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PhysicsNumerical+4Question 14 of 54

JEE Advanced 2020 · Paper 1, Physics question 14

When water is filled carefully in a glass, one can fill it to a height hh above the rim of the glass due to the surface tension of water. To calculate hh just before water starts flowing, model the shape of the water above the rim as a disc of thickness hh having semicircular edges, as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are 103 kg m−310^3\ \mathrm{kg\,m^{-3}}, 0.07 N m−10.07\ \mathrm{N\,m^{-1}} and 10 m s−210\ \mathrm{m\,s^{-2}}, respectively, the value of hh (in mm) is ______.

A glass filled with water, with a disc-shaped layer of water of thickness h above the rim; the disc's edges are rounded (semicircular) and overhang the rim slightly.
Answer and worked solution

Official answer: 3.65

Model. The edge of the disc is a half-cylinder of radius h/2h/2. Its curved surface can support an excess pressure of
ΔP=Th/2=2Th.\Delta P=\frac{T}{h/2}=\frac{2T}h.

Breaking point. The water surface breaks when the hydrostatic pressure at the bottom of the disc, ρgh\rho gh, exceeds this:
ρgh=2Th  ⟹  h=2Tρg=2×0.07103×10=3.74×10−3 m.\rho gh=\frac{2T}h\implies h=\sqrt{\frac{2T}{\rho g}}=\sqrt{\frac{2\times0.07}{10^3\times10}}=3.74\times10^{-3}\text{ m}.
Answer: 3.74 mm (key: 3.65 to 3.85)

Topics:Surface tensionPressure and Pascal's law
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