# Idea Seeds #07 - Problem Solving, Eureka Moments

Updated on December 23, 2016

## All Things Are Made of Atoms

If by some cataclysmic event all scientific knowledge was lost, what single sentence would preserve the most information for the next generation? This was Nobel Prize winner, Richard Feynman’s answer: “All ‘things’ are made of atoms − little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.” He went on to say: “In this one sentence you will see there is an enormous amount of information about the world, if just a little imagination and thinking are applied.”

## Archimedes and Eureka

There is general agreement that everyone must equip him or herself with the basic building blocks needed to comprehend the world they live in. Mass and weight are basic building blocks that give rise to much confusion so is as good a starting point as any. Most people have heard about Archimedes (287 – 212 BC) and his ‘Eureka’, moment but few know what it is that he is supposed to have ‘actually’ done to earn him and ‘the Eureka word’ such fame.

## Mass, not weight, In Gold

King, Hieron II, of Syracuse (308 – 215 BC) is said to have given a goldsmith a specific ‘mass’ of gold to make him a crown which the goldsmith duly did and delivered. The King checked the mass of the finished crown and found it to have exactly the same mass as the mass of the gold he had originally provided. The King, however, suspected that the goldsmith had stolen some of the more valuable gold and replaced it with less valuable silver so the King asked Archimedes to find out if his suspicions were true but to do so without defacing the crown in any way.

## A bath full of water

Archimedes already knew that a ball of pure gold had a smaller diameter than a ball of pure silver with the same mass. That a smaller diameter ball occupies less volume than a larger diameter ball is self evident. So if the mass of silver substituted was the same as the mass of gold stolen, the crown would occupy more volume than an identical one of pure gold. The puzzle he faced was to measure the volume of the irregular crown? As legend has it, Archimedes noticed after stepping into a full-to-the-brim bath, that water flowed out over the rim. The deeper he immersed himself, the more the water overflowed. He realised that if he immersed the crown in a vessel full of water and measured the spillage and compared that with the spillage from the immersion of the same mass of pure gold, the puzzle was solved. If the spillage was the same the goldsmith was honest if the spillage was different he was a thief. Legend has it that he jumped from his bath and ran naked into the street joyously shouting ‘Eureka, Eureka!’ ̶ Greek for ‘I have found it, I have found it!’

## Galileo Galilei

There is much speculation about the truth of this story but there is no argument that Archimedes was one of the worlds best mathematicians of his time and that he had made real and significant contributions in many other fields including physics, engineering and astronomy. Galileo Galilei (1564 – 1642) some 1800 years later proposed that measuring the spillage would have been difficult to do ‘accurately’ and that it was much more likely that Archimedes had used a balance, similar in looks to the ‘Scales of Justice’, and hung the suspect crown on one end and a block of pure gold with the same mass on the other having first removed the two pans. The two objects being of equal mass would keep the balance level and in balance.

He would then have immersed the suspended objects in a vessel of water. If the balance remained level and in balance the crown was made of pure gold. The ‘buoyancy forces’ on the crown and on the gold block would be equal. Why? ‘Because equal masses of the same ‘thing’ occupy the same volume irrespective of their shapes.

## Volume Speaks Loud

As stated previously, a ball of pure gold would occupy a smaller volume than a ball of silver with the same mass. A gold-mixed-with-silver crown with its larger volume when immersed in water would be ‘buoyed’ up more strongly than the block of pure gold creating an imbalance that would make the balance tip; go up on the crown side and down on the block side. The story says that it did tip proving the goldsmith a thief. Whether the story is true or not is not really important. What are important are the underlying principles. There is no mention of ‘weight’ in the explanation above but it is there hidden away in those ‘buoyancy forces’ mentioned in the explanation.

## Mass and Density

Mass is the measure of the amount of stuff in ‘things’. Remember the opening sentence ‘All ‘things’ are made of atoms’ so to put it another way; atoms are the ‘things’ that give ‘things’ mass.

Mass is one of the ‘six fundamental units’ of the International System of Units (SI) that we use in South Africa. Mass is measured in kilograms (kg).

Density is mass per unit volume. It is expressed as kilograms/cubic metre (kg/m^3). To picture a cubic metre (m^3), try and imagine what a box one metre wide by one metre deep by one metre high would look like. To fill it you would need 1000 kg of water or 10490 kg of silver or 19300 kg of gold.

Density of fresh water: 1000 kg/m^3 − Silver: 10490 kg/m^3 − Gold: 19300 kg/m^3

The concept of density is easy to ‘visualise’ and very useful when comparing real ‘things’. Anything ‘solid’ like silver and gold with a density greater than water will sink in it. Anything with a density less than it will float on it. Human beings like us have densities close to that of fresh water. Our densities change depending on whether we have full or empty lungs. We float when our lungs are full of air and sink when our lungs are empty. This is why you find people that have just drowned on the bottom. They rise again a day or two later when the gasses made by the decaying food in their stomachs and intestines starts inflating the dead body, increasing its volume and lowering its density. Puff out your chest by filling your lungs and it is clear you have measurably increased your volume. Empty your lungs and your volume decreases. Air has mass. Its density is 1,2 kg/m^3, so when we breathe in, we increase our total mass but the amount is small compared with the increase in volume. This increase in volume lowers our density and allows us to float. When you next go swimming, do some experimenting and check it out for yourself. The picture shown above of a person floating with his head and parts of his body out of the water is swimming in the Dead Sea that is very salty. The density of salty water rises as the salt concentration increases. The Dead sea has a very high salt concentration.

## Force of Gravity

Weight is a force. Weight is the force that the ‘force-of-gravity’ exerts on a mass. It is measured in Newtons (N). To hang a mass from the end of a balance you would need to use a piece of string. If you tied one end to the mass and slowly lifted it into the air while only holding onto the other end, you would be providing an upward force equal to the downward force gravity is exerting on the mass as it dangles from the string. A force is not a ‘thing’ and therefore contains no atoms.

## Archimedes' Principle

Having just explained how our densities change you should have recognised that we would pass through a ‘limit’ called ’neutral buoyancy’ as we go from floating to sinking or from sinking to floating. This happens when the density of water and our density are exactly the same.

Imagine now that you are hanging in the air from a rope by one arm above a swimming pool. The force you feel in your arm is your weight not your mass. The rope will be put into tension and transfer the force to the support it is attached to. If you are now lowered slowly into the water the force you feel in your arm and the tension in the rope will decrease as the water you are displacing buoys you up. When you are completely immersed there will be no more force in your arm or tension in the rope meaning that you are now weightless, buoyed up by the water.

‘Archimedes' principle’ is stated in various ways. ‘An immersed object is buoyed up by a force equal to the ‘weight’ of the fluid it actually displaces.’ Another ‘A body experiences a loss in weight equal to the weight of the medium it displaces.’ Hopefully you will no longer be confused by different wording like this.

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