A block of rectangular size of mass $m$ and area of cross-section $A$, floats in a liquid of density $\rho$. If we give a small vertical displacement from equilibrium, it undergoes $S H M$ with time period $T$, then
AIIMS 2018, Diffcult
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$M g=A l \rho g \Rightarrow m=A \rho l$
Where, $l=$ length of part immersed in liquid.
When it is given in downward displacement $y$, restoring force (upward direction) on block is
$F=-[A(l+y) \rho g-m g]$
$=-[A(l+y) \rho g-A l \rho g]$
$=-A \rho g y$
i.e. $F \propto-y$ or $a \propto-\gamma,$ so it execute SHM (inertia factor). Mass of block $=m$
Spring factor $=A \rho g$
Time period $=2 \pi \sqrt{\frac{\text { Inertia factor }}{\text { spring factor }}}$
$T=2 \pi \sqrt{\frac{m}{A \rho g}}$
$\Rightarrow T^{2} \propto \frac{m}{A \rho}$
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