A wooden cylinder of length L and cross-sectional area A is partially submerged in a liquid with the axis of the cylinder oriented straight up and down. The density of the liquid is rhoL. If the length of the cylinder that is below the surface of the liquid is d, what is the buoyant force that the liquid exerts on the cylinder? (Ignore the small buoyant force exerted by the air on the part of the cylinder above the surface of the liquid.)

The buoyant force F is equal to the weight of the displaced fluid. The weight of the displaced fluid is [tex]W=m_dg[/tex], where [tex]m_d[/tex] is the mass of the displaced fluid. The mass of the displaced fluid is [tex]m_d=\rho_LV_d[/tex], where [tex]\rho_L[/tex] is the density of the fluid and [tex]V_d[/tex] is the displaced volume, which is equal to the submerged volume of the cilinder [tex]V_d=V_s=Ad[/tex].

Putting all together we have:

[tex]F=W=m_dg=\rho_LV_dg=\rho_LAdg[/tex]