A rectangular block of mass m and cross-secti | vedclass

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(A) In equilibrium,the weight of the block is balanced by the buoyant force:$mg = A l \rho g \Rightarrow m = A \rho l$where $l$ is the length of the p

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$T^{2} \propto \frac{1}{\rho}$

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(A) In equilibrium,the weight of the block is balanced by the buoyant force:$mg = A l ho g \Rightarrow m = A ho l$where $l$ is the length of the part immersed in the liquid.When the block is given a small downward displacement $y$,the additional buoyant force (restoring force) acting upwards is:$F = -[A(l+y) ho g - mg]$$F = -[A l ho g + A y ho g - mg]$Since $mg = A l ho g$,the equation simplifies to:$F = -A ho g y$This shows that $F \propto -y$,which is the condition for $SHM$. The effective spring constant $k$ is $A ho g$ and the inertia factor is $m$.The time period $T$ of $SHM$ is given by:$T = 2 \pi \sqrt{\frac{	ext{Inertia factor}}{	ext{Spring factor}}} = 2 \pi \sqrt{\frac{m}{A ho g}}$Squaring both sides,we get:$T^{2} = 4 \pi^{2} \frac{m}{A ho g}$Thus,$T^{2} \propto \frac{1}{ho}$.

$A$ rectangular block of mass $m$ and cross-sectional area $A$ floats in a liquid of density $\rho$. If we give it a small vertical displacement from equilibrium,it undergoes $SHM$ with a time period $T$. Then:

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