A cylindrical piece of cork of base area A and height h floats in a liquid of density `rho_(1)`. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
`T=2pisqrt((hrho)/(rho_(1)g))`

Explanation

Say, inertially in equilibrium , y height of cylinder is inside the liquid . Then , weight of the cylinder =upthrust due to liquid displaced
`:. Ah rhog=Ay rho g`
when the corck cylinder is depressed slightly by `Deltay` and realsed, a restoring force, equal to additional upthrust, acts on it. the restoring force is
`F=A(y+Deltay) rho_(l)g -Ay rho_(1)g-Ayrho_(1)g=Arho_(1)gDeltay`
`:.`acceleration , `a=F/m =(Arho_(1)g Deltay)/(Ahrho)=(rho_(1)g)/(hrho).Deltay` and the acceleration is directed in the direction opposite to `Deltay`, obviously as `aprop-Detlay`, the motion of cork cylinder is SHM, whose time period is given by
`T=2pisqrt(("displacement")/("acceleration"))`
`=(2pisqrt(Deltay)/(a))`
`=2pisqrt((hrho)/(rho_(1)g))`