Academy of Geometry and Mechanics
Two of Archimedes’ “On Floating Bodies” propositions seem to almost entirely explain how the stasis of water works in correlation to displacement and density of objects. These two propositions are Proposition 6 which concerns things lighter or less dense than water and proposition 7 which concerns things heavier or denser than water.
Many of the earlier propositions are dealing with ideal volumes and weights which consist of one solid A being equal density to water and how they would cause stasis when the object is set into the water, but this is ideal. Not many objects that you would need to determine whether they would float or not would actually be the exact same density or weight over volume (W/V) as that of water. This is why propositions 6 and 7 seem to be more pertinent to understanding the actual relationship between the solid and its effect on the stasis of water.
Proposition 6 first deals with a solid that is lighter or less dense than water of the same volume.
Annunciation: “If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the difference between its weight and the weight of the fluid displaced.”
First you start out with an object A and its weight is G. Then you are given that the weight of water of an equal volume to A weighs G+H. H being some additional weight than that of A which is G. Seeing as these two objects A and the water of equal volume to A are different weights, then would cause the stasis of water to no longer exist. The next thing to determine is what would in fact put the water back into stasis. If this can be determined then there is a definite relationship between weight and volume which makes the density of things. In order to do this if A is lighter than water of the same volume, then it would make sense to add weight to A in order to make it equal to G+H without changing the volume of A. This can only be done if this additional weight is somehow not submerged in the water so it is logical to assume that this additional weight would set on top of A without being in the water. In order to more clearly understand the relationships provided in prop 6, I have created a table to help simplify them.
Object |
Weight |
Volume |
Substance |
A |
G |
A |
Solid |
Water |
G+H |
A |
Liquid |
D |
H |
0 |
Solid |
So if object A weighs G and has the volume A, and the Water of volume of A weighs G+H then in order to put the system back into stasis, an additional weight D which weighs H and take up no volume would set it back.
So: A+D=G+H ; A=G, D=H
And: water=G+H
Therefore: Objects A+D=Water (equal in volume to A)
Since: G+H=G+H
Now, as proposition 6 clearly explains the relationships between a solid lighter than water and its effect on the stasis of water, proposition 7 explains the same relationship except in relation to a solid heavier or denser than water.
Annunciation: “A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced.”
This proposition starts out with a logical explanation that if the solid is heavier than water, once dropped into water, the solid will float down to the bottom and the parts of water would move around the object until the solid was securely on the bottom of the container. Then from there we are give A, a solid heavier than water of the same volume to it with the weight of G+H, whereas G is the weight of the same volume of fluid as A. Since these two weights of G+H and H are unequal (G+H≠H) then, like the previous proposition some other weight is needed to counter-balance the weight of the water in equal volume. So B as solid lighter than the same volume of the fluid with the weight of G. This would mean that the weight of a fluid with the same volume would be its weight G divided by the weight of H the excess of the solid from the water of same volume.
Just as the previous proposition I will show a table to clarify the terms.
Object |
Weight |
Volume |
Substance |
A |
G + H |
A |
Solid |
Water |
G |
A |
Liquid |
B |
G |
B |
Solid |
Water |
G+H |
B |
Liquid |
So, A=G+H ; B=G ; Water of A=G ; Water of B=G+H
Solid B=Water A
Solid A=Water B
If A is added to B: A+B=G+G+H = 2G+H
And Water to water: W+W=G+G+H = 2G+H
And together 2G+H = 2G+H
Then therefore: A+B=W+W
If after the two solid are added together and set upon one another still taking up the same amount of volume and weighing what they weighed before, they are equal to each other’s weight of water for the opposite volumes: Solid B=Water A & Solid A=Water B. Therefore this has created stasis among the relationships once again.
Now that I have explained Archimedes’ propositions 6 & 7, though the logic holds through and can be explained with his previous assumptions, do they practically work out to be true? Fortunately, I was given the opportunity to decide this for myself in the practicals.
