A cylindrical piece of cork of density `rho`, 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))`, where `'rho'` is the density of the cork.

Explanation

Restoring force `=-` weight of liquid displaced `=-(rho_(1)Ay)g`
Applied force `F=ma`
`"i.e. "F=(rhoAh)a`
`"Hence, "rhoAha=-rhoAyg`
`"or "(y)/(a)=|(-rhoh)/(rho_(1)g)|`
We know that period of oscillation,
`T=2pisqrt(("displacement")/("accelertion"))" "therefore T=2pisqrt((rho)/(rho_(1)g))`
If `.rho.` of the floating body is equal to the density of liquid then,
`T=2pisqrt((h)/(g)).`