# Gravity at Earth's Surface

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## Physical Law

The acceleration due to gravity at the surface of Earth is approximately given by:

- $g = 9.8 \ \mathrm N \ \mathrm {kg}^{-1}$

Note that this is equivalent to

- $g = 9.8 \ \mathrm M \ \mathrm s^{-2}$

as an acceleration can be defined as the force per unit mass from Newton's Second Law of Motion.

## Proof

From Acceleration Due to Gravity, we have that:

- $g = \dfrac {G M} {r^2}$

where:

- $g$ is the acceleration on the body caused by the gravitational field of Earth
- $M$ is the mass of Earth, approximately $5.9736 \times 10^{24}\,\mathrm {kg}$
- $G$ is the gravitational constant, approximately $6.674 \times 10^{-11} \, \mathrm N \, \mathrm m^2 \, \mathrm {kg}^{-2}$
- $r$ is the radius of Earth, approximately $6.371 \times 10^{6}\,\mathrm m$.

Hence the result, by direct numerical calculation.

$\blacksquare$

Note that the value of $r$ is significantly variable, depending on where you are on Earth.

For fuller detail, see Gravity of Earth at Wikipedia.

## Sources

- 1966: Isaac Asimov:
*Understanding Physics*... (previous) ... (next): $\text {I}$: Motion, Sound and Heat: Chapter $2$: Falling Bodies: Free Fall