A solid cylinder of density rho_0,cross-secti | vedclass

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(B) Let the cylinder be in equilibrium floating in the liquid. The weight of the cylinder is balanced by the buoyant force: $mg = F_B \Rightarrow (A l

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$2 \pi \sqrt{\frac{\rho_0 l}{\rho g}}$

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(B) Let the cylinder be in equilibrium floating in the liquid. The weight of the cylinder is balanced by the buoyant force: $mg = F_B \Rightarrow (A l \rho_0) g = (A h \rho) g$,where $h$ is the submerged length. Thus,$h = \frac{\rho_0 l}{\rho}$.When the cylinder is displaced downward by a small distance $x$,the additional buoyant force acting upward is $F_{extra} = A x \rho g$.This force acts as a restoring force: $F = - (A \rho g) x$.Comparing this with the simple harmonic motion equation $F = -kx$,we get the effective spring constant $k = A \rho g$.The mass of the cylinder is $m = A l \rho_0$.The time period $T$ is given by $T = 2 \pi \sqrt{\frac{m}{k}}$.Substituting the values: $T = 2 \pi \sqrt{\frac{A l \rho_0}{A \rho g}} = 2 \pi \sqrt{\frac{\rho_0 l}{\rho g}}$.

$A$ solid cylinder of density $\rho_0$,cross-section area $A$ and length $l$ floats in a liquid of density $\rho (\rho > \rho_0)$ with its axis vertical. If it is slightly displaced downward and released,the time period of oscillation will be:

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