Question

A liquid drop splits into \(729\) identical spherical drops. If \(E\) is the original energy and \(U\) is the total resulting energy , then \(E/U = 1/x\). The value of \(x\) is.

• A. \(9\)
• B. \(7\)
• C. \(6\)
• D. \(13\)

Hint:

When a drop splits, surface energy increases by a factor of $n^{1/3}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A large drop breaks into many smaller drops. Volume remains conserved, but total surface area (and thus surface energy) increases. We need to find the ratio of initial to final surface energy.
Step 2: Key Formula or Approach:
1) Conservation of Volume: \(\frac{4}{3}\pi R^3 = n \cdot \frac{4}{3}\pi r^3 \implies R = n^{1/3} r\).
2) Surface energy: \(E = T \cdot A\).
Initial energy \(E = T \cdot 4\pi R^2\).
Final total energy \(U = n \cdot T \cdot 4\pi r^2\).
Step 3: Detailed Explanation:
Given \(n = 729 = 9^3\).
From volume conservation:
\[ R = (729)^{1/3} \cdot r = 9r \implies r = \frac{R}{9} \]
Initial surface energy:
\[ E = 4\pi R^2 T \]
Final total surface energy:
\[ U = 729 \times (4\pi r^2 T) = 729 \times 4\pi \left(\frac{R}{9}\right)^2 T \]
\[ U = 729 \times \frac{4\pi R^2 T}{81} \]
\[ U = 9 \times (4\pi R^2 T) = 9E \]
The ratio is:
\[ \frac{E}{U} = \frac{1}{9} \]
Comparing with \(\frac{1}{x}\), we find \(x = 9\).
Step 4: Final Answer:
The value of \(x\) is \(9\).