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A cylindrical block of area of cross-section $A$ and of material of density $\rho$ is placed in a liquid of density one-third of density of block. The block compresses a spring and compression in the spring is one-third of the length of the block. If acceleration due to gravity is $g$, the spring constant of the spring is:
$\rho$$ Ag$
$2\rho$$Ag$
$2\rho $$Ag/3$
$\rho $$Ag/3$
Solution
since the block is at rest, net force on the cylindrical block is zero, l.e.
net upward $=$ net downward force
$F_{s}+F_{b}=W$
where
$F_{s}=$ force due to compression in the spring
$F_{b}=$ buoyant force
$W=$ weight of the cylindrical block
$\Rightarrow k x+A L \frac{\rho}{3} g=A L \rho g$
$\Rightarrow k \frac{L}{3}+A L \frac{\rho}{3} g=A L \rho g$
$\Rightarrow k \frac{L}{3}=\frac{2 A L \rho g}{3}$
$\Rightarrow k=2 \rho A g$
