The Eureka moment: Archimedes' principle

02 March 2015

Interview with

Dr Paul Coxon, Cambridge University

Materials scientist Paul Coxon takes us back to ancient Greece, to explain the Archimedesbasis of one of the most famous experiments of all time and one which made Ancient Greek mathematician, physicist, engineer, inventor, and astronomer Archimedes a household name 2000 years later, as he explained to Chris Smith...

Paul - I'm going to show you Archimedes' principle. It's a very ancient experiment and it uses how materials behave in fluids to solve a historical fraud.

Chris - When you say Archimedes, this is the bloke that ran down the street in the nude, isn't it?

Paul - Eureka! Yes, he put himself in the bath and this is how he came to the discovery.

Chris - So, he was a eureka-streaker.

Paul - He was a eureka-streaker. I'm not naked. I'm going to show you with some plasticine.

Chris - So, we have in front of us two tanks with blue liquid and they have a sort of tap coming out of the side of the tank.

Paul - Yes and the water is just levelled with the tap. So, when we put anything in the water, the level will rise and it will trickle out. What we're going to do is we're going to immerse some materials in the water, raise the level and measure how much water is displaced, how much water is pushed out of the way.

Chris - In front of the tanks, we've got two balances, weighing scales. So, the water will come trickling out of the tank and we'll be able to weigh how much weight of water comes out when you put the objects into the tank.

Paul - Yes and if we measure the weight of the water, because we know the density of the water, we know the volume of water which is being pushed out of the way. This is a very important property of buoyancy. When you put a body in water, it displaces the water out of the way. So, I've got a little boat here and this is how a boat floats. It pushes the water out of the way and the weight of the water, which is being pushed out of the way is equivalent to the weight of the boat pushing downwards. These balance each other out and this is how the boat floats. It's called buoyancy. 

Archimedes worked for a king, a tyrant and the Greeks were fighting against the Romans and they had a military victory. So in tribute, the King made a votive wreath to be placed on the statue of a god in a temple. So, the King gave his goldsmither a known mass of gold. He knew how much gold there was and he said, "Go off, melt that, make it into a beautiful wreath for me." But he was very suspicious. He thought that the goldsmith had taken a little bit of gold and had kept some for himself and built up the rest of the gold with some cheaper material like lead or copper.

Chris - Now of course, that would mean that the crown would look the same but it wouldn't be all gold. There'd be something else in there so the weight might not be quite right.

Paul - It would look the same and it would weigh the same. So, if we just check on our balancers now, we can see that I have a mass of gold and I have a crown, and they're both exactly the same, 524 grams. So, they both look the same but they have different volumes.

Chris - So, we've got two plasticine objects, one big chunk of plasticine in a big blog, one which you've made into your crown. They both weigh 524 grams but I don't know if both of them are made of only plasticine.

Paul - That's the thing. That's the challenge which Archimedes was trying to solve. He wanted to know the purity of the crown. But he couldn't change the shape of the crown. He couldn't melt it because it was a sacred object. What he needed to do, he needed to measure the volume. So, what I'm going to do now is put in our known gold samples and it should displace a known volume of water.

Chris - So, we're putting one blob of the plasticine into one of the tanks and water is now trickling out into the bowl. Now, we've got the crown which is - we know it weighs exactly the same but we know it's made of plasticine, perhaps with something else added, and we're now going to put that in the other tank.

Paul - And we're going to displace the water and we're going to measure how much water the crown displaces.

Chris - So basically, because the objects are taking up volume or space inside the tanks, they've raised the level of the water above the level at which the pipe comes out in the water that's being pushed upwards is now flowing out and we're weighing it.

Paul - Yes and this is what Archimedes discovered in the bath. When he was given this problem to solve, he went on to one of his very rare baths in a public bath and as he lowered himself into the water, he noticed the water level rose up and it slopped over. This is what inspired him to do this experiment. We use it today in the design of ships and submarines. So, although it's a very ancient experiment, it has real life applications today.

Chris - So, this is the non-crown plasticine. This is almost finished and we're at 349, 350 grams of water. So, that's about 350 millilitres of water, isn't it? So, the volume that we've put in must be about 350 millilitres. So, the crown one, we've now got a total mass of water of 370 grams has come off. So, the crown is actually displacing more water than the original blob of plasticines. What does that tell us?

Paul - This tells us that the volume of the crown, although it weighs the same as this gold, the gold standard, its volume is bigger so its density is less. So, this was a key way of finding out that the density had been changed, had been cut and mixed with some cheaper material. Unfortunately, the poor goldsmith, he lost his head.

Chris - And you've made your one less dense because it's got some balls of wood in there - by the look of it - your crown.

Paul - Yes. So, wood is less dense. So, it bulks up the volume and it displaces more water.

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