MATHEMATICS BEHIND THE DESIGN Introductory Theories: Possessing a rudimentary diagram that depicts a particular concept, or even a more elaborate one for that matter, is a far cry from being able to appreciate exactly how that design is likely to perform. A bit of ‘number-crunching’ is therefore an essential leg of the journey towards being able to accomplish this.
The fundamental principle behind the floatation of this tower is that of the Buoyant Force, which states that the force of upthrust (the Buoyant Force) in a particular submersion is equal to the total weight of all the water that the tower displaces in that submersion. 7
Moreover, in order for the tower to stand any chance of floating, it must produce a Buoyant Force whose magnitude is equal to its own weight. In this circumstance, the tower’s weight will continue to act downwards, but the equal-magnitude Buoyant Force acting upwards in the opposite direction will effectively cancel it out (and vice versa), hence establishing an equilibrium of balanced forces, in which the tower will consequently stay afloat.
BUOYANT FORCE DIAGRAM Source: askphysics.com
Therefore, by combining this with the theory of Retrieved: April 7th 2012 Buoyant Forces, it can be deduced that the tower must displace a quantity of water with such a volume that the weight of that water collectively is equal to the weight of the tower. Step 1
This leads us to the first step of this process: calculating the weight of the tower. This was an inevitably arduous process, and one that involved breaking the structure down into separate, individual sections, and subsequently calculating the mass of each one in turn, with regards to their volumes and respective densities. Particular care had to be taken in view of the fact that the bottom two sections have a different density (2500 kg.m-3) to the rest of the tower (1800 kg.m-3). Incidentally, this did transpire as quite a major source of error in our first attempt at working this out, which we resolved in the only way we could: starting again! The cone-shaped section of the tower proved to be a particular mathematical challenge, which we tackled by way of treating the situation much as if we were looking to calculate the volume of a torus. In doing this, we were successful in taking a seemingly complex and somewhat daunting calculation, and converting it into something more familiar and therefore something we felt more comfortable in addressing, The only difference was of course that, Engineering Education Scheme: The Report | Mathematics Behind the Design
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