Gravity near the Earth: the value of g

The same acceleration for everything

Drop a heavy stone and a light one together: without air they land together. Every object near the Earth falls with the same acceleration, $g$.

Finding g

The Earth's pull on a mass $m$ is $F = \dfrac{GMm}{R^2}$, and $F = mg$. So the $m$ cancels:

That is why every object falls the same way: $g$ does not depend on the object's own mass.

The value of g

With $M = 6.0 \times 10^{24}\ \text{kg}$ and $R = 6.4 \times 10^{6}\ \text{m}$, $g = 9.8\ \text{m/s}^2$.

Does g change?

• It is a little less at the equator than at the poles, because the Earth bulges.
• It falls as you go up, because $R$ becomes the distance from the centre.
• Other worlds have other values: $1.6\ \text{m/s}^2$ on the Moon, about $25\ \text{m/s}^2$ on Jupiter.

You should find that $g$ grows in step with the mass but falls with the square of the radius. A planet with twice the mass and $\sqrt2$ times the radius has the Earth's $g$.

Watch out. $g$ (small, $9.8\ \text{m/s}^2$) is an acceleration. $G$ (capital, $6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$) is a constant. Do not mix them up.

A worked example

Find $g$ on the Moon, Mars, the Earth and Jupiter from their masses and radii.

1. Moon. With $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$, $g = \dfrac{G \times 7.35 \times 10^{22}\ \text{kg}}{(1.74 \times 10^{6}\ \text{m})^2} = 1.62\ \text{m/s}^2$.
2. Mars $3.73\ \text{m/s}^2$, Earth $9.77\ \text{m/s}^2$ (about $9.8\ \text{m/s}^2$), Jupiter $24.8\ \text{m/s}^2$.
3. So $g$ on Jupiter is about fifteen times that on the Moon.

Practise with new numbers

Try it in Python

The program works out $g$ on the four bodies.

Remember these

• $g = \dfrac{GM}{R^2} = 9.8\ \text{m/s}^2$ near the Earth.
• $g$ is the same for every object, whatever its mass.
• $g$ falls as you go higher, and differs from world to world.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why does a heavy stone not fall faster than a light one, without air?

2. A planet has the Earth's mass and half its radius. What is $g$ there?

3. Is $g$ bigger at the poles or at the equator?

4. What is the unit of $G$?