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Mechanical Properties Of Fluids

Question
CBSEENPH11020276

On the heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius R and making a circular contact 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 bubbles detach is (density of water is ρw)

  • straight R squared square root of fraction numerator 2 straight rho subscript straight w straight g over denominator 3 straight T end fraction end root
  • straight R squared square root of fraction numerator straight rho subscript straight w straight g over denominator 6 straight T end fraction end root
  • straight R squared square root of fraction numerator straight rho subscript straight w straight g over denominator straight T end fraction end root
  • straight R squared space square root of fraction numerator 3 straight rho subscript straight w straight g over denominator straight T end fraction end root

Solution

A.

straight R squared square root of fraction numerator 2 straight rho subscript straight w straight g over denominator 3 straight T end fraction end root

The bubble will detach if, 
Buoyant force ≥ Surface tension force
4 over 3 πR cubed straight rho subscript straight w straight g space greater or equal than integral straight T space straight x space dl space sin space straight theta space

left parenthesis straight rho subscript straight w right parenthesis open parentheses 4 over 3 πR cubed close parentheses straight g greater or equal than left parenthesis straight T right parenthesis left parenthesis 2 πpr right parenthesis space sin space straight theta
sin space straight theta space equals space straight r over straight R
solving space straight r space equals space square root of fraction numerator 2 straight p subscript straight w straight R to the power of 4 straight g over denominator 3 straight T end fraction end root space equals space straight R squared square root of fraction numerator 2 straight rho subscript straight w straight g over denominator 3 straight T end fraction end root