A liquid of density 2ρ is filled in a cylindrical vessel, whose cross-sectional area is 2A. A wooden cylinder of height H, cross-sectional area A and density ρ 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.

Question

A liquid of density 2ρ is filled in a cylindrical vessel, whose cross-sectional area is 2A. A wooden cylinder of height H, cross-sectional area A and density ρ 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. <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-0ukwzeb6or06o7dg.jpg" /> (A) (.g/H.)x (B) (2 g / H)x (C) (3 g / H)x (D) (4 g / H)x

Tardigrade Admin · Accepted Answer

x× A=y× (.2A-A.)⇒ x=y Where y is change in water level Extra buoyant force =wt. of extra liq. displaced (.x+y.)A× 2ρ g=ma 2xA× 2ρ g=Hρ × Aa <img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-0ukwzeb6or06ejjl.jpg" /> a=((4 g/H))x

$(g / H) x$

$(2 g / H) x$

$(3 g / H) x$

$(4 g / H) x$

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

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

(g/H)x

(2g/H)x

(3g/H)x

(4g/H)x

x×A=y×(2A−A)⇒x=y Where y is change in water level Extra buoyant force =wt. of extra liq. displaced (x+y)A×2ρg=ma 2xA×2ρg=Hρ×Aa a=(H4g​)x

$(g / H) x$

$(2 g / H) x$

$(3 g / H) x$

$(4 g / H) x$

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

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