Contents 1 History 1.1 Early history
1.2
2 Modern form 3 Bodies with spatial extent 4 Vector form 5 Gravitational field 6 Problematic aspects 6.1 Theoretical concerns with Newton's expression 6.2 Observations conflicting with Newton's formula 6.3 Newton's reservations 6.4 Einstein's solution 7 Extensions 8 Solutions of Newton's law of universal gravitation 9 See also 10 Notes 11 References 12 External links History[edit]
Early history[edit]
A recent assessment (by Ofer Gal) about the early history of the
inverse square law is "by the late 1670s", the assumption of an
"inverse proportion between gravity and the square of distance was
rather common and had been advanced by a number of different people
for different reasons".
The same author does credit Hooke with a significant and even seminal
contribution, but he treats Hooke's claim of priority on the inverse
square point as uninteresting since several individuals besides Newton
and Hooke had at least suggested it, and he points instead to the idea
of "compounding the celestial motions" and the conversion of Newton's
thinking away from "centrifugal" and towards "centripetal" force as
Hooke's significant contributions.
Newton himself gave credit in his Principia to two persons:
Bullialdus[4] (he wrote without proof that there was a force on the
earth towards the sun), and Borelli[5] (wrote that all planets were
attracted towards the sun). Whiteside wrote that the main influence
was Borelli, because Newton had a copy of his book.[6]
Every point mass attracts every single 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 them:[2] F = G m 1 m 2 r 2
displaystyle F=G frac m_ 1 m_ 2 r^ 2 where: F is the force between the masses; G is the gravitational constant (6.674×10−11 N · (m/kg)2); m1 is the first mass; m2 is the second mass; r is the distance between the centers of the masses. Error plot showing experimental values for big G. Assuming SI units, F is measured in newtons (N), m1 and m2 in
kilograms (kg), r in meters (m), and the constant G is approximately
equal to 6989667400000000000♠6.674×10−11 N m2 kg−2.[29]
The value of the constant G was first accurately determined from the
results of the
If the bodies in question have spatial extent (as opposed to being
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.[2] (This is not
generally true for non-spherically-symmetrical bodies.)
For points inside a spherically-symmetric distribution of matter,
Newton's
The portion of the mass that is located at radii r < r0 causes the same force at r0 as if all of the mass enclosed within a sphere of radius r0 was concentrated at the center of the mass distribution (as noted above). The portion of the mass that is located at radii r > r0 exerts no net gravitational force at the distance r0 from the center. That is, the individual gravitational forces exerted by the elements of the sphere out there, 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 anywhere within the hollow sphere. Furthermore, inside a uniform sphere the gravity increases linearly with the distance from the center; the increase due to the additional mass is 1.5 times the decrease due to the larger distance from the center. Thus, if a spherically symmetric body has a uniform core and a uniform mantle with a density that is less than 2/3 of that of the core, then the gravity initially decreases outwardly beyond the boundary, and if the sphere is large enough, further outward the gravity increases again, and eventually it exceeds the gravity at the core/mantle boundary. The gravity of the Earth may be highest at the core/mantle boundary. Vector form[edit]
F 21 = − G m 1 m 2
r 12
2 r ^ 12 displaystyle mathbf F _ 21 =-G m_ 1 m_ 2 over vert mathbf r _ 12 vert ^ 2 ,mathbf hat r _ 12 where F21 is the force applied on object 2 exerted by object 1, G is the gravitational constant, m1 and m2 are respectively the masses of objects 1 and 2, r12 = r2 − r1 is the distance between objects 1 and 2, and r ^ 12
= d e f
r 2 − r 1
r 2 − r 1
displaystyle mathbf hat r _ 12 stackrel mathrm def = frac mathbf r _ 2 -mathbf r _ 1 vert mathbf r _ 2 -mathbf r _ 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. Gravitational field[edit] Main article: 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 generalisation 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 r instead of r12 and m instead of m2 and define the gravitational field g(r) as: g ( r ) = − G m 1
r
2 r ^ displaystyle mathbf g (mathbf r )=-G m_ 1 over vert mathbf r vert ^ 2 ,mathbf hat r so that we can write: F ( r ) = m g ( r ) . displaystyle mathbf F (mathbf r )=mmathbf g (mathbf r ). This formulation is dependent on the objects causing the field. The field has units of acceleration; in SI, this is m/s2. 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 g ( r ) = − ∇ V ( r ) . displaystyle mathbf g (mathbf r )=-nabla V(mathbf r ). 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 V ( r ) = − G m 1 r . displaystyle V(r)=-G frac m_ 1 r . the gravitational field is on, inside and outside of symmetric masses. As per Gauss Law, field in a symmetric body can be found by the mathematical equation: ∂ V displaystyle partial V g ( r ) ⋅ d A = − 4 π G M enc , displaystyle mathbf g(r) cdot dmathbf A =-4pi GM_ text enc , where ∂ V displaystyle partial V is a closed surface and M enc displaystyle M_ text enc is the mass enclosed by the surface. Hence, for a hollow sphere of radius R displaystyle R and total mass M displaystyle M ,
g ( r )
= 0 , if r < R G M r 2 , if r ≥ R displaystyle mathbf g(r) = begin cases 0,& mbox if r<R\\ dfrac GM r^ 2 ,& mbox if rgeq Rend cases For a uniform solid sphere of radius R displaystyle R and total mass M displaystyle M ,
g ( r )
= G M r R 3 , if r < R G M r 2 , if r ≥ R displaystyle mathbf g(r) = begin cases dfrac GMr R^ 3 ,& mbox if r<R\\ dfrac GM r^ 2 ,& mbox if rgeq Rend cases Problematic aspects[edit]
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.[31] For example, Newtonian gravity provides an accurate
description of the Earth/
Φ c 2 = G M s u n r o r b i t c 2 ∼ 10 − 8 , ( v E a r t h c ) 2 = ( 2 π r o r b i t ( 1 y r ) c ) 2 ∼ 10 − 8 displaystyle frac Phi c^ 2 = frac GM_ mathrm sun r_ mathrm orbit c^ 2 sim 10^ -8 ,quad left( frac v_ mathrm Earth c right)^ 2 =left( frac 2pi r_ mathrm orbit (1 mathrm yr )c right)^ 2 sim 10^ -8 where rorbit 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 with Newton's expression[edit] There is no immediate prospect of identifying the mediator of gravity.
