A beaker is filled with two non-mixing liquids. The lower liquid has density twice that of the upper one. A cylinder of height h floats with one-fourth of its height submerged in the lower liquid and half of its height submerged in the upper liquid. Another beaker is filled with the denser of the two liquids alone. If the same cylinder is kept in the second beaker, the height of the submerged position would be -

Solution:

• Let upper liquid density = ρ, lower liquid = 2ρ
• Let cross-sectional area = A, height of cylinder = h
• Buoyant force in two liquids:
  = [Volume in lower] × 2ρg + [Volume in upper] × ρg
  = A(h/4)×2ρg + A(h/2)×ρg = Aρg h
• So, weight of cylinder = Aρg h (cylinder density = ρ)
• Now, in denser liquid (density = 2ρ):
  Let submerged height = x
  Upthrust = A x × 2ρ g
  Set equal to weight:
  A x × 2ρ g = A h × ρ g
  ⇒ x = h/2

Answer: The height of the submerged position in the denser liquid would be h/2.