![]() The integral on $\phi$ gives us a $2\pi$ factor. Now we integrate on the sphere only along $\theta$ as the force does not depend on $\phi$. ![]() At $\theta=0$ (north pole) the force pushes towards the bottom. ![]() $$dF_z=dF \cos(\theta)$$ where, as before, $\theta$ is the angle with respect to the vertical director. $\theta$ is the angle with the vertical, as usual convention, with $\theta=0$ being the north pole and $\theta=\pi$ the south pole.īecause we are only interested in the $z$ direction we only consider the part of the force directed along the z direction (the other components disappear by symmetry as they all are on the same $z$-level). This means that the force due to water acting on an infinitesimal surface of the sphere $dA$ is given byĪnd we can write $dA=R^2 \sin(\theta) d\theta d\phi$ using spherical coordinates. The pressure acting on the sphere is directed radially (it's actually directed isotropically, but only the radial part acts on the sphere). We can always set the $z$ where we want as this simply yields an extra constant external pressure $p_0$ which balances on the top and bottom, so we neglect it. It follows that the volume of the sphere is 4/6 2/3 of the volume of the circumscribed cylinder. ARCHIMEDES - Sphere and Cylinder Availability: Up to 14 days Manufacturer: Eureka Code: 5425004736055 Number of puzzles in our stock: 0 Number of. So the volume of the sphere is 1/2 - 1/3 1/6 the volume of the large cylinder, using Euclid's result. We set $z=0$ at the equatorial plane, which will be useful later on when we go to spherical coordinates, so that $p(z=R)=0$ at the north pole of the sphere and the pressure grows towards the bottom of the sphere, reaching $p(z=-R)=\rho g 2R$ at south pole (towards the top). What Archimedes gets from his Method is the equation: Vol (Sphere) Vol (Cone) (1/2)Vol (Large Cylinder). If you partially immerse the sphere, the reading on the scale decreases by the weight of the volume of water displaced by the part of the sphere that is submerged. $$p(z)=\rho g(R-z)$$ where $z$ is the height. The volume of the sphere is approximately 324 ml, so when the sphere is entirely submerged, the force registered by the scale decreases by about 3.2 N (to about 4 N). On the other hand, the force due to hydrostatic pressure is They probably have a motor attached to the. In The Blood of Olympus, while attempting to capture Nike with Frank, Hazel, and Percy, Leo creates what he calls a 'Ultimate zap-o-matic' and apparently spent only thirty seconds making it. This is the first force acting on the sphere, we will need it later. This channel is dedicated to building replicas of that kind of machines. As a talented inventor, Leo is able to use Archimedes sphere prototype to create many different gadgets throughout the series. At the end, I have formulas to compare my and yours formalism and see if your assumptions about $F_1$, $F_2$ etc are right!įirst of all, we have the weight of the sphere (with density $\rho_s$) which gives: ![]() You need to integrate the pressure at each position to recover Archimedes' law, using the fact that at a given height the pressure is $p(z)=\rho g z$ where $\rho$ is the water's density. ![]()
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