Foundations — Gravitational field intensity g = GM - r²

Before you can trust the parent note Gravitational Field Intensity, you must be able to read it. Every symbol below appears in that note. We define each in plain words, draw it, and say why the topic can't live without it. They are ordered so each one leans on the one before.

1. Mass $M$ and mass $m$ — the "how much stuff" numbers

The picture: imagine a bowling ball ($M$) sitting on a table and a marble ($m$) you hold near it. The bowling ball is the "boss"; the marble is the "probe".

Why the topic needs both: the whole point of a field is to describe what the source $M$ does to space, without caring which marble $m$ you use to test it. Keeping the two letters separate is what lets us later cancel $m$ and be left with a source-only quantity.

2. Distance $r$ — measured from the CENTRE

This is the symbol students most often get wrong, so we draw it carefully.

The picture (look at the figure): the teal dot is the centre of the planet. The orange arrow labelled $r$ runs from that centre all the way out to where the marble sits — not from the surface. If you are at altitude $h$ above the ground, then

$r=R+h$

where $R$ (capital R) is the planet's radius. The plum bracket in the figure shows $h$ starting at the surface, while $r$ starts deeper, at the core.

Why the topic needs it: gravity gets weaker the farther you are. To say "how far", you must agree on from where — and gravity acts as if all of a sphere's mass sits at its centre. So $r$ is always centre-to-you.

3. Force $\vec{F}$ and the arrow on top — vectors vs plain numbers

The picture: a scalar is a length written on a ruler; a vector is an arrow — its length is the size, its heading is the direction.

In the figure, the burnt-orange arrow is the force $\vec{F}$: it has a length (how hard) and a direction (which way — here, straight toward the planet's centre). The grey number beside it, "$5$", is what's left if you forget the direction — that's the scalar $F$ (no arrow).

Why the topic needs it: gravity always pulls toward the source. "Toward" is a direction, so force must be a vector. Later, when Earth and Moon pull on the same point from opposite sides, we can only combine their pulls correctly if we track direction — that's why the parent note subtracts the two fields instead of adding the numbers.

4. The unit vector $\hat{r}$ — "which way is out?"

The picture: stand at your point and look straight back toward the planet's centre. Now turn around 180°. The direction you now face — directly away from $M$ — is $\hat{r}$.

Why the topic needs it: to write the field as a vector, we split it into "how big" and "which way". The size is $\frac{GM}{r^2}$; the direction is carried by $\hat{r}$. Because gravity pulls inward but $\hat{r}$ points outward, we stick a minus sign in front:

$\vec{g}=-\frac{GM}{r^2}\hat{r}$

The minus is literally the phrase "opposite to $\hat{r}$" = "inward, toward $M$".

5. The gravitational constant $G$ — nature's fixed strength dial

The picture: think of $G$ as a master volume knob for all of gravity, welded in place. It is tiny, which is why you don't feel yourself pulling your friend across the room.

Why the topic needs it: the force law needs some number to turn "kilograms and metres" into "newtons of pull". $G$ is that conversion factor. Without it, $\frac{Mm}{r^2}$ would just be a shape with no physical size.

6. The exponent "${}^2$" and the fraction bar — the shape of the falloff

The picture: double the distance and the field doesn't halve — it drops to a quarter, because $2^2=4$. Triple it, and it drops to a ninth ($3^2=9$).

Look at the figure: the teal curve is $g\propto 1/r^2$. Notice how steeply it plunges near the planet and how it flattens into a long, faint tail far away — it approaches zero but never quite touches it. The dotted markers show surface value, then $1/4$ at $2R$, then $1/9$ at $3R$.

Why $r^2$ and not, say, $r$ or $r^3$? Here is the geometric reason the parent note gives, drawn out: imagine the same number of "pull-lines" leaving $M$ in all directions. At distance $r$ they are spread over the surface of a sphere. A sphere's area is $4\pi r^2$ — it grows with the square of $r$. So the lines get diluted exactly as $1/r^2$. That is why the exponent is a 2: it comes from the sphere.

7. Putting the symbols together — reading the formula as a sentence

Now every piece of $g=\frac{GM}{r^2}$ has a meaning and a picture:

The unit ${\text{N kg}}^{-1}$ is literally "newtons of force, for each kilogram of visitor" — which is exactly the definition "force per unit mass". It also equals ${\text{m s}}^{-2}$, the units of acceleration, which is why $g$ turns out to be the free-fall acceleration (that link comes from Newton's Second Law (F=ma)).

How these foundations feed the topic

Each foundation on the left is one symbol you now own. Together they assemble into the parent formula on the right.

Equipment checklist

Test yourself — cover the right side and answer before revealing.

Connections

• Parent topic — Gravitational Field Intensity — the note these foundations unlock
• Newton's Law of Universal Gravitation — where the force law $F=GMm/r^2$ comes from
• Newton's Second Law (F=ma) — why $g$ doubles as free-fall acceleration
• Gravitational Potential and Potential Energy — the next layer built on these symbols
• Gauss's Law for Gravity — the formal statement of the sphere-area argument
• Variation of g with Altitude and Depth — uses $r=R+h$ directly
• Electric Field Intensity E=kQ/r² — same symbol machinery, force per unit charge