Question

A liquid drop of radius $R$ is sprayed into 1000 identical small droplets. If $u_i$ and $u_f$ are the surface energy of the liquid in Initial and final state. Then $\frac{u_i}{u_f}$ is

• A. 10
• B. 100
• C. $\frac{1}{10}$
• D. $\frac{1}{100}$

Answer 1

Answer: (C)

C. $\frac{1}{10}$

Answer 2

The surface energy of a liquid drop is proportional to its surface area ($u = \sigma A$). Initial surface energy $u_i = \sigma \cdot 4\pi R^2$. Volume conservation: $\frac{4}{3}\pi R^3 = 1000 \cdot \frac{4}{3}\pi r^3 \implies R = 10r \implies r = \frac{R}{10}$. Final surface area $A_f = 1000 \cdot 4\pi r^2 = 1000 \cdot 4\pi (\frac{R}{10})^2 = 10 \cdot 4\pi R^2$. Thus, $A_f = 10 A_i$. Final surface energy $u_f = \sigma A_f = \sigma (10 A_i) = 10 u_i$. Therefore, $\frac{u_i}{u_f} = \frac{u_i}{10 u_i} = \frac{1}{10}$.