A tank contains two immiscible liquids of densities 6ρ and 2ρ. The higher density liquid is filled up to a height L/2 from the bottom. A thin rod of density ρ and length L is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is (2π/n)√(L/g), where g is the acceleration due to gravity. The value of n is:

Question

G1 Physics · Accepted Answer

For a small angular displacement, the net restoring torque is obtained by combining the buoyant forces on the lower and upper halves of the rod with the rod’s weight. Using I about the hinge = (1/3)mL² and linearizing the torque for small angle gives ω² = 3g/L. Therefore n = √3 ≈ 1.73.