Question

Let $R_1, R_2$ and $R_3$ be the radii of three mercury drops. A big mercury drop is formed from them under isothermal conditions. The radius of the resultant drop is ______.

• A. $(R_1^3 + R_2^3 + R_3^3)^{1/3}$
• B. $(R_1^3 + R_2^3 - R_3^3)^{1/3}$
• C. $(R_1^3 + R_2^3 + R_3^3)$
• D. $(R_1 + R_2 + R_3)^3$

Hint:

Any time liquid drops merge or split, Volume is conserved. Therefore, $R_{\text{new}}^3 = \sum R_{\text{old}}^3$. This logic applies equally to spherical capacitors combining their charges!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Three liquid drops combine to form one single large drop. We must find the radius of this new large drop using the conservation of physical volume.

Step 2: Detailed Explanation:
When liquid drops coalesce (merge together), liquids are assumed to be perfectly incompressible. Therefore, the total volume of the liquid remains completely conserved before and after the merger.
"Isothermal conditions" simply ensures the density doesn't change due to temperature fluctuations.
$\text{Total Initial Volume} = \text{Final Volume}$
The volume of a sphere is given by $V = \frac{4}{3} \pi r^3$.
Sum the volumes of the three individual initial drops:
$V_{\text{initial}} = \frac{4}{3} \pi R_1^3 + \frac{4}{3} \pi R_2^3 + \frac{4}{3} \pi R_3^3$
Let the radius of the newly formed big drop be $R_{\text{big}}$. Its volume is:
$V_{\text{final}} = \frac{4}{3} \pi R_{\text{big}}^3$
Equate the initial and final volumes:
$\frac{4}{3} \pi R_{\text{big}}^3 = \frac{4}{3} \pi R_1^3 + \frac{4}{3} \pi R_2^3 + \frac{4}{3} \pi R_3^3$
Factor out the common $\frac{4}{3} \pi$ term on the right side:
$\frac{4}{3} \pi R_{\text{big}}^3 = \frac{4}{3} \pi (R_1^3 + R_2^3 + R_3^3)$
Cancel the $\frac{4}{3} \pi$ from both sides of the equation:
$R_{\text{big}}^3 = R_1^3 + R_2^3 + R_3^3$
To solve for the resultant radius $R_{\text{big}}$, take the cube root (power of $1/3$) of both sides:
$R_{\text{big}} = (R_1^3 + R_2^3 + R_3^3)^{1/3}$

Step 3: Final Answer:
The radius of the resultant drop is $(R_1^3 + R_2^3 + R_3^3)^{1/3}$, matching option (a).