question_answer

A vessel, whose bottom has round holes with diameter of 1 mm is filled with water. Assuming that surface tension acts only at holes, then the maximum height to which the water can be filled in vessel without leakage is: (Given, surface tension of water is \[75\times {{10}^{-3}}N/m\]and \[g=10\text{ }m/{{s}^{2}}\])

A)  3 cm                                     

B)  0.3 cm

C)  3 mm                                   

D)  3 m

Correct Answer: A

Solution :

                The height (h) to which water rises in a capillary is \[h=\frac{2T\cos \theta }{r\rho g}\] Where T is surface tension, r the radius of capillary, p the density and g the acceleration due to gravity. For maximum height\[\theta =0\] \[\therefore \]  \[\cos \theta =1\] \[\Rightarrow \]               \[h=\frac{2T}{r\rho g}\] Given,   \[R=\frac{1}{2}mm=\frac{{{10}^{-3}}}{2}m\]                 \[d={{10}^{3}}\,kg/{{m}^{3}}\]                 \[h=\frac{2\times 75\times {{10}^{-3}}\times 2}{{{10}^{-3}}\times {{10}^{3}}\times 10}m\] \[\Rightarrow \]               \[h=3\times {{10}^{-2}}m\] \[\Rightarrow \]               \[h=3\,cm\]