A circular tube of uniform cross-section is filled with two liquids of densities ρ₁ and ρ₂ such that half of each liquid occupies a quarter of volume of the tube. If the line joining the free surface of the liquids makes an angle θ with horizontal, then tanθ=(ρ₁−K₁ρ₂)/(K₂ρ₁+ρ₂), where K₁ and K₂ are integers. Find the numerical value K₁/K₂.
Physics · Mixed Chapter
JEE Mains

Question

A circular tube of uniform cross-section is filled with two liquids of densities ρ₁ and ρ₂ such that half of each liquid occupies a quarter of volume of the tube. If the line joining the free surface of the liquids makes an angle θ with horizontal, then tanθ=(ρ₁−K₁ρ₂)/(K₂ρ₁+ρ₂), where K₁ and K₂ are integers. Find the numerical value K₁/K₂.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Apply static torque balance to the two liquid columns about the centre of the circular tube. Each segment has weight proportional to density and its occupied circular arc length. Step 2: Express the centre-of-mass coordinates of the two occupied arc segments in terms of the free-surface angle θ. Their combined centre of mass must lie directly beneath the circle centre at equilibrium. Step 3: Comparison of the resulting tangent expression with (ρ₁−K₁ρ₂)/(K₂ρ₁+ρ₂) gives K₁=K₂ under the pr…