MHT CET · Physics · Mechanical Properties of Fluids

A big water drop is formed by the combination of $n$ small water drops of equal radii. The ratio of the surface energy of $n$ drops to the surface energy of the big drop is

A $n^{2} : 1$
B $n : 1$
C $\sqrt{n} : 1$
D $\sqrt[3]{n} : 1$

Verified Solution

Answer & Solution

Correct Answer

(D) $\sqrt[3]{n} : 1$

Step-by-step Solution

Detailed explanation

Volume of big drop =

n

(Volume of small drop)

\frac{4}{3} π R^{3} = n . \frac{4}{3} π r^{3}

R^{3} = n r^{3}

R = n^{\frac{1}{3}} . r

E_{2} = s u r f a c e e n e r g y o f n d r o p s = n \times 4 π r^{2} \times T

E_{1} = s u r f a c e o f e n e r g y o f b i g d r o p = 4 π R^{2} T

\begin{array}{l} ∴ \frac{E_{2}}{E_{1}} = \frac{n r^{2}}{R^{2}} = \frac{n r^{2}}{{(n^{\frac{1}{3}} . r)}^{2}} = \frac{n r^{2}}{n^{\frac{2}{3}} . r^{2}} \\ ∴ \frac{E_{2}}{E_{1}} = n^{\frac{1}{3}} = \sqrt[3]{n} : 1 \end{array}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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A big water drop is formed by the combination of n small water drops of equal radii. The ratio of the surface energy of n drops to the surface energy of the big drop is

Step-by-step Solution

A big water drop is formed by the combination of $n$ small water drops of equal radii. The ratio of the surface energy of $n$ drops to the surface energy of the big drop is