Practical 20 on density fits proposition 6 adequately. This practical uses a second-class lever and an object less dense than water so that once the object is connected to the lever, it pushes the lever upwards. In the practical we used a wooden cylinder and connected it to the ruler with a straw. We set the second class lever at stasis with the cylinder hanging at the end and the two pans on each side. After I set the beaker of water underneath the cylinder and set the cylinder in the beaker, it immediately forced upwards and the system was no longer in stasis. So in accordance with the proposition, the wooden cylinder is A and the water of same volume as A was more dense and forced the cylinder out of the water. Since A is < water we needed another weight to be added to the cylinder in order to balance it out again so we added our clay weights onto the pan on the same side as the cylinder which forced the cylinder back into the water until it was just underneath the surface. The result of this was that as soon at the cylinder was under the surface, the system was back into stasis. After seeing this for ourselves, we applied our practical to the mathematics of the proposition. This is where we have technical difficulties. I still don’t believe that the proposition is wrong because it can’t be seeing as we saw the system go back to stasis. In the end our calculations ended up showing that the wooden cylinder that floated was denser than water itself, which clearly is illogical. Going back over the calculations I simply cannot discover where the problem was, but according to the known that 1g=1mL=1cm3 of water and the density of water is 1g, then our calculations would show that the proposition is true. Aside from the calculations, we were able to set the system back to stasis which at least supports the first part of the proposition in saying that if the object is lighter, then some other weight must be added without affecting the volume of the thing.
For the next proposition (7) I will show the application of this proposition to the Archimedean crown problem. First of all, the crown was composed of some percentage of lead and some percentage of zinc. Both being heavy metals, the crown quickly dropped to the bottom of the beaker after we dropped it inside. Now we have our object heavier than water since the parts moved away for it to fall to the bottom. First we weighed the actual weight of the crown in air and it came to be 174.375mere. In the crown problem, in order to determine the volume of the crown, we used the method of displacement. We filled a jar that had a spout out the side clearly up to the top until water overflowed from the spout and stopped. Then we placed another beaker at the end so that the water displaced from the crown would fall into that beaker. This would determine the weight of the water of equal volume to the crown. This weight was found with three trials then averaged and came to be 19.29mere. Clearly 174.375m>19.29 so the crown is officially heavier than water. Next the proposition claims that the crown will weigh less in water. So for the crown problem, next we weighed the crown suspended into water and once again after three trials and then an average of them we discovered that the weight of the crown in water was 153.75mere. In air the crown weighed 174.375m and in water it weighed 153.75m. Clearly 174.375m>153.75m so now the last part of the proposition claims that the difference in weight between the crown’s weight in water and air will be equal to the weight of the water displaced of equal volume to the crown.
So, 174.375m – 153.75m = 20.625 m
And our calculations showed that the weight of the water was 19.29m. So we were only off by about 1 mere. This being close enough to support the theory of Archimedes seeing as our methods of measurement were based on weights we ourselves made which shows some expected account for error. Therefore, the crown problem adequately supports the proposition 7.
Now, after evaluating the legitimacy of the two propositions and then in accordance with the data discovered by testing them, it appears that there is a correlation with logic and practicality. I believe that by using the geometrical Archimedean proofs to explain the crown problem created very much so of a happy marriage between the experiments and why they are so. For simply if you take the propositions and think about them, they surely do make sense, but there could be an underlying inherent problem with one of the assumptions that you otherwise accepted. The only way to truly be sure is to take that logic and test it in some way which would have to be physically by the practicals. On the other hand if you simply rely on sensations and the physical world, there is always room for mistakes and things that aren’t exactly equal. As Socrates believes, equality is never truly appealed to because nothing in this world is truly perfect, but in geometry; perfection is relied upon in order to make a proof. By collecting the ability to have reliable logic from geometry and the fact that it will actually show to be true by our own eyes in practicals, both aspects of truly proving something to be true are utilized.
Archimedes principle neglects the shape of body .
Hence it is generalized.
Book Einstein and Archimedes: Generalized
http://www.amazon.com/Einstein-Archimedes-Generalized-now-then/dp/3843389977/ref=sr_1_1?ie=UTF8&s=books&qid=1303484649&sr=8-1
By: Ajay on April 23, 2011
at 4:42 am