Weightlessness — true (free fall) vs apparent: Visual walkthrough

We build one single mental object: a person standing on a scale inside a box that moves. Everything follows from watching that box.

Step 1 — Draw the situation and name every letter

WHAT. A person stands on a bathroom scale inside a box (an elevator). We give names to only three things and nothing else yet:

$m$ — the mass of the person, "how much stuff they are made of" (a number of kilograms).
$g$ — the strength of gravity, "how many metres-per-second the falling speed grows each second" ( $\approx 10 m/s^{2}$ near Earth).
We pick a direction to call positive. Look at the green arrow in the figure: up is $+$ , down is $-$ . Every number after this obeys that choice.

WHY. Physics equations are silent about direction unless you fix one. Choosing "up = +" once, and never flinching, is what stops the sign mistakes the parent note warned about.

PICTURE.

Figure — Weightlessness — true (free fall) vs apparent

Step 2 — The only two forces that touch the person

WHAT. Forces are pushes and pulls, drawn as arrows from the thing pushing. On the person there are exactly two:

Gravity pulls the whole body down. Its size is $m$ times $g$ , written $m g$ . Arrow points down (so it counts as $- m g$ ).
The scale under the feet pushes up. We name that push $N$ , the normal force ("normal" is an old word for perpendicular — it presses straight out of the surface). Arrow points up ( $+ N$ ).

WHY. A normal force is the only thing you actually feel — nerves in your feet sense the floor pressing, never gravity itself. So the scale's reading is $N$ , nothing else. We must isolate exactly these two arrows before we can add them.

PICTURE.

Step 3 — Let the box move, and name that motion

WHAT. Now the box accelerates. ==Acceleration $a$ == means "how fast the velocity is changing," measured in the same up-positive convention.

$a > 0$ : velocity is being nudged upward each second (speeding up while rising, or slowing down while sinking).
$a < 0$ : velocity is being nudged downward.
$a = 0$ : velocity is not changing at all (at rest, or gliding at steady speed).

The person is locked to the box's floor, so the person shares the box's $a$ .

WHY. We introduce $a$ because the whole story is about $a$ . Notice velocity itself never entered — only its rate of change. This is the seed of the parent's warning: "moving down" ≠ "weightless."

PICTURE.

Step 4 — Add the arrows with Newton's Second Law

WHAT. Newton's Second Law says: add up the force arrows (respecting sign) and the total equals mass times acceleration. $up, + N - down, - m g = mass \times its acceleration m a$

WHY. This is the hinge of the whole derivation. The left side is "the net push-pull the person receives"; the right side is "what that net push-pull produces — an acceleration $a$ ." Setting them equal is the law. We use it here (rather than energy or momentum) because we want the instantaneous force the scale exerts, and force is exactly what $F = ma$ delivers.

PICTURE.

Step 5 — Solve for what the scale reads

WHAT. We rearrange to get $N$ alone, because $N$ is the number we actually want (the scale reading).

Start: $N - m g = ma$ . Add $m g$ to both sides: $N = m (g + a)$

WHY. Algebra just moves the known $m g$ to the other side so the unknown $N$ stands alone. Now every ingredient is on the right: your fixed weight ingredient $m g$ , adjusted by the motion ingredient $ma$ .

PICTURE.

Step 6 — Turn the dial: every case of $a$

WHAT. Feed the four meaningful values of $a$ into $N = m (g + a)$ and watch $N$ :

box's acceleration $a$	$N = m (g + a)$	feeling
$a = 0$ (rest / steady)	$N = m g$	normal weight
$a > 0$ (accel. up)	$N > m g$	heavier
$a < 0$ but $> - g$ (gentle down-accel)	$0 < N < m g$	lighter
$a = - g$ (free fall)	$N = 0$	weightless

WHY. Covering all signs of $a$ is the point — the reader must never meet a case we skipped. Each row is the same one formula with a different dial setting; there's no new physics, only new arithmetic.

PICTURE.

Step 7 — The domain of validity: $N$ can never go negative

WHAT. The formula $N = m (g + a)$ seems to keep going below zero if we push $a$ more negative than $- g$ (say the box is yanked downward faster than free fall). Plug in $a = - 1.5 g$ : the formula spits out $N = m (g - 1.5 g) = - 0.5 m g$ , a negative push. But a floor cannot pull your feet down — it can only push up or do nothing. A negative $N$ is physically impossible.

