The Geometry of a Coffee Splash
When you set your mug down too hard, the liquid forms a near-perfect annular wave. Can you derive the radius of the first ring using surface tension and your mug's diameter?
When you set your mug down too hard, the coffee erupts upward and lands back as a thin circular sheet. Within milliseconds, surface tension and inertia conspire to form a ring — a beautiful annular wavefront that expands outward.
The Setup
Assume your cylindrical mug has an inner diameter of d = 8 cm. The coffee is pushed outward with a characteristic velocity v₀ proportional to the impact speed. Surface tension γ of coffee ≈ 0.060 N/m. Density ρ ≈ 1000 kg/m³.
The Challenge
Using the capillary length formula and conservation of momentum in the thin film, derive an expression for the radius r₁ of the first visible ring as a function of d, γ, and ρ. What physical insight does your answer reveal?
The capillary length λ = √(γ/ρg) ≈ 2.5 mm. The first ring forms near r₁ ≈ d/2 + nλ where n depends on the impact energy. For a typical desktop drop, r₁ ≈ 5.5 cm. The key insight: the ring radius is dominated by the mug geometry, not the surface tension — surface tension merely sharpens the boundary.