Mass

Mass is the quantity of matter contained in an object. It is one of the most fundamental properties a physical body has, and you already met it under a different guise in the Force and Laws of Motion chapter: mass is a measure of inertia, a body's natural tendency to resist any change in its state of rest or motion. A heavier suitcase is harder to start rolling and harder to stop — that reluctance to change velocity is inertia, and mass is what quantifies it. The bigger the mass, the bigger the unbalanced force needed to produce a given change in velocity. A few defining properties are worth holding onto as a checklist.

• SI unit: the kilogram (kg), one of the seven base SI units.
• Scalar quantity: mass has magnitude only — no direction attached to it, unlike a force.
• Constant everywhere: the mass of a given object is exactly the same whether it is weighed on Earth, carried to the Moon, or floating in a spacecraft in deep space, because mass depends only on how much matter the object contains, not on any external gravitational field acting on it.
• Always positive for any physical object, and never zero for real matter.

Because mass is fixed no matter where you go, it is the more fundamental of the two quantities that sit side by side in this chapter: mass and weight. Weight is the force of gravity pulling on that fixed mass, and it does change from place to place, because the local gravitational field strength — the value of $g$ — differs on the Moon, on Mars, or in orbit from what it is on Earth's surface. Two objects can share identical mass and yet have wildly different weight on different worlds. Confusing the two is the single most common error in this topic, worth pinning down in symbols:

where: $W$ is weight (a force, in newtons), $m$ is mass (in kilograms, constant for a given object), and $g$ is the local acceleration due to gravity (in metres per second squared, different on every world). $m$ sits unchanged on the right, while $g$ does all the varying — that single observation is the whole mass–weight distinction, developed fully in the next article. Mass is measured using a common balance, or beam balance — the two-pan device that compares an unknown mass against known standard masses until it balances horizontally. This works identically anywhere in the universe because, as the derivation below shows, it never measures a force directly; it only compares two weights under the same local gravity, and that comparison cancels $g$ out entirely.

Problem. An object has a mass of 12 kg on Earth. A beam balance is separately used to compare a different unknown object against a set of standard 1 kg masses; on Earth the balance shows the object equals 5 kg. (a) What is the mass of the first object on the Moon? (b) If the same beam balance, object, and standard masses were used on the Moon, what would the balance read, and why?

Step 1. For part (a), recall that mass depends only on the amount of matter in a body, not on the local gravitational field. Taking the 12 kg object to the Moon does not remove or add any matter to it, so its mass on the Moon is still 12 kg, exactly as it was on Earth.

Step 2. For part (b), set up the balance condition on the Moon using the derivation above. If the unknown object's mass is $m_1 = 5\ \text{kg}$ and the standard masses total $m_2 = 5\ \text{kg}$, the balance condition at any location is $m_1 g = m_2 g$, and since $m_1 = m_2 = 5\ \text{kg}$ already on Earth, this equality is a statement about the masses themselves, unaffected by which value of $g$ is plugged in.

Step 3. On the Moon, $g$ is smaller than on Earth, so both pans genuinely weigh less than they did on Earth — but they weigh less by exactly the same factor, because both pans experience the same reduced lunar $g$. The beam therefore still balances level with the identical 5 kg of standard masses on the other pan.

Step 4. The beam balance reads 5 kg on the Moon as well — the same reading as on Earth. This is the direct experimental payoff of the derivation: a beam balance is a mass-measuring instrument precisely because the local value of $g$ cancels out of its balance condition, so its reading does not depend on where in the universe you use it.

Competitive stretch

If the same 5 kg object were instead placed on a spring balance (which measures weight directly through the stretch of a spring, with no second pan to cancel $g$), its reading on the Moon would drop to roughly one-sixth of its Earth reading, since lunar $g \approx 1.6\ \text{m/s}^2$ is about one-sixth of Earth's $9.8\ \text{m/s}^2$. The spring balance would show a much smaller number even though the object's mass has not changed at all — the clearest possible illustration of why mass and weight are not the same quantity.

• Confusing mass with weight in everyday speech. Saying "how much does it weigh?" when asking for mass is fine casually, but in physics they are distinct: mass is in kilograms and constant everywhere; weight is a force in newtons and varies with $g$.
• Thinking astronauts lose mass because they look weightless. An astronaut in orbit has exactly the same mass as on Earth; only their apparent weight changes, because they are in continuous free fall around the Earth, not because any matter has vanished.
• Assuming a beam balance and a spring balance measure the same thing. A beam balance compares two masses via a cancelling-$g$ trick and works correctly anywhere; a spring balance measures a single weight directly and gives different readings at different values of $g$.
• Treating mass as a vector. Mass is a scalar with magnitude only; it is weight, a force, that has direction — always vertically downward, toward the centre of the gravitating body.

Ask the tutor: Want to see exactly how weight is calculated from mass with $W = mg$, including why the same object weighs about six times less on the Moon — ask the AI tutor to walk through the mass–weight relationship with more worked numbers.