A small spherical droplet of density d
Question:

A small spherical droplet of density $d$ is floating exactly half immersed in a liquid of density $\rho$ and surface tension T. The radius of the droplet is (take note that the surface tension applies an upward force on the droplet):

1. (1) $r=\sqrt{\frac{2 \mathrm{~T}}{3(d+\rho) g}}$

2. (2) $r=\sqrt{\frac{\mathrm{T}}{(d-\rho) g}}$

3. (3) $r=\sqrt{\frac{\mathrm{T}}{(d+\rho) g}}$

4. (4) $r=\sqrt{\frac{3 \mathrm{~T}}{(2 d-\rho) g}}$

Correct Option: , 4

Solution:

(4) For the drops to be in equilibrium upward force on

drop $=$ downward force on drop

$T .2 \pi R=\frac{4}{3} \pi R^{3} d g-\frac{2}{3} \pi R^{3} \rho g$

$\Rightarrow T(2 \pi R)=\frac{2}{3} \pi R^{3}(2 d-\rho) g$

$\Rightarrow T=\frac{R^{2}}{3}(2 d-\rho) g \Rightarrow R=\sqrt{\frac{3 T}{(2 d-\rho) g}}$