Question

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 $\rho_{w}$ )

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

Answer 1

Answer: (A)

When the bubble gets detached,
Buoyant force = force due to surface tension

$\int T \times dl \,\sin \,\theta=\frac{4}{3} \pi R^{3}\, \rho_{w} g $
$\Rightarrow T \times 2 \pi r \times \frac{r}{R}=\frac{4}{3} \pi R^{3} \rho_{w} \,g $
$\Rightarrow r^{2}=\frac{2 R^{4} \rho_{w} g}{3} $
$\Rightarrow r=R^{2} \sqrt{\frac{2 \rho_{w D} g}{3 T}}$