A spring balance

You need: a soft spring from an old pen, a thread, a ruler and ten small weights that are all alike, each of ten grams.

You need: a soft spring, a thread, a ruler and $10$ weights that are alike, each $10$ g.

Number of weights $n=10$, each $m_0=10\text{g}$.

Hang the spring from a nail and mark where its end rests. Hang one weight and the end sinks two millimetres. A second weight: four millimetres. Every weight adds the same two millimetres.

Hang the spring and mark its end. With $1$ weight, $x=2$ mm; with $2$, $x=4$ mm. Each weight adds the same $2$ mm.

$x(n)=2n\text{mm}$.

The stretch is proportional to the load: two millimetres for every ten grams, which is two tenths of a millimetre for every gram. Ten weights stretch it twenty millimetres. Do not go past that: a spring pulled too far stays long, and the rule stops being true.

$x\propto m$, that is $x=km$ with $k=2/10=0.2$ mm per g. With $10$ weights, $x=0.2\times 100=20$ mm. Stop there: past its limit the spring stays long and $x/\propto m$.

$x\propto m$, $x=km$, $k=0.2\text{mm/g}$; $x(100\text{g})=20\text{mm}$; $x>20\Rightarrow x/\propto m$.

Now hang something unknown. If the end sinks fourteen millimetres, its mass is fourteen divided by two tenths: seventy grams. The surprise: take your balance to the moon and the same lump stretches it only a sixth as far, about two point three millimetres. The balance measures the pull of the earth, not the amount of stuff.

Unknown lump: $x=14\Rightarrow m=14/0.2=70$ g. Surprise: on the moon $x=14/6\approx 2.3$ mm. The balance measures the pull, not the amount.

$x=14\Rightarrow m=14/0.2=70\text{g}$; on the moon $x_M=14/6\approx 2.3\text{mm}$.