Consider a very wide water tank with a vertical wall at x = L (see figure). A liquid of surface tension S and density ρ has a surface y(x) making a small angle θ₀ with the x-axis at the wall. For small slopes, take θ ≈ sin θ ≈ tan θ = dy/dx. Which differential equation describes the surface? Image
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Question

Consider a very wide water tank with a vertical wall at x = L (see figure). A liquid of surface tension S and density ρ has a surface y(x) making a small angle θ₀ with the x-axis at the wall. For small slopes, take θ ≈ sin θ ≈ tan θ = dy/dx. Which differential equation describes the surface? Image

Options

G1 Physics · Accepted Answer

Step 1: The curvature of a gently sloping liquid surface gives a Laplace pressure difference ΔP = S/R. Step 2: The hydrostatic pressure change over height y is ΔP = ρgy. Equating: ρgy = S/R. Step 3: For small slopes, 1/R ≈ d²y/dx² because [1 + (dy/dx)²]³ᐟ² ≈ 1. Step 4: Substitute into the pressure balance: ρgy = S(d²y/dx²). Final answer: d²y/dx² = (ρg/S)y, Option B.