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A cubical block of density $\rho $ is floating on the surface of water. Out of its height $\mathrm{L}$, fraction $\mathrm{x}$ is submerged in water. The vessel is in an elevator accelerating upward with acceleration $\mathrm{a}$. What is the fraction immersed ?
Solution
Let density of water is $\rho_{w}$. A block of height $\mathrm{L}$ float on it. $x$ be the height of block submerged in water.
Volume of block $\mathrm{V}=\mathrm{L}^{3}$
Mass of block $m=\mathrm{V} \rho=\mathrm{L}^{3} \mathrm{pg}$
Weight of the block $=m g=\mathrm{L}^{3} \rho g$
First Case : Volume of part of cube submerged in water $=x \mathrm{~L}^{2}$
$\therefore$ Weight of water displaced by block $=x \mathrm{~L}^{2} \rho_{\mathrm{w}} g$
Weight of block = weight of water displaced by block.
$\mathrm{L}^{3} \rho g=x \mathrm{~L}^{2} \rho_{w} g$
$\therefore \frac{x}{\mathrm{~L}}=\frac{\rho}{\rho_{\mathrm{w}}}$
$\therefore x=\left(\frac{\rho}{\rho_{w}}\right) \mathrm{L} \quad \ldots(1)$
Second Case : When vessel is placed in an elevator moving upward with acceleration $a$, then
effective acceleration $=g^{\prime}=(g+a)$
(More acceleration $a$ is due to Pseudo force)
$\therefore$ Weight of block $=m g^{\prime}$
Suppose, an elevator is moving upward. Let new fraction of block submerged in water is $x_{1}$
