Question

In the figure shown, the heavy cylinder (radius R) resting on a smooth surface separates two liquids of densities 2ρ and 3ρ . The height h for the equilibrium of cylinder must be

Answer 1

we know the formula,
                       at Equilibrium,
                           ρ₁L₁A₁ = ρ₂L₂A₂ 
where ρ is the density of liquid of liquid,
           L is the height of liquid,
           A is the area.
A₁ = h × l 
A₂ = R × l  { where l is the breadth of liquid which is same for both sides of liquid } 
   now,
           2ρ × h ×( h × l) = 3ρ × R × ( R × l) 
             2h² = 3R² 
              h = R√(3/2)
hence, h = $R\sqrt{\frac{3}{2}}$