter of unknown composition that does not emit or reflect enough electromagnetic radiation to be observed directly, but whose presence can be inferred from gravitational effects on visible matter.**.....**Click the link for more information.**Friedmann-Lemaître-Robertson-Walker (FLRW) metric**is an exact solution of the Einstein field equations of general relativity; it describes a , isotropic expanding or contracting universe.

**.....**Click the link for more information.

The

**Friedmann equations**are a set of equations in cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity.**.....**Click the link for more information.**formation of galaxies**is still one of the most active research areas in astrophysics; and, to some extent, this is also true for

**galaxy evolution**. Some ideas, however, have gained wide acceptance.

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**Hubble's law**is a statement in physical cosmology which states that the redshift in light coming from distant galaxies is proportional to their distance. The law was first formulated by Edwin Hubble and Milton Humason in 1929

^{[1]}after nearly a decade of observations.

**.....**Click the link for more information.

In physical cosmology,

**cosmic inflation**is the idea that the nascent universe passed through a phase of exponential expansion that was driven by a negative-pressure vacuum energy density.**.....**Click the link for more information.**large-scale structure**refers to the characterization of observable distributions of matter and light on the largest scales (typically on the order of billions of light-years).

**.....**Click the link for more information.

**ΛCDM**or

**Lambda-CDM**is an abbreviation for

**Lambda-Cold Dark Matter**. It is frequently referred to as the

**concordance model**of big bang cosmology, since it attempts to explain cosmic microwave background observations, as well as large scale structure

**.....**Click the link for more information.

The

**metric expansion of space**is a key part of science's current understanding of the universe, whereby spacetime itself is described by a metric which changes over time in such a way that the spatial dimensions appear to grow or stretch as the universe gets older.**.....**Click the link for more information. In physical cosmology,

**Big Bang nucleosynthesis**(or**primordial nucleosynthesis**) refers to the production of nuclei other than those of H-1 (i.e. the normal, light isotope of hydrogen, whose nuclei consist of a single proton each) during the early phases of the**.....**Click the link for more information.**This article or section may contain inappropriate or misinterpreted which do not the text.**

Please help [ improve this article] by checking for inaccuracies. This article has been tagged since October 2007.

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**redshift**occurs when the electromagnetic radiation, usually visible light, that is emitted from or reflected off an object is shifted toward the (less energetic) red end of the electromagnetic spectrum.

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The

**shape of the Universe**is an informal name for a subject of investigation within physical cosmology. Cosmologists and astronomers describe the geometry of the universe which includes both local geometry and global geometry.**.....**Click the link for more information.**Structure formation**refers to a fundamental problem in physical cosmology. The universe, as is now known from observations of the cosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately 13.7 Gyr ago.

**.....**Click the link for more information.

**Physical cosmology**

- Age of the universe
- Big Bang
- Blueshift
- Comoving distance
- Cosmic microwave background
- Dark energy
- Dark matter
- FLRW metric
- Friedmann equations
- Galaxy formation
- Hubble's law
- Inflation

**.....**Click the link for more information.

For a timeline of the cosmos (or universe), see .

This

**timeline of cosmological theories**and discoveries is a chronological catalog of the evolution of humankind's understanding of the cosmos over the last two-plus millennia.

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The

**ultimate fate of the universe**is a topic in physical cosmology. Many possible fates are predicted by rival scientific theories, including futures of both finite and infinite duration.**.....**Click the link for more information. The

**Universe**is defined as the summation of all particles and energy that exist and the space-time in which all events occur. Based on observations of the portion of the Universe that is observable, physicists attempt to describe the whole of space-time, including all matter and**.....**Click the link for more information.**Astronomy**is the scientific study of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earth's atmosphere (such as the cosmic background radiation).

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**General relativity**(GR) (aka

**general theory of relativity**(GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16.

^{[1]}It unifies special relativity, Newton's law of universal gravitation, and the insight that gravitational

**.....**Click the link for more information.

**Particle physics**is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. It is also called

**"high energy physics"**

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**Quantum gravity**is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity.

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**Sir Isaac Newton**

Isaac Newton at 46 in

Godfrey Kneller's 1689 portrait

Born 4 January 1643 [OS: 25 December 1642]

^{}

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