A sheet of rubber is stretched out over a ring of radius 0.25 m . I pour liquid water at 20^∘ C

step 1 In this question, force increases is equal to force decreases, hence force increases is equal to

A sheet of rubber is stretched out over a ring of radius...

A sheet of rubber is stretched out over a ring of ra$\operatorname{dius} 0.25 \mathrm{m} .$ I pour liquid water at $20^{\circ} \mathrm{C}$ on it, as in Fig. $\mathrm{P} 4.77,$ so that the rubber forms a half sphere (cup). Neglect the rubber mass and find the surface tension near the ring.

Key Concepts

Spherical Geometry and Curvature

In problems involving spherical or hemispherical shapes, the curvature is constant and determined by the radius of the sphere. Understanding spherical geometry is important because the uniform curvature simplifies the application of the Young–Laplace equation. The symmetry inherent in a sphere or hemisphere allows for straightforward calculation of forces and pressures that arise due to surface tension.

Force Equilibrium at Boundaries

Force equilibrium in systems involving surface tension requires that the net forces, including those due to the surface tension and any other acting forces, balance at the boundaries. In problems where a liquid is held in a curved film or membrane, analyzing the balance of forces—often along a circular boundary—is essential. This concept underpins the method used to determine unknown quantities such as surface tension when the system is in mechanical equilibrium.

Surface Tension

Surface tension is a physical property of liquids that represents the force per unit length along the liquid’s surface. It arises from the cohesive forces between molecules and leads to phenomena such as the minimization of surface area, droplet formation, and capillary action. This concept is fundamental in explaining why fluids form spherical shapes in the absence of other forces.

Young?Laplace Equation

The Young–Laplace equation describes the pressure difference across a curved liquid interface in terms of the surface tension and the curvature of the interface. For a surface with constant curvature, like that of a sphere, the pressure difference is directly proportional to the surface tension and inversely proportional to the radius. This equation is crucial in predicting the behavior of bubbles, droplets, and other curved fluid surfaces.

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