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Question Number 155293 by peter frank last updated on 28/Sep/21

Answered by mr W last updated on 28/Sep/21

Commented by mr W last updated on 28/Sep/21

$$T_1=T_2=T_3=T=membranetension$$$$p_1,p_2,p_3=resultantpressureonmembrane$$$$p_1=\frac{2T_1}{r_1}$$$$p_2=\frac{2T_2}{r_2}$$$$p_3=\frac{2T_3}{r_3}$$$$p_3=p_1-p_2$$$$\frac{2T_3}{r_3}=\frac{2T_1}{r_1}-\frac{2T_2}{r_2}$$$$\Rightarrow \frac{1}{r_3}=\frac{1}{r_1}-\frac{1}{r_2}=\frac{r_2-r_1}{r_1{r}_2}$$$$\Rightarrow r_3=r=\frac{r_1{r}_2}{r_2-r_1}$$

Commented by peter frank last updated on 28/Sep/21

$$\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{much}.$$$$\mathrm{but}\mathrm{soap}\mathrm{bubble}\mathrm{has}\mathrm{two}\mathrm{surface}$$$$\frac{4\mathrm{T}}{\mathrm{r}}$$

Commented by mr W last updated on 28/Sep/21

$$ireferetotheresultantforceasif$$$$itisamembranelikeaballoon.$$

Commented by peter frank last updated on 28/Sep/21

$$\mathrm{okay}.$$

Commented by Tawa11 last updated on 28/Sep/21

$$\mathrm{Great}\mathrm{sirs}$$

Answered by peter frank last updated on 28/Sep/21

Commented by peter frank last updated on 28/Sep/21

$$\mathrm{S}=\mathrm{common}\mathrm{interface}$$$$\mathrm{H}=\mathrm{outside}\mathrm{pressure}$$$${\mathrm{P}}_1\mathrm{\&}{\mathrm{P}}_2=\mathrm{inside}\mathrm{pressure}\mathrm{for}\mathrm{small}$$$$\mathrm{and}\mathrm{large}\mathrm{bubble}$$$$\gamma =\mathrm{surface}\mathrm{tension}$$$$\underset{―}{\mathit{F}\mathit{o}\mathit{r}\mathit{s}\mathit{m}\mathit{a}\mathit{l}\mathit{l}\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}\mathit{e}}$$$${\mathrm{P}}_1-\mathrm{H}=\frac{4\gamma }{{\mathrm{r}}_1}....\left(\mathrm{i}\right)$$$$\underset{―}{\mathit{F}\mathit{o}\mathit{r}\mathit{l}\mathit{a}\mathit{r}\mathit{g}\mathit{e}\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}\mathit{e}}$$$${\mathrm{P}}_2-\mathrm{H}=\frac{4\gamma }{{\mathrm{r}}_2}....\left(\mathrm{ii}\right)$$$$\underset{―}{\mathit{F}\mathit{o}\mathit{r}\mathit{t}\mathit{h}\mathit{e}\mathit{i}\mathit{n}\mathit{t}\mathit{e}\mathit{r}\mathit{f}\mathit{a}\mathit{c}\mathit{e}}$$$${\mathrm{P}}_1-{\mathrm{P}}_2=\frac{4\gamma }{\mathrm{r}}....\left(\mathrm{iii}\right)$$$$\left(\mathrm{i}\right)-\left(\mathrm{ii}\right)$$$${\mathrm{P}}_1-{\mathrm{P}}_2=\frac{4\gamma }{{\mathrm{r}}_1}-\frac{4\gamma }{{\mathrm{r}}_2}$$$$\mathrm{but}{\mathrm{P}}_1-{\mathrm{P}}_2=\frac{4\gamma }{\mathrm{r}}$$$$\frac{4\gamma }{\mathrm{r}}=\frac{4\gamma }{{\mathrm{r}}_1}-\frac{4\gamma }{{\mathrm{r}}_2}$$$$\frac{1}{\mathrm{r}}=\frac{1}{{\mathrm{r}}_1}-\frac{1}{{\mathrm{r}}_2}$$$$\mathrm{r}=\frac{{\mathrm{r}}_1{\mathrm{r}}_2}{{\mathrm{r}}_{2-}{\mathrm{r}}_1}$$

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