Acceleration due to gravity

In the sciences, the term acceleration due to gravity refers to a constant g describing the magnitude of the gravitation on Earth, the planets, and on other extraterrestrial bodies. The constant has dimension of acceleration, i.e., m/s² (length per time squared) whence its name.

In the article on gravitation it is shown that for a relatively small altitude h above the surface of a large, homogeneous, massive sphere (such as a planet) Newton's gravitational potential V is to a good approximation linear in h: V(h) = g h, where g is the acceleration due to gravity. This aproximation relies on h << R_sphere (where R_sphere is the radius of the sphere). The exact gravitational potential is not linear, but has an inverse squared dependence on the distance.

On Earth, the term standard acceleration due to gravity refers to the value of 9.80656 m/s² and is denoted as g_n. That value was agreed upon by the 3rd General Conference on Weights and Measures (Conférence Générale des Poids et Mesures, CGPM) in 1901. The actual value of acceleration due to gravity varies somewhat over the surface of the Earth; g is referred to as the local gravitational acceleration .

Any object of mass m near the Earth (for which the altitude h << R_Earth) is subject to a force m g in the downward direction that causes an acceleration of magnitude g_n toward the surface of the earth. This value serves as an excellent approximation for the local acceleration due to gravitation at the surface of the earth, although it is not exact and the actual acceleration g varies slightly between different locations around the world.

More generally, the acceleration due to gravity refers to the magnitude of the force on some test object due to the mass of another object. Under Newtonian gravity the gravitational field strength, due to a spherically symmetric object of mass M is given by:

$f=G{\frac {M}{r^{2}}}.$

The magnitude of the acceleration f is expressed in SI units of meters per second squared. Here G is the universal gravitational constant G = 6.67428×10⁻¹¹ Nm²/kg² and $r$ is the distance from the test object to the centre of mass of the Earth and M is the mass of the Earth.

In physics, it is common to see acceleration as a vector, with an absolute value (magnitude, length) f and a direction from the test object toward the center of mass of the Earth (antiparallel to the position vector of the test object), hence as a vector the acceleration is:

${\vec {f}}=-G{\frac {M}{r^{2}}}{\vec {e}}_{r}\quad {\hbox{with}}\quad {\vec {e}}_{r}\equiv {\frac {\vec {r}}{r}}.$