Figure shows a soap film formed between two square figures made of a uniform wire. The bigger square is held while keeping it in a horizontal plane and the smaller square is slowly allowed to drop ver

Question

Figure shows a soap film formed between two square figures made of a uniform wire. The bigger square is held while keeping it in a horizontal plane and the smaller square is slowly allowed to drop vertically. It reaches an equilibrium state after dropping a height h . Let surface tension of soap = T . Mass per unit length of wire = λ . Acceleration due to gravity = g . Given that h = 4 T 2 − λ 2 g 2 ​ n λ ga ​ ; find the value n.

TestHub · Accepted Answer

Each side of the film (two surfaces) pulls the small square with force $2 T a$ along the film; vertical component $2 T a sin θ$

Equilibrium: $4 \times 2 T a sin θ = 4 aλ g \Rightarrow sin θ = \frac{λ g}{2 T}$

$tan θ = \frac{λ g}{4 T ^{2} - λ ^{2} g ^{2}}$

Geometry: horizontal run per side $= \frac{4 a - a}{2} = \frac{3 a}{2}$ , so $h = \frac{3 a}{2} tan θ$

$h = \frac{1.5 λ g a}{4 T ^{2} - λ ^{2} g ^{2}} \Rightarrow n = 1.50$

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