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Chapter 7: Problem 1001

The radii of two soap bubbles are \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2} .\) In isothermal conditions two meet together is vacuum Then the radius of the resultant bubble is given by (A) \(\mathrm{R}=\left[\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right) / 2\right]\) (B) \(\mathrm{R}=\mathrm{r}_{1}\left(\mathrm{r}_{1}+\mathrm{r}_{2}+\mathrm{r}_{3}\right)\) (C) \(\mathrm{R}^{2}=\mathrm{r}_{1}^{2}+\mathrm{r}_{2}^{2}\) (D) \(\mathrm{R}=\mathrm{r}_{1}+\mathrm{r}_{2}\)

Short Answer

Expert verified
The radius of the resultant soap bubble is given by the formula \(R = \sqrt{r_1^2 + r_2^2}\). The correct answer is (C) \(R^2 = r_1^2 + r_2^2\).

Step by step solution

01

Given information

We are given the radii of two soap bubbles, \(r_1\) and \(r_2\), and need to find the radius of the resultant soap bubble, R, when they merge. Step 2: Understand the concept of surface area conservation
02

Surface area conservation

Under isothermal conditions and in a vacuum, the surface area of the two soap bubbles (assuming they are spherical) will be conserved during the merging process. The surface area of a soap bubble is given by the formula: \(A = 4\pi r^2\) Step 3: Write the equation for the conservation of surface area
03

Conservation of surface area equation

Using the surface area formula, we can set up the equation for the conservation of surface area: \(A_1 + A_2 = A_R\) Substituting the respective radii and the surface area formula, we get: \(4\pi r_1^2 + 4\pi r_2^2 = 4\pi R^2\) Step 4: Simplify and solve for R
04

Solve for R

Now, we need to solve for R. First, simplify the equation by dividing both sides by \(4\pi\): \(r_1^2 + r_2^2 = R^2\) Now, take the square root of both sides of the equation: \(\sqrt{r_1^2 + r_2^2} = R\) The radius of the resultant soap bubble, R, is given by the formula: \(R = \sqrt{r_1^2 + r_2^2}\) Comparing with the given options, the correct answer is: (C) \(R^2 = r_1^2 + r_2^2\)

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