Archimedes and the golden crown.pdf

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www.iop.org/journals/physed
Archimedes and the golden crown
Frank Thompson
Joule Centre, University of Manchester, Manchester M60 1QD, UK
Abstract
Archimedes (287–212 BC) is well known for his explanation of buoyancy,
and in particular for his ‘eureka’ moment. This experiment uses his density
measurement method to find the purity of gold, and additional confirmation
of the findings are given by x-ray fluorescence.
Preliminaries
Just recently I have convinced myself that I am on
nodding terms with this great gentleman. Two or
three times a week I now pass him and nod to him
in a respectful and reverent manner, but he seems
to be lost in a world of his own, with an ecstatic
look on his face, as we can see in figure 1.
What can be in the mind of this person who
sits naked in his bath in a busy thoroughfare of
Manchester?
The story goes that King Hiero II in Syracuse,
Sicily, was presented with a new gold crown
in the shape of a laurel wreath, and he asked
Archimedes to determine whether it was solid
gold or some cheap imitation made from base
metals. Archimedes knew that he could not melt
the crown down to form a regular shape, say, a
cube, and thus calculate the density from mass
and volume measurements, so he had to devise
another method. Archimedes is then supposed to
have had his ‘eureka’ moment in his bath when
he realized that he could use the buoyancy force
exerted on the crown when it was submerged in
water to make his measurement, and, since that
time, many generations of scientists have used this
same liquid immersion method as the standard for
density measurement.
About fifty years ago in Libya, I, too, was
presented with a gift of gold, namely, cuff-links!
With this recent intimate Archimedean bonding,
I felt that I should use density measurement to
determine the quality of my gift. After all,
the cuff-links had been fashioned from gold lira
Figure 1. A sculpture of Archimedes frozen in time.
Photo by Dan Karran.
of the Ottoman Empire as the markings showed
(figure 2), and they could well have a high gold
purity.
Recently, this journal had a special issue
regarding water [1], where topics concerning
buoyancy, Archimedes, and clouds (to name but
three) were discussed in detail. My main aim
396 P HYSICS E DUCATION 43 (4) 0031-9120/08/040396+04$30.00 © 2008 IOP Publishing Ltd

Figure 2. Coins fashioned into cuff-links.
twin cup arrangement
gantry supporting cups
resting on balance pan
liquid providing
bouyancy
support for liquid
resting on balance
casting
Figure 3. The twin cup arrangement for density
measurement.
in this article is to give one of many practical
uses of buoyancy for density measurement, and to
undertake a personal quest to determine the quality
of gold in a pair of cuff-links.
It is hoped that students partaking of such
an experiment will imbibe some of the ecstasy
of Archimedes, who initiated this work and many
other mathematical, scientific and engineering
discoveries [2].
Experimentation
The usual arrangement for density measurement
by liquid immersion is given in figure 3.
A beaker of liquid is supported by the casing
of an electronic balance (in the present case, model
PM 460 from Mettler-Toledo Ltd) whereas the
twin cups have a gantry which rests on the balance
pan. An illustration of the apparatus is presented
in figure 4, with one of the cuff-links in the lower
cup and the other resting on the beaker support.
The balance is tared (zeroed) with the cups
being empty. The mass is then recorded for the
sample in the upper cup. The cup stirrup is
removed from the gantry, the sample is placed in
the lower cup, and the stirrup replaced. Ensuring
that the sample is fully covered with liquid and
Archimedes and the golden crown
Figure 4. A Mettler balance with a twin cup assembly
for density measurement.
that no bubbles are trapped around it, a second
mass is recorded. For the case of the liquid
being water at a temperature in the region of
20 ◦ C, the density is simply the ratio of the initial
measurement to the difference in the initial and
final mass measurements. (At a temperature of
16 ◦ C the density of water is 999 kg m −3 and even
at 22 ◦ C it has only fallen to 998 kg m −3 : i.e.,
very close to 1 g cm −3 [3]). The length of the
stirrup is not critical, but a compact apparatus is
usually desirable so that the stirrup is as short
as is permitted to allow the sample to be fully
submerged.
Suitable shielding of the balance from
draughts is recommended. For samples that have
a density less than that of the liquid providing
buoyancy, a metal weight (anti-float cap) is placed
on the bottom cup, and the initial tare of the
balance will compensate for this added mass.
Surface tension effects are likely to be small,
as the wire forming the stirrup has a small
diameter. Also, it will be seen that the stirrup is
in the same position when the two measurements
are made, and therefore any surface tension effect
will be cancelled out.
July 2008 P HYSICS E DUCATION 397

