Isaac Newton's theory of universal gravitation (part of classical mechanics) states the following: Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... For other uses, see Gravitation (disambiguation). ... Classical mechanics (also called Newtonian mechanics) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ...

Every single point mass attracts every other point mass by a force pointing along the line combining the two. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses: A point mass in physics is an idealisation of a body whose dimensions can be neglected compared to the distances of its movement. ... In physics, force is anything that can cause a massive body to accelerate. ... “Line” redirects here. ... In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. ... This article or section is in need of attention from an expert on the subject. ... In algebra, the square of a number is that number multiplied by itself. ...

where:

• F is the magnitude of the gravitational force between the two point masses,
• G is the gravitational constant,
• m₁ is the mass of the first point mass,
• m₂ is the mass of the second point mass,
• r is the distance between the two point masses.

Assuming SI units, F is measured in newtons (N), m₁ and m₂ in kilograms (kg), r in metres (m), and the constant G is approximately equal to 6.67 × 10⁻¹¹ N m² kg⁻². G was first accurately measured in the Cavendish experiment by the British scientist Henry Cavendish in 1798, it was also the first test of Newton's theory of gravitation between masses in the laboratory. This was 111 years after the publication of "Philosophiae Naturalis Principia Mathematica" and 71 years after Newton's death, so all of Newton's calculations could not use the value of G; instead he could only calculate a force relative to another force. According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... Cover of brochure The International System of Units. ... The newton (symbol: N) is the SI derived unit of force, named after Sir Isaac Newton in recognition of his work on classical mechanics. ... “Kg” redirects here. ... ‹ The template below (Unit of length) is being considered for deletion. ... In physics, the Cavendish experiment was the first experiment to accurately measure the gravitational constant by measuring the force of gravity between two masses in the laboratory. ... For other persons named Henry Cavendish, see Henry Cavendish (disambiguation). ... Year 1798 (MDCCXCVIII) was a common year starting on Monday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Friday of the 11-day slower Julian calendar). ... Newtons own copy of his Principia, with handwritten corrections for the second edition. ...

Newton's law of gravitation resembles Coulomb's law of electrical forces. Newton's law is used to calculate the Gravitational force between two masses; similarly Coulomb's Law is used to calculate the magnitude of electrical force between two charged bodies. Coulomb's Law's equation has the product of two charges in place of the product of the masses which is in Newton's Law of Gravitation. Hence, according to Coulomb's Law, the electrical force is proportional to the product of the charged bodies divided by the distance between them. Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated as follows: This is analogous to Newtons third law of motion in mechanics. ...

Acceleration due to gravity

Let a₁ 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 = m₁ a₁, meaning that a₁ = F / m₁. Substituting F from the earlier equation gives: Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ...

and similarly for a₂.

Assuming SI units, gravitational acceleration (as acceleration in general) is measured in metres per second squared (m/s² or m s^-2). Non-SI units include galileos, gees (see later), and feet per second squared. The International System of Units (symbol: SI) (for the French phrase Système International dUnités) is the most widely used system of units. ... Meters per second squared is the SI derived unit of acceleration, defined by distance or displacement in metres divided by time in seconds and again divided by time in seconds. ... The galileo or gal is the CGS unit of acceleration. ... The term g force or gee force refers to the symbol g, the force of acceleration due to gravity at the earths surface. ... For other uses, see Foot (disambiguation). ...

The force attracting a mass to the earth also attracts the earth to the mass, so that their acceleration to each other is given by:

If m₁ is negligible compared to m₂, small masses would have approximately the same acceleration. However, for appreciably large m₁, the combined acceleration, should be considered.

If r changes proportionally very little during an object's travel – such as an object 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, the near and far side of the earth are around 6,350 km different distance from the Moon; although a small difference compared to the 385,000 km average separation, this is enough to cause a slightly different gravitational force by the moon on the earth's oceans on each side compared to that exercised on the earth itself, and hence give rise to the tides. Precise values of g vary depending on the location on the Earths surface. ... Comet Shoemaker-Levy 9 after breaking up under the influence of Jupiters tidal forces. ... This article is about Earths moon. ... This article is about tides in the Earths oceans. ...

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. The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... A physical body is an object which can be described by the theories of classical mechanics, or quantum mechanics, and experimented upon by physical instruments. ...

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^[1]. (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 r₀ from the center of the mass distribution: In Newtonian physics, the shell theorem states that the gravity due to a uniform spherical shell is zero on an object inside the shell, and acts on an object outside the shell as if the entire mass of the shell were at its center. ...

• The mass located at a radius r < r₀ causes the same force at r₀ as if all of the mass enclosed within a sphere of radius r₀ were concentrated at the center of the mass distribution (as noted above).
• The mass located at a radius r > r₀ exerts no net gravitational force at r₀. I.e., the individual forces exerted by the elements of the sphere on the point at r₀ cancel each other out.

As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration.

Vector form

Gravity on Earth from a macroscopic perspective.

Gravity in a room: the curvature of the Earth is negligible at this scale, and the force lines can be approximated as being parallel and pointing straight down to the center of the Earth

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ...

where

is the force applied on object 2 due to object 1

G is the gravitational constant

m₁ and m₂ 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 F₁₂ = − F₂₁. In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ...

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. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In physics, gravitational acceleration is the acceleration of an object caused by the force of gravity from another object. ...

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 m instead of m₂ 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/s². Look up si, Si, SI in Wiktionary, the free dictionary. ...

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 In vector calculus, an irrotational or conservative vector field is a vector field whose curl is zero. ...

.

If m₁ 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/c² and (v/c)² are both much less than one, where φ is the gravitational potential, v is the velocity of the objects being studied, and c is the speed of light. ^[2] For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since In physics, gravitational potential is the measure of potential energy an object possesses due to its position in a gravitational field. ... “Lightspeed” redirects here. ...

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. For a less technical and generally accessible introduction to the topic, see Introduction to 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.

This article or section is in need of attention from an expert on the subject. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. ...

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^[3]. 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.

The observed fact that gravitational and inertial masses are the same for all bodies is unexplained within Newton's system. General relativity takes this as a postulate. See equivalence principle. Precession of a gyroscope Precession refers to a change in the direction of the axis of a rotating object. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... The eight planets and three dwarf planets of the Solar System. ... This article is about the planet. ... A second of arc or arcsecond is a unit of angular measurement which comprises one-sixtieth of an arcminute, or 1/3600 of a degree of arc or 1/1296000 ≈ 7. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ...

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.^[4]

Hypotheses non fingo is a Latin phrase attributed to Isaac Newton often translated as, I make no hypotheses. The context is that he is refusing to speculate as the the whys of the laws of nature, confining himself entirely to their phenomenal description. ...

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. For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ...

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 at it applies to orbits

Newtons cannonball was a thought experiment Isaac Newton used to hypothesize that the force of gravity was universal, and it was the key force for planetary motion. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... This article or section should be merged with Celestial Mechanics Astrodynamics is the study and creation of orbits, especially those of artificial satellites. ...

Notes

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