Weight

Where it comes from

Galileo's falling-body experiments showed something startling: everything falls at the same rate (air resistance aside). A cannonball and a musket ball hit the ground together. But a cannonball is clearly pulled harder — it hurts more to catch. How can the pull differ while the fall is the same?

Newton's answer: gravity pulls proportionally to mass — F_g ∝ m — but mass also resists acceleration proportionally (inertia). Apply the second law to a falling object:

The vacuum verdict: they land together. The iron ball feels a bigger pull, but it also carries proportionally more inertia, and the two effects cancel exactly. Near Earth's surface that universal fall rate is g = 9.8 m/s² — and multiplying it back by the mass gives the pull itself: W = mg.

Derivation

Weight needs no new physics: it is Newton's second law applied to an object in free fall. The only input from experiment is the value of g.

Notice the direction: weight always points toward the planet's center — straight down for everyday purposes. And notice what the formula doesn't say: it never claims your mass changes. Take the same 70 kg to the Moon and the Moon pulls with g = 1.62 m/s², so you weigh less while your mass stays 70 kg.

Mass vs weight, once and for all: mass (kg) is stuff + inertia — same everywhere. Weight (N) is a force — it depends on where you stand. Saying “I weigh 70 kg” mixes them up: 70 kg is your mass, and your weight is 686 N.

How to use it

The procedure, every time:

1. Get the mass in kilograms. Grams? Divide by 1000 first. Pounds? Convert: 1 lb of mass = 0.454 kg. W = mg needs kg.
2. Multiply by the local g. Earth: 9.8. Moon: 1.62. Mars: 3.71. Jupiter: 24.8. (Unless told otherwise, assume Earth.)
3. Label the units: newtons. A “weight” in kg is a mass, not a weight — examiners hunt for this.
4. Point it down. In a free-body diagram, the weight arrow always aims at the planet's center.

Reading it backwards

The formula also finds mass from a measured weight: m = W/g. A 490 N object on Earth has m = 490/9.8 = 50 kg. And it finds g itself: on an unknown planet, g = W/m — weigh a known mass and you have measured the planet's gravity.

Worked examples

Four problems, easiest first. In each one, read every step — the why of each move is the lesson.

Memorization tips

• Say it aloud: “weight equals mass times g.” Then add the footnote: “mass is portable, weight is local.”
• The 10× estimate: g ≈ 10 for quick mental math — a 70 kg person weighs ~700 N. Refine to 9.8 for the final answer.
• Units as memory: W in newtons, m in kilograms. If your weight comes out in kg, you computed a mass.
• The Moon test: whenever mass and weight blur, ask “what happens on the Moon?” The thing that changes is the weight; the thing that doesn't is the mass.
• The cancellation story: a = mg/m = g. The heaviest and lightest objects fall together because W = mg — the formula explains the experiment.
• Scale wisdom: a scale reads the normal force, not gravity directly. Standing still, they match — that is why “weighing yourself” works.

Final challenge

Five mixed questions — Earth, other planets, elevators, and the traps, all in one. Score 5/5 and weight is yours.