5.5 — Weight and mass

Two different concepts

Mass (m): an intrinsic property of an object — amount of matter. Unit kg. Constant — independent of location.

Weight (\vec W): the gravitational force on the object. Unit newton (N). Variable — depends on local gravitational acceleration.

\[ \vec W = m\vec g \]

At Earth's surface: |\vec W| = mg = m \cdot 9.8\ m/s^2. For m = 60 kg: W = 588\ N.

Why we confuse them

Everyday speech uses "weight" and "mass" interchangeably. "I weigh 70 kg" — physically incorrect (kg measures mass). Correct: "I have a mass of 70 kg" or "I weigh 686 N".

Weight in different places

For an object of mass 60 kg:

Place	g	Weight
Earth (sea level)	9.8	588 N
Everest summit	9.78	587 N
Pole	9.83	590 N
Moon	1.62	97 N
Mars	3.7	222 N
Jupiter	24.8	1488 N
ISS (h = 400 km)	8.7	522 N
Deep space	~0	~0 N

Mass is 60 kg everywhere.

Equivalence principle — why is m in F = ma the same as in F = mg?

Fundamental question: why does gravitational mass (m in F_g = mg) equal inertial mass (m in F = ma)?

Newton's answer: coincidence. Einstein's answer: intrinsic. The equivalence principle is the foundation of general relativity.

Experimental consequence: all objects fall with the same acceleration — because a = F/m = mg/m = g. Mass cancels. Galileo's experiment ~1600 (Galileo article) and Apollo 15 on the Moon.

Weightlessness

What does "weightless" mean? Two interpretations:

1. Intuitive (wrong): "no gravity". ISS astronauts are weightless because gravity is zero. Wrong! g ≈ 8.7 m/s² at ISS — about 90% of surface.

2. Physical (correct): "in free fall". The astronaut and the station accelerate identically. The astronaut feels no normal force from the station — hence "weightless".

Similar feeling: an elevator in free fall, or the top of a roller-coaster loop. Maximum weightlessness during vertical free fall.

Measuring mass vs weight

Balance scale: compares against a standard mass. Independent of g — same reading on the Moon.

Spring scale: measures weight via spring deflection. g-dependent — gives 1/6 on the Moon.

Note: if your household scale reads "70 kg", it's actually measuring 686 N and dividing by g = 9.8. On the Moon, the same spring would read 114 N and (with the same formula) 11.6 kg — which is wrong.

A few notes and common mistakes

1. Mass is a scalar, weight is a vector. Mass is just a number; weight has direction (down).

2. g is not constant. Decreases with altitude (Gravity Article 3). Different on other planets.

3. "Feeling lighter in an elevator" is only apparent. Your mass hasn't changed. The normal force from the seat has decreased (because you're accelerating).

4. "Weightless" satellites are under strong gravity. Without gravity, they wouldn't orbit — they'd go in a straight line.

What you should be able to do

• Distinguish mass from weight carefully
• Compute weight with W = mg on different planets
• Explain the equivalence principle and its consequence
• Define weightlessness correctly as free fall

Preview of §5.6

The force that makes real problems tricky: friction. Two kinds (static and kinetic), two coefficients (\mu_s, \mu_k), broad applications.

📚 See also: Halliday Vol 1, Ch 5, §5.5.

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