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Gravitational Force
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Gravity
Gravitation is the force of attraction that exists between all particles with mass in the universe. It is the force of gravity which is responsible for holding objects onto the surface of planets and, with Newton's law of inertia is responsible for keeping objects in orbit around one another.
"Gravity is the force that pulls you down."  Merlin in Disney's The Sword in the Stone
Merlin was right, of course, but gravity does much more than just hold you in your chair. It was the genius of Isaac Newton to recognize that. Newton recalled in a late memoir that while he was trying to figure out what kept the Moon in the sky, he saw an apple fall to the ground in his orchard, and he realized that the Moon was not suspended in the sky, but continuously falling, like a cannon ball that was shot so fast that it continuously misses the ground as it falls away due to the curvature of the Earth.
If one wishes to be precise, one should distinguish between gravitation, the universal force of attraction, and gravity, which is the resultant, on the Earth's surface, of the attraction by the earth's masses, and the centrifugal pseudoforce caused by the Earth's rotation. In casual discussion, gravity and gravitation are often used interchangeably.
By Newton's third law, any two objects exert equal and oppositely directed gravitational pull on each other.
Speed of gravity: Einstein's theory of relativity predicts that the speed of gravity (defined as the speed at which changes in location of a mass are propagated to other masses) should be consistent with the speed of light. In 2002, the FomalontKopeikin experiment produced measurements of the speed of gravity which matched this prediction. However, this experiment has not yet been widely peerreviewed, and is facing criticism from those who claim that FomalontKopeikin did nothing more than measure the speed of light in a convoluted manner.
Table of contents 
Newton's Law of Universal Gravitation
Newton explains, "Every object in the Universe attracts every other object with a force directed along the line of centers for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects."
Newton eventually published his still famous law of universal gravitation in his Principia Mathematica as follows:
where:

F = gravitational force between two objects

m_{1} = mass of first object

m_{2} = mass of second object

r = distance between the objects

G = universal constant of gravitation
Strictly speaking, this law applies only to pointlike objects. If the objects have spatial extent, the true force has to be found by integrating the forces between the various points.
Vector Form
The above form is a simplified version. It is more properly expressed as vector equation. (All quantities in bold represent vector quantities in what follows.) The form below is vectorially complete:
where:

is the force on m_{1} by m_{2}

m_{1} and m_{2} are the masses

and are the position vectors of their respective masses

G is the gravitational constant
For the force on mass two, simply multiply by 1.
The primary difference between the two formulations is that the second form uses the difference in position to construct a vector that points from one mass to the other, and then divides that vector by its length to prevent it from changing the magnitude of the force.
Newton's Reservations
It's important to understand that while Newton was able to formulate his law of gravity in his monumental work, he was not comfortable with it because 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 (see below) 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, does not enable Einstein to assign the 'cause of this power' to curve space anymore than the Law of Universal Gravitation enabled Newton to assign its cause. In his own words:

