Question

On 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 ${\rho }_w$)

• A. $R^2\sqrt{\frac{{\rho }_{w}g}{T}}$
• B. $R^2\sqrt{\frac{3{\rho }_{w}g}{T}}$
• C. $R^2\sqrt{\frac{2{\rho }_{w}g}{3T}}$
• D. $R^2\sqrt{\frac{{\rho }_{w}g}{6T}}$

Answer 1

Answer: (C)

$R^2\sqrt{\frac{2{\rho }_{w}g}{3T}}$

$\left(2\pi rT\right)sin\theta =\frac{4}{3}\pi R^3{\rho }_{\omega }.g$
$T\times \frac{r}{R}\times 2\pi r=\frac{4}{3}\pi R^3{\rho }_{w}g$
$r^2=\frac{2}{3}\frac{R^4{\rho }_{w}g}{T}$
$r=R^2\sqrt{\frac{2{\rho }_{\omega }g}{3T}}$