Force

=== Gravitational ===
{{main|Gravity}}
[[File:Falling ball.jpg|upright|thumb|Images of a freely falling basketball taken with a [[stroboscope]] at 20 flashes per second. The distance units on the right are multiples of about 12&nbsp;millimeters. The basketball starts at rest. At the time of the first flash (distance zero) it is released, after which the number of units fallen is equal to the square of the number of flashes.]]
What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the [[acceleration]] of every object in [[free-fall]] was constant and independent of the mass of the object. Today, this [[Gravitational acceleration|acceleration due to gravity]] towards the surface of the Earth is usually designated as <math> \vec{g}</math> and has a magnitude of about 9.81 [[meter]]s per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth.<ref>{{cite journal |last=Cook |first=A. H. |journal=Nature |title=A New Absolute Determination of the Acceleration due to Gravity at the National Physical Laboratory |date=1965 |doi=10.1038/208279a0 |page=279 |volume=208 |bibcode=1965Natur.208..279C |issue=5007 |s2cid=4242827 |doi-access=free }}</ref> This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of <math>m</math> will experience a force:
<math display="block">\vec{F} = m\vec{g}.</math>

For an object in free-fall, this force is unopposed and the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a [[normal force]] (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a [[Newton's law of gravity|law of gravity]] that could account for the celestial motions that had been described earlier using [[Kepler's laws of planetary motion]].<ref name=uniphysics_ch4 />

Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an [[inverse square law]]. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body.<ref name=uniphysics_ch4 /> Combining these ideas gives a formula that relates the mass (<math> m_\oplus</math>) and the radius (<math> R_\oplus</math>) of the Earth to the gravitational acceleration:
<math display="block" qid=Q30006>\vec{g}=-\frac{Gm_\oplus}{{R_\oplus}^2} \hat{r},</math>
where the vector direction is given by <math>\hat{r}</math>, is the [[unit vector]] directed outward from the center of the Earth.<ref name="Principia"/>

In this equation, a dimensional constant <math>G</math> is used to describe the relative strength of gravity. This constant has come to be known as the [[Newtonian constant of gravitation]], though its value was unknown in Newton's lifetime. Not until 1798 was [[Henry Cavendish]] able to make the first measurement of <math>G</math> using a [[torsion balance]]; this was widely reported in the press as a measurement of the mass of the Earth since knowing <math>G</math> could allow one to solve for the Earth's mass given the above equation. Newton realized that since all celestial bodies followed the same [[Kepler's laws|laws of motion]], his law of gravity had to be universal. Succinctly stated, [[Newton's law of gravitation]] states that the force on a spherical object of mass <math>m_1</math> due to the gravitational pull of mass <math>m_2</math> is
<math display="block" qid=Q11412>\vec{F}=-\frac{Gm_{1}m_{2}}{r^2} \hat{r},</math>
where <math>r</math> is the distance between the two objects' centers of mass and <math>\hat{r}</math> is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.<ref name="Principia"/>

This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the 20th century. During that time, sophisticated methods of [[perturbation analysis]]<ref>{{cite web |last=Watkins |first=Thayer |title=Perturbation Analysis, Regular and Singular |work=Department of Economics |publisher=San José State University |url=http://www.sjsu.edu/faculty/watkins/perturb.htm |access-date=2008-01-05 |archive-date=2011-02-10 |archive-url=https://web.archive.org/web/20110210010802/http://www.sjsu.edu/faculty/watkins/perturb.htm |url-status=dead }}</ref> were invented to calculate the deviations of [[orbit]]s due to the influence of multiple bodies on a [[planet]], [[moon]], [[comet]], or [[asteroid]]. The formalism was exact enough to allow mathematicians to predict the existence of the planet [[Neptune]] before it was observed.<ref name='Neptdisc'>{{cite web |url=http://www.ucl.ac.uk/sts/nk/neptune/index.htm |title=Neptune's Discovery. The British Case for Co-Prediction. |access-date=2007-03-19 |last=Kollerstrom |first=Nick |year=2001 |publisher=University College London |archive-url= https://web.archive.org/web/20051111190351/http://www.ucl.ac.uk/sts/nk/neptune/index.htm |archive-date=2005-11-11}}</ref>