A closed circular glass tube of radius 1 m contains two immiscible liquids filling equal semicircles, with densities ρ₁ = 700 kg m⁻³ and ρ₂ = 1000 kg m⁻³. Determine period of small oscillations (s, approximate). [See Image.]
Physics · Mixed Chapter
JEE Main Practice

Question

A closed circular glass tube of radius 1 m contains two immiscible liquids filling equal semicircles, with densities ρ₁ = 700 kg m⁻³ and ρ₂ = 1000 kg m⁻³. Determine period of small oscillations (s, approximate). [See Image.]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Let angle φ describe a small displacement of liquid interfaces around the tube. The density difference produces restoring pressure ΔP ≈ 2(ρ₂−ρ₁)gR φ. Step 2 — Fluid length 2πR has inertia m = πRA(ρ₁+ρ₂). Tangential coordinate is s = Rφ. Step 3 — The angular equation is φ̈ + [2(ρ₂−ρ₁)g/(πR(ρ₁+ρ₂))]φ = 0. Step 4 — Period T = 2π√[πR(ρ₁+ρ₂)/(2g(ρ₂−ρ₁))]. Step 5 — With R=1 m, ρ₁=700, ρ₂=1000 kg m⁻³, g=9.8, T≈5.99 s, nearest integer = 6.