question_answer

On heating water, bubbles being formed at the bottom of the vessel detatch and rise. Take the bubbles to be spheres of radius R and making a circularcontact of radius r with the bottom of the vessel. If r << R, and the surface tension of water is T, value of r just before bubblesdetatch is : (density of water is \[{{\rho }_{w}}\])   JEE Main  Solved  Paper-2014

A) \[{{R}^{2}}\sqrt{\frac{{{\rho }_{w}}g}{T}}\]                         

B) \[{{R}^{2}}\sqrt{\frac{3{{\rho }_{w}}g}{T}}\]

C) \[{{R}^{2}}\sqrt{\frac{{{\rho }_{w}}g}{3T}}\]                       

D)  \[{{R}^{2}}\sqrt{\frac{{{\rho }_{w}}g}{6T}}\]

Correct Answer: D

Solution :

\[\left[ r=\sqrt{\frac{2}{3}\frac{\rho g}{T}}\cdot {{R}^{2}} \right]\] The down ward force on the bubble due to surface tension \[=2\pi r.T\sin \theta \]\[=\frac{2\pi T{{r}^{2}}}{R}\]The upward buoyant force exceeds, the surface tension force then the bubbledetaches. \[\therefore \]\[\frac{4}{3}\pi {{R}^{3}}\rho g=\frac{2\pi T{{r}^{2}}}{R}\]\[\Rightarrow \]\[\left[ r=\sqrt{\frac{2}{3}\frac{\rho g}{T}}\cdot {{R}^{2}} \right]\]