Newton's law of universal gravitation

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Isaac Newton's law of universal gravitation states the following:

Every point mass attracts every other point mass by a force directed along the line connecting the two. This force is proportional to the product of the masses and inversely proportional to the square of the distance between them:

[F = -G \frac]

where:

F is the magnitude of the (repulsive) 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 (newtons times metres squared per kilogram squared).

It can be seen that the repulsive force F is always negative, which means that the net attractive force is positive. (This sign convention is adopted in order to be consistent with Coulomb's Law, where a positive force means repulsion between two charges.)

Contents

1 Acceleration due to gravity
2 Bodies with spatial extent
3 Vector form
4 Gravitational field
5 Problems with Newton's theory

5.1 Theoretical concerns
5.2 Disagreement with observation
5.3 Newton's reservations

6 Notes
7 See also

Acceleration due to gravity

Let a₁ be the acceleration due to gravity experienced by the first point mass. Newton's second law states that F = m₁ a₁, meaning that a₁ = F / m₁. Substituting F from the earlier equation gives:

[a_1 = G \frac]

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.

Notice in the above equation that a₁, the acceleration of the mass m₁, does not actually depend on the magnitude of m₁. One consequence is that all bodies, regardless of their mass, fall to earth at the same rate (ignoring air resistance).

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 The Earth's gravity). Across a large body, variations in r, and the consequent variation in gravitational strength, can create a significant tidal force.

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^Principia. (This is not generally true for non-spherically-symmetrical bodies.

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.

[ \mathbf_ = - G ^2} \, \mathbf}_] or [ \mathbf_ = G ^2} \, \mathbf}_]

where

F₁₂ is the force on object 2 due to object 1

G is the gravitational constant

m₁ and m₂ are respectively the masses of objects 1 and 2

r₂₁ = | r₂ − r₁ | is the distance between objects 2 and 1

[ \mathbf}_ \equiv \frac_2 - \mathbf_1}_2 - \mathbf_1\vert} ] 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.

The vector formula for gravitational acceleration is similarly analogous to the scalar formula:

[ \mathbf_1 = G } \, \mathbf}_ ]

Gravitational field

Globular Cluster M13 demonstrates 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 1 is a rocket, object 2 the Earth), we simply write [\mathbf r] instead of [\mathbf r_] and [m] instead of [m_1] and define the gravitational field [ \mathbf g(\mathbf r) ] as:

[ \mathbf g(\mathbf r) = -G \, \mathbf}]

so that we can write:

[\mathbf( \mathbf r) = m \mathbf g(\mathbf r) ]

This formulation is independent of the objects causing the field. The field has units of force divided by mass; in SI, this is N·kg⁻¹.

Problems with Newton's theory

Although Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used, it is limited to domains where gravitational potential is a small fraction of speed of light squared. The more accurate general relativity theory of gravity must be used in general cases. General relativity results in Newtonian gravity in the limit of small potential, so Newton's law of gravitation is often said to be low-gravity limit of general relativity.

Theoretical concerns

There is no prospect of identifying the mediator of gravity. 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, this is necessary to preserve the conservation of angular momentum observed by Johannes Kepler. However, it is in direct conflict with Einstein's theory of special relativity which places an upper limit—the speed of light in vacuum—on the velocity at which signals can be transmitted.

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^Precession. There is a 43 arcsecond per century discrepancy between the Newtonian prediction (resulting 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.

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 the fact 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. While it is true that Einstein's hypotheses are successful in explaining the effects of gravitational forces more precisely than Newton's in certain cases, he too never assigned the cause of this power in his theories. It is said that in Einstein's equations, "matter tells space how to curve, and space tells matter how to move", but this new idea, completely foreign to the world of Newton, did not enable Einstein to assign the "cause of this power" to curved space any more than the Law of Universal Gravitation enabled Newton to assign its cause. In Newton's own words:

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.

If science is eventually able to discover the cause of the gravitational force, Newton's wish could eventually be fulfilled as well.

It should be noted that the word "cause" here is not being used in the same sense as "cause and effect" or "the defendant caused the victim to die". Rather, when Newton uses the word "cause," he (apparently) is referring to an "explanation". In other words, a phrase like "Newtonian gravity is the cause of planetary motion" means simply that Newtonian gravity explains the motion of the planets. See Causality and Causality (physics).

Notes

Note Principia: - 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 0-520-08817-4
Note Precession: - 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.)