section_path: Examples of forces in classical mechanics/Gravitational force or Gravity
section_name: Gravitational force or Gravity
paragraph_text: Gravitational force or Gravity
  
  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 acceleration due to gravity towards the surface of the Earth is usually designated as g {\displaystyle \mathbf {g} } and has a magnitude of about 9.81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth. 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 m {\displaystyle m} will experience a force: F = m g. {\displaystyle \mathbf {F} =m\mathbf {g}.}
  
  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.
  
  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 law of gravity that could account for the celestial motions that had been described earlier using Kepler's laws of planetary motion.
  
  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. Combining these ideas gives a formula that relates the mass (m ⊕ {\displaystyle m_{\oplus }}) and the radius (R ⊕ {\displaystyle R_{\oplus }}) of the Earth to the gravitational acceleration: g = − G m ⊕ R ⊕ 2 r ^, {\displaystyle \mathbf {g} =-{\frac {Gm_{\oplus }}{{R_{\oplus }}^{2}}}{\hat {\mathbf {r} }},} where the vector direction is given by r ^ {\displaystyle {\hat {\mathbf {r} }}}, is the unit vector directed outward from the center of the Earth.
  
  In this equation, a dimensional constant G {\displaystyle G} 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 G {\displaystyle G} using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing G {\displaystyle G} could allow one to solve for the Earth's mass given the above equation. Newton realized that since all celestial bodies followed the same 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 m 1 {\displaystyle m_{1}} due to the gravitational pull of mass m 2 {\displaystyle m_{2}} is F = − G m 1 m 2 r 2 r ^, {\displaystyle \mathbf {F} =-{\frac {Gm_{1}m_{2}}{r^{2}}}{\hat {\mathbf {r} }},} where r {\displaystyle r} is the distance between the two objects' centers of mass and r ^ {\displaystyle {\hat {\mathbf {r} }}} is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.
  
  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 were invented to calculate the deviations of orbits 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.