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Question Number 57114 by Tinkutara last updated on 30/Mar/19

Commented by ajfour last updated on 30/Mar/19

Commented by mr W last updated on 31/Mar/19

$$nicesolutionsir!$$$${let}^{\prime }ssaym>m_{min}.whatistheperiode$$$$oftheswingingmotionofthestick?$$

Commented by ajfour last updated on 31/Mar/19

$$\mathrm{should}\mathrm{viscosity}\mathrm{be}\mathrm{taken}\mathrm{into}$$$$\mathrm{consideration},\mathrm{Sir}?$$

Commented by mr W last updated on 31/Mar/19

$$no.onlyweightofstickandbuoyant$$$$forceact.stabilitymeansthebuoyant$$$$forcebringsthestickalwaystoits$$$$originalposition.$$

Answered by ajfour last updated on 30/Mar/19

$$\mathrm{for}\mathrm{stability}$$$$\mathrm{centre}\mathrm{of}\mathrm{buoyancy}\mathrm{must}\mathrm{be}\mathrm{above}$$$$\mathrm{or}\mathrm{at}\mathrm{same}\mathrm{level}\mathrm{as}\mathrm{centre}\mathrm{of}\mathrm{gravity}.$$$$\Rightarrow \frac{\mathrm{x}}{2}⩾\frac{\mathrm{m}\times 0+\rho \pi {\mathrm{R}}^2\mathrm{L}\left(\frac{\mathrm{L}}{2}\right)}{\mathrm{m}+\rho \pi {\mathrm{R}}^2\mathrm{L}}$$$$\mathrm{and}$$$$\mathrm{m}+\rho \pi {\mathrm{R}}^2\mathrm{L}=\sigma \pi {\mathrm{R}}^2\mathrm{x}$$$$\Rightarrow \frac{\mathrm{m}+\rho \pi {\mathrm{R}}^2\mathrm{L}}{2\sigma \pi {\mathrm{R}}^2}⩾\frac{\rho \pi {\mathrm{R}}^2\mathrm{L}\left(\frac{\mathrm{L}}{2}\right)}{\mathrm{m}+\rho \pi {\mathrm{R}}^2\mathrm{L}}$$$$\mathrm{let}\pi {\mathrm{R}}^2\mathrm{L}=\mathrm{c},\mathrm{then}$$$${(\mathrm{m}+\rho \mathrm{c})}^2⩾\sigma \rho {\mathrm{c}}^2$$$$\Rightarrow \mathrm{m}⩾\left(\sqrt{\sigma \rho }\right)\mathrm{c}-\rho \mathrm{c}$$$$\Rightarrow {\mathrm{m}}_{\mathrm{minimum}}=(\sqrt{\sigma \rho }-\rho )\pi {\mathrm{R}}^2\mathrm{L}.$$

Commented by Tinkutara last updated on 30/Mar/19

Thank you Sir! But what was wrong with previous answer?

Commented by ajfour last updated on 30/Mar/19

$$\mathrm{previous}\mathrm{answer}\mathrm{should}\mathrm{go}\mathrm{for}$$$$\mathrm{maximum}\mathrm{mass}\mathrm{that}\mathrm{be}\mathrm{attached}$$$$\mathrm{to}\mathrm{stick}\mathrm{to}\mathrm{keep}\mathrm{it}\mathrm{afloat}\mathrm{still},\mathrm{i}\mathrm{think}.$$

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