A large tank of cross-section area A contains liquid of density $ρ$ . A cylinder of density $\frac{ρ}{4}$ , length $l = 40 c m$ , and cross-section area $a (< < A)$ floating in the liquid. It is pushed inside the liquid so that it is just submerged and then released at $t = 0$ . Assuming that the water level in the tank remains constant, determine the speed (in m /s) of the cylinder when it reaches its equilibrium position for the first time. Neglect viscous forces. $[Take g = 10 {ms}^{- 2}]$

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Detailed Solution

The motion of the cylinder will be simple harmonic. In equilibrium, the block will float with 1/4^th of its length submerged. If further pushed by a distance x,
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$\begin{array}{l} (\frac{ρ}{4}) ℓa \times {acc}^{n} = - (ρag) x \\ \Rightarrow ω = \sqrt{\frac{4 g}{ℓ}} \cdot At equilibrium V_{max} = ω \times Amplitude \\ \Rightarrow V_{max} = \sqrt{\frac{4 g}{ℓ}} \times \frac{3 ℓ}{4} = \frac{3}{2} \sqrt{gℓ} = 3 m / s \end{array}$