I wish we could derive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto attempted the search of nature in vain.
If science is eventually able to discover the cause of the gravitational force, Newton's wish could eventually be fullfiled as well.
Comparison with electromagnetic force
The gravitational attraction of protons is approximately a factor 10^{36} weaker than the electromagnetic repulsion. This factor is independent of distance, because both forces are inversely proportional to the square of the distance. Therefore on an atomic scale mutual gravity is negligible. However, the main force beween common objects and the earth and between celestial bodies is gravity; this is due to the fact that they (at least one of the two) are electrically neutral to a high degree: even if in both bodies there were a surplus or deficit of only one electron for every 10^{18} protons and neutrons this would already be enough to cancel gravity (or in the case of a surplus in one and a deficit in the other: double the attraction).
The relative weakness of gravity can be demonstrated with a small magnet picking up pieces of iron. The small magnet is able to overwhelm the gravitational force of the entire earth.
Gravity is small unless at least one of the two bodies is large, but the small gravitational force exerted by bodies of ordinary size can fairly easily be detected through experiments such as the Cavendish torsion bar experiment.
Selfgravitating system
A selfgravitating system is a system of masses kept together by mutual gravity. An example is a star.
History
Nobody knows for sure if Newton's recollection about the apple was accurate, but his insight is the same nevertheless. Philosophers had thought since the Greeks that the "natural" movement of stars, planets, the Sun and the Moon was circular, Kepler established that orbits are actually elliptical, but still thought that the movements of the planets was dictated by some "divine force" emanated from the sun, but Newton realized that the same force that causes a thrown rock to fall back to the Earth keeps the planets in orbit of the Sun, and the Moon in orbit of the Earth.
Newton was not alone in making significant contributions to the understanding of gravity. Before Newton, Galileo Galilei corrected a common misconception, started by Aristotle, that objects with different mass fall at different rates. To Aristotle, it simply made sense that objects of different mass would fall at different rates, and that was enough for him. Galileo, however, actually tried dropping objects of different mass at the same time. Aside from differences due to friction from the air, Galileo observed that all masses accelerate the same. Using Newton's equation, F = ma, it is plain to us why:
The above equation says that mass m_{1} will accelerate at acceleration a_{1} under the force of gravity, but divide both sides of the equation by m_{1} and:
Nowhere in the above equation does the mass of the falling body appear. When dealing with objects near the surface of a planet, the change in r divided by the initial r is so small that the acceleration due to gravity appears to be perfectly constant. The acceleration due to gravity on Earth is usually called g, and its value is about 9.8 m/s^{2} (or 32 ft/s^{2}). Galileo didn't have Newton's equations, though, so his insight into gravity's proportionality to mass was invaluable, and possibly even affected Newton's formulation on how gravity works.
However, across a large body, variations in r can create a significant tidal force.
Einstein's General Theory of Relativity
Newton's formulation of gravity is quite accurate for most practical purposes. It has a few problems with it though:

It assumes that changes in the gravitational force are transmitted instantaneously when positions of gravitating bodies change. However, this contradicts the fact that there exists a maximum velocity at which signals can be transmitted (speed of light in vacuum).

Assumption of absolute space and time contradicts Einstein's theory of Special relativity.

It predicts that light is deflected by gravity only half as much as observed.

It does not explain gravitational waves or black holes.

Under newtonian gravity (with instantaneous transmission of gravitational force), if the Universe is Euclidean, static, of uniform, average, positive density and infinite, then the total gravitational force on a point is a divergent series. In other words, newtonian gravity is inconsistent with a Universe which is Euclidean, static, of uniform, average, positive density and infinite.
For the first two of these reasons, Einstein and Hilbert developed a new theory of gravity called General Relativity, published in 1915. This theory predicts that the presence of matter "warps" the local spacetime environment, so that apparently "straight" lines through space and time have the properties we think of "curved" lines as having.
Although General Relativity is, as a theory, more accurate than Newton's law of gravity, it also requires a significantly more complicated mathematical formalism. Instead of describing the effect of gravitation as a "force", Einstein introduced the concept of curved spacetime in which bodies move along curved trajectories.
Today General Relativity is accepted as the standard description of classical gravitational phenomena. (Alternative theories of gravitation exist but are more complicated than General Relativity.) General Relativity is consistent with all currently available measurements. For weak gravitational fields and bodies moving at slow speeds at small distances, Einstein's General Relativity gives almost exactly the same predictions as Newton's law of gravitation. Crucial experiments that justified the adoption of General Relativity over Newtonian gravity were the gravitational redshift, the deflection of light rays by the Sun, and the precession of the orbit of Mercury.
More recent experimental confirmations of General Relativity were gravitational waves from orbiting binary stars, the existence of neutron stars and black holes, gravitational lensing, and the convergence of measurements in observational cosmology to an approximately flat model of the observable Universe, with a matter density parameter of approximately 30% of the critical density and a cosmological constant of approximately 70% of the critical density.
Quantum Mechanics and Waves
Gravity is the only one of the four fundamental forces of nature that stubbornly refuses to be quantised (the other three: Electromagnetism, the Strong Force, and the Weak Force, can be quantised). Quantisation means that the force is measured in discrete steps that cannot be reduced in size, no matter what; alternatively, that gravitational interaction is trasmitted by particles called gravitons. Scientists have theorized about the graviton for years, but have been frustrated in their attempts to find a consistent quantum theory for it. Many believe that string theory holds a great deal of promise to unify general relativity and quantum mechanics, but this promise has yet to be realized.
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