Question

A water droplet splits into $27$ identical small droplets. The pressure difference between the inner and outer surface of the big droplet will be

• A. Same as that of smaller droplet
• B. $1/3\text{rd}$ of the pressure difference for smaller droplet.
• C. $1/4\text{th}$ of the pressure difference for smaller droplet.
• D. None of these

Answer 1

The correct option is B $1/3\text{rd}$ of the pressure difference for smaller droplet.

Let the radius of bigger droplet is $R$ and radius of smaller droplets be $r$.
Applying volume conservation for bigger and smaller droplets,
$\Rightarrow \frac{4}{3}\pi R^3=27\times \frac{4}{3}\pi r^3$

$\Rightarrow {\left(\frac{R}{r}\right)}^3=27$
$\Rightarrow \frac{R}{r}=3$
$\Rightarrow r=\frac{R}{3}$
The pressure difference between the inner and outer surface of bigger water droplet( consists of single surface) is given as,
$\Delta P_B=\frac{2T}{R}$
similarly for smaller water droplet,
$\Delta P_S=\frac{2T}{r}=\frac{2T}{\left(\frac{R}{3}\right)}=\frac{6T}{R}$
Hence,
$\Delta P_B=\frac{\Delta P_S}{3}$
$\therefore$Option $\left(b\right)$ is correct.