Attempts by physicists 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:
Observations conflicting with Newton's formula[edit] Newton's Theory does not fully explain the precession of the perihelion of the orbits of the planets, especially of planet Mercury, which was detected long after the life of Newton.[32] There is a 43 arcsecond per century discrepancy between the Newtonian calculation, which arises only from the gravitational attractions from the other planets, and the observed precession, made with advanced telescopes during the 19th century. The predicted angular deflection of light rays by gravity that is calculated by using Newton's Theory is only one-half of the deflection that is actually observed by astronomers. Calculations using General Relativity are in much closer agreement with the astronomical observations. In spiral galaxies, the orbiting of stars around their centers seems to strongly disobey Newton's law of universal gravitation. Astrophysicists, however, explain this spectacular phenomenon in the framework of Newton's laws, with the presence of large amounts of dark matter. Newton's reservations[edit]
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" that his equations implied. In 1692, in his
third letter to Bentley, he wrote: "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."
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 (although he invented two mechanical hypotheses in 1675 and
1717). 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 has yet to be found. And in
Newton's 1713
Part of a series on Spacetime General relativity
General relativity Introduction to general relativity Mathematics of general relativity Einstein field equations Classical gravity Introduction to gravitation Newton's law of universal gravitation Relevant mathematics
Four-vector
Derivations of relativity
v t e These objections were explained 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, energy and momentum 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. In general relativity, the gravitational force is a fictitious force due to the curvature of spacetime, because the gravitational acceleration of a body in free fall is due to its world line being a geodesic of spacetime. Extensions[edit] Newton was the first to consider in his Principia an extended expression of his law of gravity including an inverse-cube term of the form F = G m 1 m 2 r 2 + B m 1 m 2 r 3
displaystyle F=G frac m_ 1 m_ 2 r^ 2 +B frac m_ 1 m_ 2 r^ 3 , B displaystyle B a constant attempting to explain the Moon's apsidal motion. Other extensions were proposed by Laplace (around 1790) and Decombes (1913):[34] F ( r ) = k m 1 m 2 r 2 exp ( − α r ) displaystyle F(r)=k frac m_ 1 m_ 2 r^ 2 exp(-alpha r) (Laplace) F ( r ) = k m 1 m 2 r 2 ( 1 + α r 3 ) displaystyle F(r)=k frac m_ 1 m_ 2 r^ 2 left(1+ alpha over r^ 3 right) (Decombes) In recent years, quests for non-inverse square terms in the law of gravity have been carried out by neutron interferometry.[35] Solutions of Newton's law of universal gravitation[edit] Main article: n-body problem The n-body problem is an ancient, classical problem[36] of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem — from the time of the Greeks and on — has been motivated by the desire to understand the motions of the Sun, planets and the visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem too.[37] The n-body problem in general relativity is considerably more difficult to solve. The classical physical problem can be informally stated as: given the quasi-steady orbital properties (instantaneous position, velocity and time)[38] of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.[39] The two-body problem has been completely solved, as has the restricted three-body problem.[40] See also[edit] Book: Isaac Newton Physics portal Bentley's paradox Gauss's law for gravity Jordan and Einstein frames Kepler orbit Newton's cannonball Newton's laws of motion Static forces and virtual-particle exchange Notes[edit] ^ It was shown separately that large, spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers. References[edit] ^ Isaac Newton: "In [experimental] philosophy particular propositions
are inferred from the phenomena and afterwards rendered general by
induction": "Principia",
External links[edit] Feather & Hammer Drop on Moon on YouTube
Newton‘s Law of Universal
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