Consider two identical cylinders [each of mass $m$ density $\rho _0$ horizontal cross-section area $s$] in equilibrium, partially submerged in two containers filled with liquids of densities $\rho_1$ and $\rho_2$ as shown in figure. Find the period of small oscillations of this system about its equilibrium. Neglect the changes in the level of liquids in the containers. Neglect mass of the strings. acceleration due to gravity is $g$ . ($v$ is volume of each block)
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On displasing cylinder by $x.$ Net force on cylinders $F_{\mathrm{N}}=\rho_{1} \mathrm{xsg}+\rho_{2} \mathrm{xsg}=2 \mathrm{ma}=\left(\rho_{1}+\rho_{2}\right) \mathrm{xsg}$
$\overrightarrow{\mathrm{a}}=-\left(\frac{\rho_{1}+\rho_{2}}{\rho_{0}}\right) \frac{\mathrm{sg}}{2 \mathrm{v}} \vec{\mathrm{x}}$
$\omega=\sqrt{\left(\frac{\rho_{1}+\rho_{2}}{\rho_{0}}\right) \frac{\operatorname{sg}}{2 v}}$
$\mathrm{T}=\frac{2 \pi}{\omega}=2 \pi \sqrt{\left.\frac{\rho_{0}}{\rho_{1}+\rho_{2}}\right) \frac{2 \mathrm{v}}{\mathrm{sg}}}$
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