Foundations — Weightlessness — true (free fall) vs apparent

Before you can trust the parent note's punchline — $N=m(g+a)$, and $N=0$ in free fall — every letter in that little equation has to mean something to you. This page builds each symbol from nothing, in the order they lean on each other, so no notation ever appears before it is earned.

0 — What is a "force"? (the arrow that pushes or pulls)

The picture: stand a block on a table. Two arrows act on it — one pulling it down (gravity), one pushing it up (the table). See the figure below.

Why the topic needs it: the entire subject is a competition between two arrows (gravity down, support up). If you can't picture arrows adding and cancelling, none of the algebra will feel like anything.

1 — Mass $m$ (how much stuff, how stubborn)

The picture: imagine shoving two trolleys with the same hand-push. The heavy one (big $m$) barely moves; the light one shoots off. Same push, different response — that "stubbornness" is mass.

Why the topic needs it: $m$ appears in both the pull of gravity ($mg$) and the resistance to acceleration ($ma$). It's the one quantity in both sides of the story.

2 — Acceleration $a$ (how fast the speeding-up happens)

We first need the idea of velocity, then acceleration.

The picture: a lift moving down but braking. The blue velocity arrow points down; the orange acceleration arrow points up (it opposes the motion to slow it). Look at how they point opposite ways.

Why the topic needs it: weightlessness is defined by a specific value of acceleration ($a=-g$), not by which way you're travelling. Getting $a$ right is the whole game.

3 — Sign convention: what "up positive" means

The picture: a vertical number line drawn beside the lift. Above the arrow-tail is $+$, below is $-$. Every force and acceleration just becomes a signed number on that line.

Why the topic needs it: $N=m(g+a)$ only works if signs are consistent. Gravity's acceleration then reads $-g$ (it points down), which is exactly why a free-falling lift uses $a=-g$, not $+g$. The parent note's fourth "mistake" is entirely a sign-convention error.

4 — Gravitational acceleration $g$ and true weight $mg$

The picture: every object in the frame carries a fixed downward arrow of length $mg$. It never disappears while you're near Earth — even in orbit ($g\approx 8.7$) it's almost as long.

Why the topic needs it: this is true weight, the thing people confuse with what a scale shows. Keeping $mg$ (the real pull) separate from $N$ (the felt push) is the heart of the whole chapter. See Gravitation — Variation of g with altitude for why $g$ shrinks with height but never reaches zero at the ISS.

5 — Normal force $N$ (the push you actually feel)

The picture: two arrows on the standing person — gravity $mg$ pointing down, the support arrow $N$ pointing up from the floor. The scale's number is the length of that upward arrow.

Why the topic needs it: the parent note's entire thesis is "$N$ = apparent weight = what you feel." When $N=0$ you feel weightless. Full detail lives in Normal Force.

6 — Newton's Second Law: tying $m$, $a$, and forces together

Applying it to the standing person with up-positive: the up force is $+N$, the down force is $-mg$, and the body accelerates with the lift at $a$: $N-mg=ma⟹N=m(g+a)$

The picture: the two arrows ($N$ up, $mg$ down) don't cancel unless $a=0$. Their difference is what drives the acceleration. When $a=-g$, the difference is zero and $N$ itself is zero.

Why the topic needs it: this is literally the derivation the parent note runs. Deeper foundations of $F_{\text{net}}=ma$ are in Newton's Second Law.

7 — Free fall and orbit as the same thing

An orbiting astronaut is also in free fall: gravity is the only force, giving acceleration $g$ — but pointed toward Earth's centre, curving the path into a circle instead of a straight drop. See Circular Motion & Centripetal Acceleration and Free Fall Kinematics. Inside the falling lift, if you pretend you're standing still, gravity appears to "vanish" — that trick is formalised in Non-inertial Frames & Pseudo-forces.

How the foundations feed the topic

Equipment checklist

Use these ::: reveals to self-test before reading the main note.