Eureka!

You’re struggling on a difficult problem, trying to rack your brains around how to approach it. However, every path you try leads to a dead-end, leaving you shaking your fists in frustration. Casting your attention elsewhere, you notice a flock of birds flying past outside your window. Suddenly, for no apparent reason, the solution strikes you and you want to cry out in joy. But what word could possibly fit the occasion?

Some of you may be familiar with a certain word having been uttered famously by the Greek scholar Archimedes a while back. The story goes that he burst out of his house and ran down the street fully naked, shrieking the word “Eureka!”, which can be translated literally as ‘I have found it!’. Archimedes had finally realised, having stepped into his bathtub and noticed the water level rising, that the volume of water displaced must be equal to the volume of his submerged body. Using this concept, he was able to deduce a way to measure the volume of irregular objects.

King Hiero II of Syracuse used Archimedes’ insight to determine whether his suspicion about his goldsmith using impure gold was true. By lowering the gold crown into some water and observing the displaced amount, Archimedes was able to work out the density of the crown by using its measured mass and volume. I’m sure Archimedes was overwhelmed with his new discovery, but let’s just say it didn’t end quite so well for the goldsmith…

Archimedes went on to write a duo of books which cover his discoveries on the physics involved with bodies submerged in water, aptly named On Floating Bodies. From this we find his most famous contribution to science, Archimedes’ principle. This states that, any object, wholly or partially immersed in a liquid, is buoyed up by a force equal to the weight of the fluid displaced by the object. From this we can now decide whether an object will float or sink.

Let’s consider a block of solid iron and the vertical forces acting on it whilst it is submerged. The only downwards force is its own weight, and the upwards force is the buoyant force of the water. Therefore, if the buoyant force is smaller than the weight of the block, it will surely sink. However, if the buoyant force is equal to its weight, it will float. Using the above principle, we can therefore say that for the block to float, it must displace its own weight in water.

Since iron is about 8 times denser than water, it will not displace enough water to provide a suitable buoyant force to keep it afloat, and, as we’d expect, the block sinks. If, however, we reshape the block into a bowl and lower it slowly into the water, it will eventually displace an amount of water which is equal to the weight of the boat itself, and therefore stay afloat. By stretching out the iron, we have effectively decreased its density since it occupies a larger volume but still has the same mass.

Having equipped you with this hydrostatic toolset, I want you to spend a few minutes considering the following question, which is commonly used to trick the hydro-unaware.

If I’m in a boat on a lake, and I throw a rock from inside the boat into the water, what will happen to the level of the water?

Give me a shout if you want an immediate solution, or wait patiently for me to reveal the solution in next week’s post. Whatever floats your boat.

Harvey

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