Final answer:
The fraction of the cylinder that is submerged is equal to the ratio of the cylinder's density to the water's density, rhoC/rho. The difference between the pressure at the cylinder's lower surface and the atmospheric pressure is found using the hydrostatic pressure formula, which yields P = h rho g plus atmospheric pressure.
Step-by-step explanation:
To find the fraction of the cylinder that is submerged, we can use the principle of buoyancy, which states that the buoyant force on an object in a fluid is equal to the weight of the fluid displaced by the object. When the object is floating, the buoyant force is equal to the weight of the object. Therefore, for a floating object, the weight of the fluid displaced must be equal to the weight of the object. For a cylinder of density rhoC, radius R, height H, and the density of water rho, the fraction submerged can be expressed as the ratio of the submerged volume to the total volume of the cylinder. Let's call the submerged height of the cylinder h, then the submerged volume Vsubmerged = πR2h, and the total volume of the cylinder V = πR2H. Using the principle of buoyancy:
Weight of Fluid Displaced = Weight of Cylinder
rho x Vsubmergedg = rhoC x Vg
rho x πR2h = rhoC x πR2H
rho x h = rhoC x H
h/H = rhoC/rho
The fraction of the cylinder that is submerged is thus rhoC/rho.
To derive the expression for the difference between the pressure at the cylinder's lower surface (submerged) and the atmospheric pressure, we use the hydrostatic pressure formula P = hpg. The pressure at the bottom of the submerged part of the cylinder is P = h rho g. This is the pressure due to the water above it plus the atmospheric pressure, Patm. Therefore, the total pressure at the bottom is Ptotal = P + Patm = h rho g + Patm.