# Acceleration Due to Gravity

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The Acceleration Due to Gravity calculator computes the acceleration due to gravity (g) based on the mass of the body (m), the radius of the body (R) and the Universal Gravitational Constant (G) .

INSTRUCTIONS: Choose units and enter the following:

Acceleration Due to Gravity (g): The acceleration is returned in meters per second.   However, this can be automatically converted to compatible units via the pull-down menu.

Note: vCalc provides special mass and distance units for application in space sciences.  The mass units include Earth Mass, Jupiter Mass and Solar Mass.  The distances include Astronomical Units (ua), Light Seconds, Minutes, Hours, Days and Years, Parsecs, Kilo-Parsecs and Kilo-Light Years.

### The Science

Acceleration due to gravity is the instantaneous change in downward velocity (acceleration) caused by the force of gravity toward the center of mass.  Acceleration due to gravity is typically experienced on large bodies such as stars, planets, moons and asteroids but can occur minutely with smaller masses.  The acceleration due to gravity is different based on the mass of the star, planet, moon or asteroid and the distance from its center of mass and its surface.  For that reason, gravity has a lesser pull on bodies of lesser mass or density than the Earth such as the moon.  The formula for acceleration due to gravity is:

g = (G•M)/R²

Where:

#### Acceleration Due to Gravity in the Solar System:

Newton's law of universal gravitation states that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. (Separately it was shown that large spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers.) This is a general physical law derived from empirical observations by what Isaac Newton called induction. It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687. (When Newton's book was presented in 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him) In modern language, the law states the following:

 ,where:F is the force between the masses,G is the gravitational constant (6.673×10−11 N·(m/kg)2),m1 is the first mass,m2 is the second mass, andr is the distance between the centers of the masses.

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 6.674×10−11 N m2 kg−2. However, vCalc enables the user enter units in any of the applicable mass units (see pull-down list).  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, although 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 required only when there is a need for extreme precision, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at very close distances (such as Mercury's orbit around the sun).

### Newton's theory of gravitation

Main article: Newton's law of universal gravitation   Sir Isaac Newton, an English physicist who lived from 1642 to 1727

In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. In his own words, "I deduced that the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly."[3]

Newton's theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted for by the actions of the other planets. Calculations by both John Couch Adams and Urbain Le Verrier predicted the general position of the planet, and Le Verrier's calculations are what led Johann Gottfried Galle to the discovery of Neptune.

A discrepancy in Mercury's orbit pointed out flaws in Newton's theory. By the end of the 19th century, it was known that its orbit showed slight perturbations that could not be accounted for entirely under Newton's theory, but all searches for another perturbing body (such as a planet orbiting the Sun even closer than Mercury) had been fruitless. The issue was resolved in 1915 by Albert Einstein's new theory of general relativity, which accounted for the small discrepancy in Mercury's orbit.

Although Newton's theory has been superseded, most modern non-relativistic gravitational calculations are still made using Newton's theory because it is a much simpler theory to work with than general relativity, and gives sufficiently accurate results for most applications involving sufficiently small masses, speeds and energies.

# Reference

• http://en.wikipedia.org/wiki/Gravitation
• http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation