A cylindrical block of density $\rho$ is partially immersed in a liquid of density $3\rho .$ The plane surface of the block remains parallel to the surface of the liquid. The height of the block is $60\, cm.$ The block performs $SHM$ when displaced from its mean position. [Use $ g = 9.8\, m/s^2$]
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When cylindrical block is partially immersed
$F_B=m g \Rightarrow 3 \rho A y\,g=\rho A\left(60 \times 10^{-2}\right)\,g$
$y=20\,cm$
$\Rightarrow \text { Maximum amplitude }=20\,cm$
Restoring force when it is slightly depressed by an amount of $x$.
$F =-(\Delta V \sigma g )=-( A \sigma g ) x$
$T =2 \pi \sqrt{\frac{ m }{ A \sigma g }}=2 \pi \sqrt{\frac{\rho A h}{3 A \rho g}}=2 \pi \sqrt{\frac{ h }{3 g}}$
$=2 \pi \sqrt{\frac{60 \times 10^{-2}}{3 \times 9.8}}=\frac{2 \pi}{7}$
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