Editing Mass (section)

=== Universal gravitational mass ===
[[File:Universal gravitational mass.jpg|An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center.|left|thumb]]
In contrast to earlier theories (e.g. [[celestial spheres]]) which stated that the heavens were made of entirely different material, Newton's theory of mass was groundbreaking partly because it introduced [[Newton's law of universal gravitation|universal gravitational mass]]: every object has gravitational mass, and therefore, every object generates a gravitational field.  Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object. If a large collection of small objects were formed into a giant spherical body such as the Earth or Sun, Newton calculated the collection would create a gravitational field proportional to the total mass of the body,<ref name="principia"/>{{rp|397}} and inversely proportional to the square of the distance to the body's center.<ref name="principia"/>{{rp|221}}<ref group="note">These two properties are very useful, as they allow spherical collections of objects to be treated exactly like large individual objects.</ref>

For example, according to Newton's theory of universal gravitation, each carob seed produces a gravitational field.  Therefore, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere.  Hence, it should be theoretically possible to determine the exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun.  In fact, by [[unit conversion]] it is a simple matter of abstraction to realize that any traditional mass unit can theoretically be used to measure gravitational mass.

[[File:Cavendish Experiment.png|thumb|right|Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper.]]

Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice.  According to Newton's theory, all objects produce gravitational fields and it is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere.  However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the [[Cavendish experiment]], did not occur until 1797, over a hundred years later. [[Henry Cavendish]] found that the Earth's density was 5.448 ± 0.033 times that of water.  As of 2009, the Earth's mass in kilograms is only known to around five digits of accuracy, whereas its gravitational mass is known to over nine significant figures.{{clarify|reason="mass in kilograms" does not specify an experimental method. What's the distinction between the methods?|date=January 2014}}

Given two objects A and B, of masses ''M''<sub>A</sub> and ''M''<sub>B</sub>, separated by a [[Displacement (vector)|displacement]] '''R'''<sub>AB</sub>, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude
: <math>\mathbf{F}_{\text{AB}}=-GM_{\text{A}}M_{\text{B}}\frac{\hat{\mathbf{R}}_{\text{AB}}}{|\mathbf{R}_{\text{AB}}|^2}\ </math>,
where ''G'' is the universal [[gravitational constant]]. The above statement may be reformulated in the following way: if ''g'' is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass ''M'' is
: <math>F=Mg</math>.
This is the basis by which masses are determined by [[weighing]]. In simple [[spring scales]], for example, the force ''F'' is proportional to the displacement of the [[spring (device)|spring]] beneath the weighing pan, as per [[Hooke's law]], and the scales are [[calibration|calibrated]] to take ''g'' into account, allowing the mass ''M'' to be read off. Assuming the gravitational field is equivalent on both sides of the balance, a [[Beam balance|balance]] measures relative weight, giving the relative gravitation mass of each object.