A rectangular block of mass $m$ and area of cross-section $A$ floats in a liquid of density $\rho $. If it is given a small vertical displacement from equilibrium it undergoes with a time period $T,$ then
AIPMT 2006, Diffcult
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Let $l$ be the length of block immersed in liquid as shown in the figure. When the block is floating,
$\therefore \quad m g=A l \rho m$
If the block is given vertical displacement $y$ then the effective restoring force is
$F =-[A(l+y) \rho g-m g]=-[A(l+y) \rho g-A l \rho g] $
$=-A l \rho g y$
$Restoring\, force =-[A l \rho g] y .$ As this $F$ is directed towards its equilibrium position of block, so if the block is left free, it will execute simple harmonic motion. $Here\, inertia\, factor = mass \,of \,block =m$
$Spring ,factor =A \rho g$
$\therefore \quad$ Time period $=T=2 \pi \sqrt{\frac{m}{A \rho g}}$ i.e. $T \propto \frac{1}{\sqrt{A}}$
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