Question 137 of 192

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):

A$r=\sqrt{\dfrac{2T}{3(d+\rho )g}}$B$r=\sqrt{\dfrac{T}{(d-\rho )g}}$C$r=\sqrt{\dfrac{T}{(d+\rho )g}}$D$r=\sqrt{\dfrac{3T}{(2d-\rho )g}}$

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Answer: (D) $r=\sqrt{\dfrac{3T}{(2d-\rho )g}}$

$dVg=p\left(\dfrac{V}{2}\right)g+T\left(2\pi r\right)$
$\Rightarrow d.\dfrac{4}{3}\pi {r}^{3}g=p.\dfrac{2}{3}\pi {r}^{3}g+2\pi rT$
$\Rightarrow \dfrac{2}{3}{r}^{2}g\left(2d-p\right)=2T$
$\Rightarrow r=\sqrt{\dfrac{3T}{\left(2d-p\right)g}}$