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Isaac Newton's theory of universal gravitation (part of classical mechanics) states the following:
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:
is the magnitude of the gravitational force between the two point masses G is the gravitational constant m<sub>1</sub> is the mass of the first point mass m<sub>2</sub> 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), m<sub>1</sub> and m<sub>2</sub> in kilograms (kg), r in metres (m), and the constant G is approximately equal to 6.67 × 10<sup>−11</sup> N m<sup>2</sup> kg<sup>−2</sup>.
Let a<sub>1</sub> be the acceleration due to gravity experienced by the first point mass. Newton's second law states that F = m<sub>1</sub> a<sub>1</sub>, meaning that a<sub>1</sub> = F / m<sub>1</sub>. Substituting F from the earlier equation gives:
and similarly for a<sub>2</sub>.
Assuming SI units, gravitational acceleration (as acceleration in general) is measured in metres per second squared (m/s<sup>2</sup> or m s<sup>-2</sup>). Non-SI units include galileos, gees (see later), and feet per second squared.
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 m<sub>1</sub> is negligible compared to m<sub>2</sub>, small masses would have approximately the same acceleration. However, for appreciably large m<sub>1</sub>, 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.
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[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 r<sub>0</sub> from the center of the mass distribution:
As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration.
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.
where
F<sub>12</sub> is the force applied on object 2 due to object 1 G is the gravitational constant m<sub>1</sub> and m<sub>2</sub> are respectively the masses of objects 1 and 2 = | r<sub>2</sub> − r<sub>1</sub> | 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<sub>12</sub> = − F<sub>21</sub>.
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<sup>2</sup>.
Gravitational fields are also conservative, that is, the work done by gravity from one position to another is path-independent.
Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used. It can be used when the dimensionless quantities φ/c<sup>2</sup> and <em>(v/c)</em> are both small, where φ is the gravitational potential, <em>v</em> is the velocity of the objects being studied, and <em>c</em> is the speed of light. As an example, Newtonian gravity provides an accurate description of the Earth/Sun system, since where <em>r</em><sub>orbit</sub> 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.
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.[3]
These objections were mooted by Einstein's general relativity theory in which gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. However, there is now the question of why mass and energy curve spacetime.
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