A rectangular block of ice with cross-sectional area A and height h floats on water of density rho0. The density of ice is rho1. Find the work that must be done to push the ice block until it is just completely submerged in the water.

Question

Accepted Answer

(rho0 - rho1)² g A h² / (2 rho0). Floating: weight = buoyancy, so rho1*A*h*g = rho0*A*d*g, giving submerged depth d = (rho1/rho0)h. The length above water is x0 = h - d = h(rho0 - rho1)/rho0. To submerge fully you must push down by x0. When pushed down an extra distance x, the net extra upward force is rho0*A*x*g (acts like a spring with k = rho0*A*g). Work = integral from 0 to x0 of rho0*A*g*x dx = (1/2) rho0*A*g*x0² = (1/2) rho0*A*g*[h(rho0 - rho1)/rho0]² = (rho0 - rho1)² g A h² / (2 rho0).

A rectangular block of ice with cross-sectional area A and height h floats on water of density rho0. The density of ice is rho1. Find the work that must be done to push the ice block until it is just completely submerged in the water.

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