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(C) Let $l$ be the length of the block immersed in the liquid at equilibrium. When the block is floating,the weight of the block is balanced by the bu

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$T \propto \frac{1}{\sqrt{A}}$

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(C) Let $l$ be the length of the block immersed in the liquid at equilibrium. When the block is floating,the weight of the block is balanced by the buoyant force:$mg = A l \rho g$If the block is given a small vertical displacement $y$ downwards,the additional buoyant force acting upwards is:$F_{restoring} = -(A(l+y)\rho g - mg) = -(Al\rho g + Ay\rho g - mg)$Since $mg = Al\rho g$,we get:$F_{restoring} = -A\rho g y$This is the equation of simple harmonic motion $F = -ky$,where the spring constant $k = A\rho g$.The time period $T$ of the oscillation is given by:$T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{m}{A\rho g}}$From this expression,we can see that $T \propto \frac{1}{\sqrt{A}}$.

$A$ 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 simple harmonic motion with a time period $T$. Then:

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