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Isaac Newton 's theory of universal Gravitation (part of Classical Mechanics ) states the following:
Assuming SI Units , ''F'' is measured in Newton s (N), ''m''1 and ''m''2 in Kilogram s (kg), ''r'' in Metre s (m), and the constant ''G'' is approximately equal to 6.67 × 10−11 N m2 kg−2. ''G'' was first accurately measured in the Cavendish Experiment by the British scientist Henry Cavendish in 1798 , it was also the first test of Newton's theory of gravitation between masses in the laboratory. This was 111 years after the publication of " Philosophiae Naturalis Principia Mathematica " and 71 years after Newton's death, so all of Newton's calculations could not 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. Newton's law is used to calculate the Gravitational force between two masses; similarly Coulomb's Law is used to calculate the magnitude of Electrical Force between two charged bodies. Coulomb's Law's equation has the product of two charges in place of the product of the masses which is in Newton's Law of Gravitation. Hence, according to Coulomb's Law, the electrical force is proportional to the product of the charged bodies divided by the distance between them. 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''1 = ''F'' / ''m''1. Substituting ''F'' from the earlier equation gives: : and similarly for ''a''2. Assuming SI Units , gravitational acceleration (as acceleration in general) is measured in Metres Per Second Squared (m/s2 or m s-2). Non-SI units include Galileo s, Gees (see later), and Feet per second squared. The force attracting a mass to the earth also attracts the earth to the mass, so that their acceleration to each other is given by: : If ''m''1 is negligible compared to ''m''2, small masses would have approximately the same acceleration. However, for appreciably large ''m''1, the combined acceleration, should be considered. If ''r'' changes proportionally very little during an object's travel – such as an object 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, the near and far side of the earth are around 6,350 km different distance from the Moon ; although a small difference compared to the 385,000 km average separation, this is enough to cause a slightly different gravitational force by the moon on the earth's oceans on each side compared to that exercised on the earth itself, and hence give rise to the Tide s. 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- Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: . Preceded by ''A Guide to Newton's Principia'', by I.Bernard Cohen. University of California Press 1999 ISBN 0-520-08816-6 ISBN 0-520-08817-4. (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:
As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration. VECTOR FORM 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. : where : is the force applied on object 2 due to object 1 : is the gravitational constant : and are respectively the masses of objects 1 and 2 : is the distance between objects 1 and 2 : 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 instead of and instead of and define the gravitational field as: : so that we can write: : 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 :. If ''m''1 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 : 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 .
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