A container of width 2a is filled with a liquid. A thin wire of mass per unit length λ is gently placed across the middle of the liquid surface. The surface is depressed by a small distance y (y ≪ a). If the surface tension is T = αλag/(2y), find α. Image
Physics · Properties of solid and fluid
JEE Advanced Practice

Question

A container of width 2a is filled with a liquid. A thin wire of mass per unit length λ is gently placed across the middle of the liquid surface. The surface is depressed by a small distance y (y ≪ a). If the surface tension is T = αλag/(2y), find α. Image

Physics · Properties of solid and fluid

G1 Physics · Accepted Answer

Step 1: For small depression, the surface on each side makes a small angle θ with the horizontal, with sinθ ≈ tanθ ≈ y/a. Step 2: Vertical surface-tension force per unit wire length is 2T sinθ ≈ 2Ty/a. Equating it to the wire weight per unit length λg gives T = λag/(2y). Therefore α = 1.