Explanation

(d) Let $l$ be the length of block immersed in liquid as shown in the figure. When the block is floating.
[image:14_image60.emf]
$\therefore mg = Al\rho g$
If the block is given vertical displacement $y$ then the effective restoring force is
$F = - \left\lbrack A(l + y)\rho g - mg \right) = - \left\lbrack A(l + y)\rho g - Al\rho g \right\rbrack = - (Al\rho g)y$
$i.e.,\ F \propto y.$ As this $F$ is directed towards its equilibrium position of block, so if the block is left free, it will execute simple harmonic motion.
Here inertia factor = mass of block $= m$
Spring factor $= A\rho g$
$\therefore Time\ period = T = 2\pi\sqrt{\frac{m}{A\rho g}}$
$i.e.\ T \propto \frac{1}{\sqrt{A}}$