FThompson
Table 1. Density values, given in units of g cm −3 .
Material Measured density Accepted value [4]
Aluminium 2.70 (±0.03) 2.71
Steel 7.76 (±0.07) 7.80
Cuff-links 17.1 (±0.2) 19.3 (pure gold)
Wood 0.60 (±0.05) 0.70 (soft wood)
Results
Ten readings were taken for each of the samples of
aluminium, steel and the cuff-links, and the results
are given in table 1. The density of a sample of
wood is also included, though this result must be
counted as, at best, very approximate.
An accuracy of about ±1% was attained with
the metal samples, and there is good agreement
between the measured values of aluminium and
steel and the published values. But a major
problem with the wood sample was that water
soaked into the wood pores, and one could see
bubbles of air being ejected from the sample.
Some thin coating of varnish would have to be
applied to the wood to prevent water ingress if a
serious study was to be undertaken. (Note that the
balance gives a negative reading for the submerged
wood and so, on subtraction, the two readings have
to be added.)
It is fair to conclude that the cuff-links are
not pure gold, but that they must contain an
appreciable amount of gold to have a density value
of 17.1gcm−3 .
Conclusions and discussion
In the world of jewellery the purity of gold is based
on a carat number, with 100% gold being equal
to 24 carats. A gold sample of 22 carats will
have 22 parts gold and two parts of one or more
additional metal(s), and this scale goes to 10 carat
gold, which would have 10 parts gold and 14 parts
of additional metals.
If, say, silver (density 10.5 gcm−3 ) was used
as the alloying metal, then an 18 carat gold would
have a density close to 17 g cm−3 (18/24 × 19.3 +
6/24 × 10.5 = 16.9) and therefore on the basis
of this hypothetical notion one could conclude that
the cuff-links are 18 carat gold, and not very close
to the ultimate 24 carats.
(The epilogue shows that the alloying metal is
copper rather than silver.)
Epilogue
Staff at Tech Ed. Systems Ltd kindly offered to
analyse the cuff-links using x-ray fluorescence,
with apparatus from Leybold Didactic GmbH, and
their response is given below.
‘Some background to the method used and
principles of this analytical technique
The central component of our detector is a Si PIN
photodiode, which is cooled by means of a Peltier
element in order to reduce the leakage current. In
the Si PIN diode, the incoming x-rays produce
electron–hole pairs through interaction with the
crystal atoms. These pairs are separated in an
electric field and then converted into a voltage
pulse. The number of electron pairs and the pulse
height are proportional to the x-ray energy. The
pulse height analysis is carried out with a multichannel
analyser (MCA), which is interfaced to a
PC.
Result
I used a galvanized steel sample to calibrate
the detector (FeZn). On analysis your cufflinks
showed the presence of only two elements.
Gold, of course, and also copper. There seemed
to be no other metals present. I analysed the
sample using different geometric configurations
to establish whether the link between the coins
contained any other elements (these were not
minted in the Ottoman period). I found no
difference in the composition.
It seems then, Frank, that your cuff-links
are 18 carat red gold with a composition of
approximately 75% Au and 25% Cu. If you look
at the height of the Cu (Kα) peak and the Au (Lα)
peak you will see that they show a ratio of 3:1.
This fits nicely with your findings using the density
measurement.’
Acknowledgments
My thanks to Salvatori, Darren and Jennie (Tech
Ed. Systems, The Pump House, Rowdens Road,
Wells, Somerset BA5 1TU) for their helpful
discussions and analysis carried out.
Thanks to Dan Karran for his photograph
of the statue of Archimedes. At present, this
sculpture is suffering from disfigurement.
398 P HYSICS E DUCATION July 2008

Received 25 February 2008, in final form 27 March 2008
doi:10.1088/0031-9120/43/4/007
References
[1] 2007 Phys. Educ. 42 Special issue on water
[2] Zannos S 2005 Life and Times of Archimedes
(Hastings, UK: Mitchell Lane)
[3] Worsnop B L and Flint H T 1942 Advanced
Practical Physics 7th edn (London: Methuen
and Co Ltd) p 186
Archimedes and the golden crown
[4] Tennent R M 1971 Science Data Book (Edinburgh:
Oliver and Boyd)
Dr Thompson joined the Physics School
at Manchester Polytechnic after spending
some years working in oil prospecting in
Libya. In the late 1990s he took early
retirement, and he has done consultancy
work since then. At present, he is
working at the Joule Centre, Manchester
University.
July 2008 P HYSICS E DUCATION 399