Question

A liquid of density $2\rho $ is filled in a cylindrical vessel, whose cross-sectional area is $2A.$ A wooden cylinder of height $H,$ cross-sectional area $A$ and density $\rho $ is floating in the liquid at equilibrium with its axis vertical. The cylinder is pushed down by a small distance $x$ from its equilibrium position and released. Find its initial acceleration.

• A. $\left(\right.g/H\left.\right)x$
• B. $\left(2 g / H\right)x$
• C. $\left(3 g / H\right)x$
• D. $\left(4 g / H\right)x$

Answer 1

Answer: (D)

$x\times A=y\times \left(\right.2A-A\left.\right)\Rightarrow x=y$
Where $y$ is change in water level
Extra buoyant force $=wt.$ of extra liq. displaced
$\left(\right.x+y\left.\right)A\times 2\rho g=ma$
$2xA\times 2\rho g=H\rho \times Aa$

$a=\left(\frac{4 g}{H}\right)x$