Question 149 of 192

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 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 detatch is :
(density of water is ${\rho }_{\text{w}}$)

A${\text{R}}^{2}\sqrt{\dfrac{{2\rho }_{\text{w}}\text{g}}{3T}}$​B${\text{R}}^{2}\sqrt{\dfrac{{\rho }_{\text{w}}\text{g}}{6T}}$C${\text{R}}^{2}\sqrt{\dfrac{{\rho }_{\text{w}}\text{g}}{T}}$D${\text{R}}^{2}\sqrt{\dfrac{{3\rho }_{\text{w}}\text{g}}{T}}$

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Answer: (A) ${\text{R}}^{2}\sqrt{\dfrac{{2\rho }_{\text{w}}\text{g}}{3T}}$​

When the bubble gets detached,

Buoyant force = force due to surface tension


$\int \text{T}\times dl \text{sin} \theta =\dfrac{4}{3}\pi {\text{R}}^{3}{\rho }_{\text{w}}\text{g}$

$\Rightarrow \text{T}\times 2\pi \text{r}\times \dfrac{\text{r}}{\text{R}}=\dfrac{4}{3}\pi {\text{R}}^{3}{\rho }_{\text{w}}\text{g}$
$\Rightarrow {\text{r}}^{2}=\dfrac{2{\text{R}}^{4}{\rho }_{\text{w}}\text{g}}{3T}$