Question

One thousand small water drops of equal radii combine to form a big drop. The ratio of final surface energy to the total initial surface energy is

• A. \( 1 : 1000 \)
• B. \( 1 : 1 \)
• C. \( 1 : 10 \)
• D. \( 1 : 100 \)

Hint:

When drops merge, volume is conserved but surface area decreases significantly.

Answer 1

The Correct Option is C

Step 1: Use volume conservation.
Let radius of each small drop be \( r \).
Total volume of 1000 drops \[ 1000 \times \frac{4}{3}\pi r^3 = \frac{4}{3}\pi R^3 \] \[ \Rightarrow R^3 = 1000 r^3 \Rightarrow R = 10r \]
Step 2: Write surface energy relation.
Surface energy is proportional to surface area.
Initial surface energy: \[ E_i \propto 1000 \times 4\pi r^2 \]
Final surface energy: \[ E_f \propto 4\pi R^2 = 4\pi (10r)^2 = 400\pi r^2 \]
Step 3: Find ratio.
\[ \frac{E_f}{E_i} = \frac{400\pi r^2}{1000 \times 4\pi r^2} = \frac{1}{10} \]
Step 4: Conclusion.
The ratio of final surface energy to initial surface energy is \( 1 : 10 \).