Instead of computing Newton's Law of Gravitation for every object, describe what a planet does to the space around it: the force on each kilogram placed there. That is the gravitational field strength, and near Earth's surface you already know its value.
Key idea
$g = \dfrac{F}{m} = \dfrac{GM}{r^2}$, in N/kg, which is the same as m/s². At Earth's surface it is $9.8\ \text{N/kg}$; it shrinks with distance from Earth's center but never reaches zero.
Why it works
Divide Newton's law by the small mass: $\dfrac{GMm/r^2}{m} = \dfrac{GM}{r^2}$. What's left depends only on the planet and the distance, not on the object. And by the second law, an object in free fall accelerates at $F/m$, which is $g$. So the field strength and the free-fall acceleration are the same number.
Worked examples
Worked example: Check the surface value
Use $G$, Earth's mass $5.972 \times 10^{24}\ \text{kg}$ and radius $6.371 \times 10^{6}\ \text{m}$ to find $g$ at the surface.
- $$g = \dfrac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{(6.371 \times 10^{6})^2}.$$
- $g \approx 9.82\ \text{N/kg}$, matching the familiar $9.8$.
Answer: about $9.8\ \text{N/kg}$
Worked example: At the space station
Find $g$ at $400\ \text{km}$ altitude.
- $r = 6.771 \times 10^{6}\ \text{m}$.
- $$g = \dfrac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{(6.771 \times 10^{6})^2} \approx 8.69\ \text{N/kg},$$ about 89% of the surface value.
Answer: about $8.69\ \text{N/kg}$
Worked example: Another planet
A planet has twice Earth's mass and twice its radius. What is $g$ at its surface?
- $g \propto \dfrac{M}{R^2}$: factor $\dfrac{2}{2^2} = \dfrac12$.
- $g = 4.9\ \text{N/kg}$.
Answer: $4.9\ \text{N/kg}$
| Body | g at the surface (about) |
|---|---|
| Earth | 9.8 N/kg |
| Moon | 1.62 N/kg |
| Mars | 3.71 N/kg |
| Jupiter (cloud tops) | 24.8 N/kg |
Common mistake
Thinking $g$ is zero in orbit, or the same at every height. Astronauts float because they and their station are falling together around Earth, not because gravity is gone: at the station's height it is still about 89% as strong. Check: $g = GM/r^2$ is zero only as $r \to \infty$.
Try it in the Science Lab's Orbit Lab.