Newton's law of universal gravitation

The mechanisms of Newton's law of universal gravitation; a point mass m₁ attracts another point mass m₂ by a force F₂ which is proportional to the product of the two masses and inversely proportional to the square of the distance (r) between them. Regardless of masses or distance, the magnitudes of |F₁| and |F₂| will always be equal. G is the gravitational constant.

Isaac Newton's theory of universal gravitation is a physical law describing the gravitational attraction between bodies with mass. It is a part of classical mechanics and was first formulated in Newton's work Philosophiae Naturalis Principia Mathematica, first published on July 5 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:

$F=G{\frac {m_{1}m_{2}}{r^{2}}},$

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

Newton's law has since been superseded by Einstein's theory of general relativity, but it continues to be used as an excellent approximation of the effects of gravity.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.

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 $a=F/m_{1}$. Substituting $F$ from the earlier equation gives:

$a_{1}=G{\frac {m_{2}}{r^{2}}}$     and similarly     $a_{2}=G{\frac {m_{1}}{r^{2}}}$.

Assuming SI units, gravitational acceleration (as acceleration in general) is measured in metres per second squared (m/s² or ms^-2). Non-SI units include gals, gees, and feet per second squared.

The force of gravity attracting a mass $\ m_{1}$ to another mass $\ m_{2}$ will also be accompanied by a force attracting $\ m_{2}$ to $\ m_{1}$. Therefore the position of one mass from the second mass gravitationally accelerates according to:

$\left|{\frac {d^{2}\ {\vec {r}}}{dt^{2}}}\right|=a_{1}+a_{2}=G{\frac {m_{1}+m_{2}}{r^{2}}}$

When $m_{1}$ is negligible compared to $m_{2}$, the acceleration will be approximately the same regardless of the value of $m_{1}$. So the acceleration will be approximately the same for all small masses. 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, 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. (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:

• 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 in the hollow section.

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.

$\mathbf {F} _{12}=-G{m_{1}m_{2} \over {\vert \mathbf {r} _{12}\vert }^{2}}\,\mathbf {\hat {r}} _{12}$

where

$\mathbf {F} _{12}$ is the force applied on object 2 due to object 1

$G$ is the gravitational constant

$m_{1}$ and $m_{2}$ are respectively the masses of objects 1 and 2

$\vert \mathbf {r} _{12}\vert \ =\vert \mathbf {r} _{2}-\mathbf {r} _{1}\vert$ is the distance between objects 1 and 2

$\mathbf {\hat {r}} _{12}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{\vert \mathbf {r} _{2}-\mathbf {r} _{1}\vert }}$ 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

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 $\mathbf {r}$ instead of $\mathbf {r} _{12}$ and $m$ instead of $m_{2}$ and define the gravitational field $\mathbf {g} (\mathbf {r} )$ as:

$\mathbf {g} (\mathbf {r} )=-G{m_{1} \over {{\vert \mathbf {r} \vert }^{2}}}\,\mathbf {\hat {r}}$

so that we can write:

$\mathbf {F} (\mathbf {r} )=m\mathbf {g} (\mathbf {r} )$

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

$\mathbf {g} (\mathbf {r} )=-\mathbf {\nabla } V(\mathbf {r} )$.

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

$V(r)=-G{\frac {m_{1}}{r}}.$

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. For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since

${\frac {\Phi }{c^{2}}}={\frac {GM_{\mathrm {sun} }}{r_{\mathrm {orbit} }c^{2}}}\sim 10^{-8},\quad \left({\frac {v_{\mathrm {Earth} }}{c}}\right)^{2}=\left({\frac {2\pi r_{\mathrm {orbit} }}{(1\ \mathrm {yr} )c}}\right)^{2}\sim 10^{-8}$

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 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: 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 orbits of the planets, especially of planet Mercury. 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.

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.

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.^{(unreferenced)}

See also

• Newton's cannonball
• Newton's laws of motion
• Orbital mechanics - the analysis of Newton's laws as it applies to orbits
• Gauss's law for gravity

Notes