Newton's law of gravitation — universal, action at distance: Foundations

This page is the "unpack the toolbox" page. The parent note (the topic) throws around $m$ , $r$ , $F$ , $\overset{r}{^}$ , $G$ , $g$ , exponents like $r^{2}$ , and words like "vector" and "field". If any of those are fuzzy, the whole topic collapses. So we build each one from absolute zero, in an order where every item only uses items already defined.

1. Mass — the amount of "stuff"

Picture it. Imagine two identical boxes. Fill one with feathers and one with lead. The lead box holds more matter → bigger $m$ . Mass is NOT about size or how it feels to hold — it is a count of the underlying "stuff".

Why the topic needs it. Gravity is a pull between two piles of matter. If you don't have a number for "how much matter", you cannot even start. That is why the formula has TWO mass symbols: one for each object.

The little numbers in $m_{1}$ and $m_{2}$ are subscripts — just name-tags. $m_{1}$ means "the mass of object number 1"; it is not $m$ multiplied by $1$ .

2. Distance and the meaning of $r$

Figure — Newton's law of gravitation — universal, action at distance

Look at the figure. Two balls sit apart. The green line is $r$ — it runs from the centre of one ball to the centre of the other, not surface-to-surface. The letter $r$ is chosen because this distance is like the radius of an imaginary sphere centred on one object (we use that sphere in the next section).

Why the topic needs it. Gravity weakens with distance. To say "how much weaker", we need a single number for "how far apart" — that number is $r$ .

3. Powers and exponents — what $r^{2}$ really says

Picture it. $r^{2}$ is literally the area of a square whose side is $r$ . If $r = 3 m$ , then $r^{2} = 9 m^{2}$ — a $3 \times 3$ tile.

Why " $\div r^{2}$ " and not " $\div r$ "? Because gravity's influence spreads over the surface of a sphere, and that surface area is $4 π r^{2}$ — an area, hence the square. The next section paints that picture.

Recall If

$r$ triples, what happens to $r^{2}$ ? It multiplies by $3^{2} = 9$ . So a $1/ r^{2}$ pull becomes $9$ times weaker.

4. Why area $= 4 π r^{2}$ — the sphere that dilutes gravity

Look at the figure. Two concentric spheres. The same number of orange arrows (the "influence") pierce both. On the small sphere they are crowded; on the big sphere they are sparse. The surface area of a sphere of radius $r$ is $4 π r^{2}$ , so the influence per unit area falls as $1/ r^{2}$ .

Why the topic needs it. This is the reason the exponent is exactly $2$ — not a guess, but geometry. The same picture explains why light and sound also fade as $1/ r^{2}$ .

Here $π$ (pi) is just the fixed number $\approx 3.14159$ that connects a circle's size to its width — it always shows up when spheres and circles appear.

5. Proportionality — the " $\propto$ " arrow

Picture it. A straight line through the origin on a graph: double the horizontal, double the vertical. No bends, no offset.

Why the topic needs it. The parent builds the law in pieces: first $F \propto m_{1} m_{2}$ (force follows the masses), then $F \propto 1/ r^{2}$ (force follows the diluting). Proportionality lets us state each true relationship before we know the exact numerical constant.

6. The constant $G$ — turning " $\propto$ " into " $=$ "

Why it exists. $F \propto \frac{m _{1} m _{2}}{r ^{2}}$ tells us the shape of the law but not the scale. Nature picks the scale; we measure it (Cavendish's experiment). Multiply by $G$ and the arrow $\propto$ becomes an equals sign: $F = \frac{G m _{1} m _{2}}{r ^{2}}$

The tiny exponent $1 0^{- 11}$ means "move the decimal point 11 places left" — an absurdly small number, which is why gravity feels weak between everyday objects.

7. Vectors — giving the force a direction

Look at the figure. The force on ball 1 is an arrow pointing toward ball 2 (gravity pulls them together). The force on ball 2 is an equal-length arrow pointing toward ball 1. Same length (equal strength), opposite directions — this equal-and-opposite pairing is Newton's Third Law.

