# Archimedes principle: think carefully

Benjamin has recently acquired a 'new' book from Grandma and Grandad: Mr Archimedes' Bath (by Pamela Allen – here's the amazon link – the reviews are as interesting as the content). The story-line is reasonable guessable from the title. Mr Achimedes puts water into his bath, gets in, and the water overflows. What's going on? So we've been doing some copycat experiments – not by filling the bath right up and having it slosh all over the bathroom floor (Waipa District Council – you can rest easy about water usage)  but filling up rather more sensible-sized containers and dropping objects in.

Archimedes principle is actually a little more involved than simply saying that putting an object in the water will raise the water level. It says that the weight of water displaced is equal to the force of buoyancy acting on the object.  This picture summarizes it. That is, if an object of 2 kg floats, then 2 kg of water will be displaced. If an object is unable to displace enough water for this to be the case, it will sink. That still should be pretty easy to get, especially if you've done some experimenting. However, it can still be the basis of some really hard questions. I had one in my third year  physics exams at Cambridge. In our 'paper 3', as it was called then, the examiners had free reign to ask about ANYTHING that was on the core curriculum from any of our years of study – plus ANYTHING that was considered core knowledge for entry into the degree (which meant basically anything at all you were taught in physics or general science from primary school upwards). This paper was feared like anything – it was basically impossible to revise for*.

Here is a question then, as I recall it from the exam.

An ice cube contains a coin. The ice completely surrounds the coin. The cube is floating in a container of water.  The cube melts. Does the water level rise, fall, or stay the same?

Now, the icecube melting question is one that is often banded about. A floating icecube will displace its own mass of water (so says Mr Archimedes). When it melts, this water will occupy the 'space' that is displaced by the cube. Consequently, the water level will stay the same. A practical example of this is in the estimation of sea-level rises due to global climate change. When the ice floating on the Arctic Ocean melts, it does not cause a sea-level rise, since it is already displacing its own weight. However, the icecap on Greenland will cause a sea-level rise as it melts, since it is currently not displacing any of the sea (since it is sitting on land.)

However, that is not the question that is asked. Our icecube has a coin inside it. What difference does it make? Well, the icecube-and-the-coin will still displace its weight of water since it floats. However, when the icecube melts, the coin sinks and no longer displaces the same amount of water as it did when it was frozen into the cube. Therefore the water level falls. That's quite a subtle application of Archimedes principle. After the exam, a group of us sat arguing about it, till we collectively worked out what the right answer was (see – exams can be good learning experiences!). Unfortunately, at this point I realized my answer was wrong. Even still, I managed to get out of the degree with a first-class honours, so I couldn't have done too badly on this exam overall.

*The other question I remember from this paper is 'What is Cherenkov Radiation?' I didn't have a clue what Cherenkov radiation was when I sat the paper – I made up some waffly words and wrote them down and almost certainly received zero for the question.'  Later, one of my friends found a single, incidental sentence in a handout that was given out by our nuclear physics lecturer that identified what it was. That's how nasty this exam was.

## 13 thoughts on “Archimedes principle: think carefully”

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