WHY. The moment the required contact push would turn negative, the feet simply lift off the floor: contact is lost and $N = 0$ . So the honest statement of our result carries a domain: $N = m (g + a) valid for a \geq - g, N = 0 for a \leq - g .$ $a = - g$ (free fall) is exactly the boundary — the last moment feet and floor still touch with zero force. Below it, they part company.

PICTURE.

Step 8 — Free fall, term by term (the boundary case)

WHAT. Set $a = - g$ exactly — the edge of the valid domain from Step 7: $N = m (g + (- g)) = m \cdot 0 = 0$ The upward push shrinks to nothing. Gravity's $m g$ is still a full-strength force pulling down (that is precisely why the box accelerates downward at $g$ ), but the floor and the feet now fall together, so the floor no longer needs to press.

WHY. This is the whole topic in one substitution. Weightlessness is not "gravity off" — it is "support force off." Gravity ( $m g$ ) is still one of the two force arrows from Step 2; what has vanished is only the contact arrow $N$ . One force went to zero; the other is untouched.

PICTURE.

Step 9 — Sanity-check with the parent's numbers

WHAT. A $60 kg$ person, $g = 10$ :

Up at $a = + 2$ : $N = 60 (10 + 2) = 720 N$ — heavier than true weight $600 N$ .
Cable snaps, $a = - 10$ : $N = 60 (10 - 10) = 0 N$ — weightless.
Descending but slowing at $3 m/s^{2}$ (so $a = + 3$ , acceleration points up): $N = 60 (10 + 3) = 780 N$ .

WHY. Numbers close the loop: the same $N = m (g + a)$ reproduces every worked example in the parent, and the third one re-confirms Step 3 — a downward trip with upward acceleration feels heavy, because only $a$ matters.

PICTURE.

The one-picture summary

Read the figure left to right, panel by panel — each sub-panel is spelled out here so nothing is decorative:

Left panel (the free body). The person with two arrows: teal $N$ pushing up from the floor, ink $m g$ pulling down. These are the only two forces (Step 2).
First arrow → middle text. Newton's law adds those arrows: $N - m g = ma$ (Step 4).
Middle text. Rearranged, this becomes the boxed result $N = m (g + a)$ (Step 5).
"slide $a$ …" label. We turn the single dial $a$ through all its values (Step 6), staying inside the valid range $a \geq - g$ (Step 7).
Right panel. At the boundary $a = - g$ the motion bit cancels the weight bit, giving $N = 0$ — plum "weightless" (Step 8). Push $a$ any lower and contact is simply lost, still $N = 0$ .

Read as one sentence: two arrows on a person → Newton's law adds them → $N = m (g + a)$ , valid for $a \geq - g$ → slide the dial $a$ → at $a = - g$ the scale reads zero.

Recall Feynman: the whole walkthrough in plain words

Put a person on a scale in a lift. Two things touch them: gravity pulling down (size $m g$ ) and the floor pushing up (size $N$ — that's the only thing they can actually feel). Newton's rule says: add those two, and you get mass times how fast the lift is speeding up or slowing down, $N - m g = ma$ . Untangle it and you get $N = m (g + a)$ — the scale reading equals your weight plus an extra bit from the lift's acceleration. If the lift accelerates up, $N$ grows, you feel heavy. If it accelerates down, $N$ shrinks. And if the rope snaps so the lift falls freely, its acceleration is exactly $- g$ : the two bits cancel and the scale reads zero. A floor can only push, never pull, so it can't drive the reading below zero — if the lift somehow fell even faster, your feet would just lift off and float. Gravity never left — it's the reason you're falling — but with the floor dropping just as fast as your feet, nothing pushes back, so you float. That's weightlessness.

Connections

Newton's Second Law — the $N - m g = ma$ hinge in Step 4.
Normal Force — the arrow $N$ that is apparent weight, and why it can't go negative (Step 7).
Newton's Third Law — why the scale displays the force you push on it.
Non-inertial Frames & Pseudo-forces — the falling-lift view where gravity looks "cancelled."
Circular Motion & Centripetal Acceleration — orbital free fall with $a = g$ inward.
Free Fall Kinematics — the $a = - g$ motion behind Step 8.
Gravitation — Variation of g with altitude — why $g \approx 8.7$ at ISS, not zero.