Why the topic needs it. "The Earth pulls the apple" is incomplete — pulls it which way? Downward, toward Earth's centre. Force without direction is meaningless for predicting motion, so we need vectors.

A plain $F$ (no arrow) means just the magnitude — the length of the arrow, a positive number.

8. Unit vectors — the pure "which-way" arrow $\overset{r}{^}$

Picture it. Think of a compass needle: it tells you which way but not how far. To build a full force vector we take the direction ( $\overset{r}{^}_{12}$ ) and stretch it by the magnitude ( $\frac{G m _{1} m _{2}}{r ^{2}}$ ): $F_{12} = how strong \frac{G m _{1} m _{2}}{r ^{2}} which way \overset{r}{^}_{12}$

Why the topic needs it. It cleanly splits the force into "how hard" times "which direction", so the attractive nature is encoded purely by making $\overset{r}{^}$ point toward the other mass.

9. Local field strength $g$ — gravity you can feel

Why the $m$ cancelling matters. Because $m$ vanishes, every object accelerates at the same $g$ — a feather and a hammer, ignoring air. That is Galileo's tower experiment falling out of the algebra.

Why the topic needs $g$ . $G$ is too tiny and abstract to feel. $g$ is the everyday, on-a-planet version — the number that says "9.8 metres-per-second faster every second when you drop something." See Gravitational Field & Potential for the full field idea and Weight vs Mass for $W = m g$ .

The symbol $R_{\oplus}$ (with the ⊕ Earth-circle) means "radius of the Earth"; $M_{\oplus}$ means "mass of the Earth".

10. Circular motion words (for the "falling Moon" check)

Why the topic needs it. Newton's proof of universality compares the Moon's gravity-acceleration ( $g /6 0^{2}$ ) with its circular-motion acceleration ( $4 π^{2} r / T^{2}$ ). They match — the falling apple force IS the orbiting Moon force. The circular-motion machinery lives in Circular Motion & Centripetal Force, and the orbits it produces are Kepler's Laws.

Prerequisite map

Equipment checklist

Cover the right side and test yourself — you are ready for the parent note only if each is instant.

What does

m

measure and in what unit?

The amount of matter (stuff-count) in an object, in kilograms.

What does

r

measure for two spheres?

The straight-line distance between their centres, in metres.

What does

r^{2}

mean, and what shape is it?

r \times r

; the area of a square of side

r

Why is the sphere's surface area

4 π r^{2}

important here?

Gravity's influence spreads over that area, so it dilutes as

1/ r^{2}

What does the symbol

\propto

mean?

"Is proportional to" — both quantities scale up and down by the same factor.

What turns

\propto

into

=

in the gravitation law?

Multiplying by the fixed constant

G

State

G

's value and units.

6.674 \times 1 0^{- 11} N m^{2} k g^{- 2}

What is a vector, and what does the arrow

F

add over plain

F

A quantity with size AND direction; the arrow says "direction included", plain

F

is just the size.

What does the unit vector

\overset{r}{^}_{12}

carry, and where does it point?

Pure direction (length 1), pointing from object 1 toward object 2.

How is local

g

built from

G

, and why do all objects share it?

g = GM / R^{2}

; the object's own mass cancels, so acceleration is the same for all.

What are

T

and centripetal acceleration used for in this topic?

To check the Moon's orbital inward acceleration

4 π^{2} r / T^{2}

equals its gravity acceleration

g /6 0^{2}

Connections

Parent topic — assembles all these symbols into the full law.
Newton's Third Law — the equal-and-opposite arrows in the vector figure.
Shell Theorem — why $r$ is centre-to-centre for spheres.
Gravitational Field & Potential — the field view of $g$ .
Weight vs Mass — separates the stuff-count from the pull.
Circular Motion & Centripetal Force — supplies $T$ and $a$ for the falling-Moon check.
Kepler's Laws — the orbits inverse-square gravity produces.
General Relativity — replaces "instant action at a distance" with curved spacetime.