Gravity

(Redirected from Newtons Laws of Gravity)

HomePage | Recent changes | View source | Discuss this page | Page history | Log in |

Printable version | Disclaimers | Privacy policy

Introduction

"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. Legend has it that Newton was sitting in an (?his parents'?) apple orchard, trying to figure out what kept the moon in the sky. He saw an apple fall to the ground, and he realized that the Moon wasn't 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.

Newton's Law of Gravitation

Newton eventually published his still famous law of universal gravitation in his Principia as follows:

F = G m1m2/r2

Where F is the force mass one and mass two exert on each other, m1 and m2 are their respective masses, r is the distance between their centers of mass, and G is the constant of proportionality that is called the Gravitational Constant.

Vector Form

The above form is nice, but a tad oversimplified. It should really be expressed as a vector equation. (All quantities in bold represent vector quantities in what follows.) The below form is vectorially complete:

Fon 1 from 2 = G m1m2|r2 - r1|-3 (r2 - r1)

where Fon 1 from 2 is the force on m1 by m2, m1 and m2 are the masses, r1 and r2 are the position vectors of their respective masses, and G is the gravitational constant. For the force on mass two, simply multiply Fon 1 from 2 by negative one.

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 it's length to prevent it from changing the magnitude of the force.

History

Nobody knows for sure if the story about Newton sitting under an apple tree is true, but Newton's insight is the same nevertheless. Scientists had previously thought that the movement of stars, planets, and the Sun were all dictated by some "divine force" (?name? someone check out Kepler to verify please), 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 wasn't 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 minor 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:

F = G m1m2/r2 = m1a1

The above equation says that mass m1 will accelerate at acceleration a1 under the force of gravity, but divide both sides of the equation by m1 and:

a1 = G m2/r2

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 here on earth is usually called g, and its value is about 9.8 m/s2 (or 32 ft/s2). 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:

  1. 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).
  2. Assumption of absolute space and time contradicts Einstein's theory of Special relativity.
  3. It fails to answer why a massless photon is deflected by a gravitational field.
  4. It does not explain gravitational waves or black holes.

For the first two of these reasons, Einstein in 1915 developed a new theory of gravity called General Relativity. Today GR is accepted as the standard description of classical gravitational phenomena. (Alternative theories of gravitation exist but are more complicated than GR.) GR is consistent with all currently available measurements. For weak gravitational fields and bodies moving at slow speeds at small distances, Einstein's GR gives almost exactly the same predictions as Newton's law of gravitation. Crucial experiments that justified the adoption of GR 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 GR were gravitational waves from orbiting binary stars and existence of neutron stars and black holes.

Although GR is a theory superior to 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 space-time in which bodies move along curved trajectories.

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 being Electromagnetism, the Strong Force, and the Weak Force). What quantisation means, is 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.


/Talk