Pressure Forces on Cylinder

Consider a solid, closed, right circular cylinder of length $2 \, \text{m}$ and radius $1 \, \text{m}$ . The cylinder is placed below the surface of a body of water.

The water surface is the plane $z = 0$ . The center of the cylinder is at $(C_x,C_y,C_z) = (0,0,-2)$ . The central axis of the cylinder (along its length) points in the direction of the vector $(v_x,v_y,v_z) = (1,1,1)$ .

The density of the water is $1000 \, \text{kg/m}^3$ , and the ambient gravitational acceleration is $10 \, \text{m/s}^2$ in the negative $z$ direction. Neglect atmospheric pressure.

The total water pressure force acting on the cylinder can be represented as a vector sum of the pressure forces on the top disk face, bottom disk face, and side:

$\large{\vec{F}_{water} = \vec{F}_T + \vec{F}_B + \vec{F}_S }$

Determine the following ratio of force magnitudes:

$\large{\text{Ratio} = \frac{\Big|\vec{F}_T\Big| + \Big|\vec{F}_B\Big| + \Big|\vec{F}_S\Big|}{\Big|\vec{F}_{water}\Big|} = \frac{\Big|\vec{F}_T\Big| + \Big|\vec{F}_B\Big| + \Big|\vec{F}_S\Big|}{\Big|\vec{F}_T + \vec{F}_B + \vec{F}_S\Big|} }$