Can density replace buoyancy

Pressure and buoyancy

But what was it really like?

In the first century BC, the Roman architect Vitruvius tells us, Archimedes discovered a fraud in the manufacture of a golden crown intended for Hiero II, King of Syracuse. The crown (Latin corona) probably had the shape of a wreath. Hiero probably put such a wreath on the statue of a god or goddess. Suspecting that the goldsmith had replaced some of the gold given him with the same amount of silver, Hiero asked Archimedes to check whether the wreath was made of pure gold. Since the wreath was a sanctuary dedicated to the gods, it could not be damaged in any way. (In modern terms: non-destructive investigation). Archimedes' solution to the problem, as described by Vitruvius, is briefly summarized in the following text.

"He found the solution when he went into the bathroom and overflowed it. He took a piece of gold, the weight of which matched the crown and which he knew was made of pure gold, and put it in a piece filled to the brim with water Bowl. Then he took out the gold piece and put the king's crown in its place. An alloy with the lighter silver would increase the volume and cause the water to overflow from the bowl. "

Although theoretically correct, this method has been criticized for various reasons. At first, according to Vitruvius' description, it was the "result of a genius spirit", while the method described is much easier to present than one is used to with the detailed descriptions of Archimedes. Second, this method does not use the laws of buoyancy or leverage found by Archimedes. Third, and this is probably the decisive argument, the method described by Vitruvius would be far too imprecise for Archimedes, who was known for his great precision, because of the means available to Archimedes.

The third point needs explanation. The largest gold wreath from the time of Archimedes is the depicted wreath by Vergina. It has a largest diameter of 18.5 centimeters and a mass of 714 grams, even if some of its leaves are missing. To imagine this, let's assume that Hiero's wreath weighed 1000 grams and that a water container with a diameter of 20 centimeters was used. The opening of the container had an area of ​​314 cm². (All calculations are based on three important conditions.)

Since gold has a density of 19.3 g / m³, 1000 g gold would have a volume of 1000 g: 19.3 g / cm3 = 51.8 cm3;. Such an amount of gold would have raised the water level by 51.8 cm when immersed in the container3 : 314cm2 = 0.165 cm raised.

Let us further assume that the dishonest goldsmith has replaced 30% (300 g) of the gold intended for the crown with silver. Silver has a density of 10.6 g / cm³ and the gold-silver crown would then have a volume of 700g: 19.3g / cm3 + 300g: 10.6g / cm3 = 64.6 cm3. Such a crown would have the water level at the opening by 64.6 cm3/ 314cm2 = Let rise 0.206 cm.

The difference in the water level changes caused by the crown or the gold nugget is 0.206 cm - 0.165 cm = 0.41 mm. This is far too low to be able to be read directly or measured by overflowing, if you consider the sources of error, which include the surface tension of the water, water droplets that stick to the gold nugget when it is removed and air bubbles on the finely crafted gold wreath arise. In addition, the difference in the water level change would be less than 0.41 mm if the crown were lighter than 1000 g, or its diameter was greater than 30 cm or the silver content was less than 30%.

A more conceivable and technically easy to implement method is the following, which uses Archimedes' two laws of lift and lever. Hang the wreath on one end of a pair of scales and balance them with a lump of gold of equal mass on the other end of the scale. Then submerge the wreath and gold crown hanging on it in a container of water. If the balance remains in equilibrium, the wreath and gold nuggets have the same volume and thus the Krabz the same density as pure gold. But if the balance sinks in the direction of gold, then the volume of the wreath is greater than that of gold and its density less than that of gold. It must then be an alloy of gold and a lighter material.

In order to check the practicability of this technique we want to assume again a 1000 g wreath made of an alloy with 70% gold and 30% silver. It has a volume of 64.6 cm3, and displaces 64.6 g of water. (Water has a density of 1.00 g / cm3.) Its mass of 1000 g is reduced in the water by the buoyancy of 64.6 g, so that the balance is loaded with 935.4 g. On the other hand, 1000 g of pure gold has a volume of 51.8 cm3, and the scales are loaded with 1000g - 51.8 g, i.e. 948.2 g. So if both ends of the scale are submerged in the water, one end is loaded with 935.4 g, the other with 948.2 g and there is a difference of 12.8 g. Archimedes' time scales could easily determine such differences. In addition, the sources of error of the Vitruvius method (surface tension and adhering water) do not play a role in this balance method.

It should be noted that the Libra method will work even if the masses of the wreath and gold piece are not the same. You only have to change the distance between the crown or gold nugget from the pivot point of the balance so that it is in equilibrium before you immerse it in the water.

The two methods described above can be summarized as follows: Assuming a 1000g crown consisting of 700g gold and 300g silver, there is a difference in volume of 12.8 cm3 into a 1000 g lump of gold. The method of Vitruvius tries to recognize this volume difference through the overflowing water. 12.8 cm³ of water form a cube with an edge length of 2.34 cm and would be easy to measure in this form. But if this 12.8 cm3 Water can be distributed over a surface large enough to submerge the crown (in our example 314 cm2) this means a height difference of only 0.41 mm. Such a difference in height is too small to be precisely determined by direct observation or by crossing over. The Libra method translates this difference in volume of 12.8 cm3 exactly into a load difference of 12.8 g on the scales, which was also easily measurable with the antique scales.

Translated from English from a site at Drexel University Philadelphia, USA.