question_answer

A small amount of water is poured from the top and the water surface assumes the profile as shown in the figure. Assume that water does not wet the vertical part of the tube. The angle of contact \[\alpha \] is \[120{}^\circ \]and the surface tension of water is\[0.070\text{ }N/m\]. The thickness h of the layer of water is (Assume the thickness of the layer of water substantially smaller than the diameter of opening of tube and\[g=10\text{ }m/{{s}^{2}}\])

A) \[2.3\text{ }mm\] 

B)                    \[4.6\text{ }mm\]        

C)    \[6.2\text{ }mm\]             

D)        \[8.4\text{ }mm\]

Correct Answer: B

Solution :

[b] Consider a thin plane layer of water, which is located near the opening, whose thickness in the direction of normal to the plane of growing is a. In equilibrium, \[\sigma a+\sigma a\cos \theta =\frac{\rho gh}{2}ha\] \[\sigma a(1+\cos \theta )=\frac{g\rho {{h}^{2}}a}{2}\]             \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,h=\sqrt{\frac{2\sigma (1+cos\theta )}{\rho g}}=\sqrt{\frac{0.07\times 3}{{{10}^{3}}\times 10}}\]                         \[=4.6\times {{10}^{-3}}=4.6mm\]