Ment
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Hermit, is the radius of the hemisphere known? If so, here it is:
mg = integrate (h from 0 to H) ρgh*dS(h) = integrate (α from 0 to arcsin(H/R)) ρgRsin(α)*2π(Rcos(α))*dα = integrate (α from 0 to arcsin(H/R)) ρgR*π*sin(2α)*dα = -0.5πρgRcos(2α) (α from 0 to arcsin(H/R)) = -0.5πρgR (cos(2arcsin(H/R))-1) = -0.5πρgR (1-2(H/R)^2-1) = πρgR(H/R)^2 = πρgH^2/R
H = sqrt(mR/πρg)