We have to devise a method of weighing very small amounts of gold. A simple balance seems the easiest route.
How does a balance work?
When the distances from the centre are equal:
If you have a straight arm pivoted in the centre, with two identical weights hanging off at the same distance from the centre, it's quite easy to imagine that it will be balanced.
When the distances from the centre are not equal:
If you move weight A towards the centre, the balance would tip upwards from A.
The closer that a force acts to the centre of rotation, the less impact it has. It's like turning a spanner: if you grab it close to the head, it's much harder to turn than if you grab the end of the spanner. This is why the balance is most accurate when the pans are a long way from the centre.
Building our balance
So we needed a long thin 'arm' of some kind, with two identical pans to put the weights in which are both the same distance from the centre of the arm. The weights should be of a known weight which can be added to the other pan until the whole thing balances.
But we didn't have any known weights: we were just given a 500g bag of sugar. We knew we had to weigh some gold, but didn't know how much it would be. It could be twenty grams or a few milligrams.
Making your own balance
a) Decide whether you want to weigh tiny things or heavy things.
Weighing tiny things is harder. Keep the whole balance really light, and keep friction down to a minimum so the balance is responsive.
Weighing heavy things means making a robust balance that won't bend much or break when you put on the weights.
b) Choosing an arm
Find something long and thin that won't bend when you hang weights at both ends.
The longer the arm, the more accurately you can weigh things. But the longer it is, the easier it is to bend when weights are put at the ends.
For weighing small things, choose a light but stiff material for the arm, like a drinking straw or light piece of wood. For heavier things, test out possible arms before deciding which to use. It can be hard to find something that doesn't bend. We looked for something at least a metre long, and eventually used a wooden pole, as all the metal ones bent too much.
Have a small pointer at one end of the balance (eg just a bit of wire) which you can line up against a strip of paper. Then you can draw a line to represent the balance point onto the paper before starting to weigh something.
c) Balancing the arm - the pivot point
Find the centre of the pole, by balancing it on a pivot, like a nail, or a sharp knife edge, until you find the point where it stays horizontal. Pivot the arm at this point.
The final pivot, shouldn't have too much friction. A 'sticky' pivot means the balance isn't completely free to move where necessary, so the readings will be less accurate.
Ideally, make a hole through the centre of the balance and hang it by putting a nail or pin right through. The balance is held in position, reducing the chance of losing what you're weighing.
Make the hole slightly higher than the centre of the arm - it's more stable that way.
Mark a length-scale onto your balance arm, so you can easily measure off distances from the centre.
d) Making the pans
Ideally the pans should be the same weight.
For weighing light things, you have to make really light pans. We folded up square pieces of tin foil, then used small pieces of wire to hang them from the arm. You could use paper and glue, or two small containers. To keep the weights the same, start off with two pieces of tin foil or paper or whatever that are the same size. Also use bits of wire of the same length or try to use similar amounts of glue.
For weighing heavy things, the pans need to be bigger. Again, try to get two pans of the same weight. You can use string or thin rope to hook them onto the balance arm, but make sure it’s strong enough to hold the things you want to weigh. Sounds obvious, but it's easy to get wrong!
e) Obtaining known weights
Most foods come in known weights. So you could use bags of flour, sugar, salt, etc. Water could be useful, if you’re able to measure volumes of it easily. Every 1cm of water weighs a gram.
You may want to make some weights. Start off with a known amount of something, and using your balance to find something of the same weight that's easier to handle.
We started off with 500g of sugar and used our balance to find how many nails weighed the same amount. Then we counted the nails to work out how much each one weighed. We then found how long a bit of wire needed to be to balance against 50 nails, and so worked out how much the wire weighed per cm. Then we were able to cut up the wire into lengths of 1g, 0.5g, 0.1g etc to give small weights.
f) Using your balance
Now, get the arm of your balance pivoted: put your two pans an equal distance apart from the centre of the balance. Make sure the arm is horizontal, and it's worth trying to do this when the pans are at several different distances from the centre.
If the pans aren't an equal weight, add small weights to one of them until they balance. Then mark off where the pointer is pointing on to a clean strip of paper.
Put your unknown weight into one pan, and add weights to the other pan until you reach a balance point. Start with bigger weights and go on to smaller ones. Add up all the weights to get the final amount.
You may need to start with the pans quite close to the centre to make sure that the arm doesn't bend when you add the weights. Ideally, the final reading should come when the pans are quite a long way from the centre, as this will give a more accurate reading.