A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ρ1. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period T = 2π √hρ / ρ1g where ρ is the density of cork. (Ignore damping due to viscosity of the liquid)
Base area of the cork =A Height of the cork =h Density of the liquid =ρ1 Density of the cork=ρ In equilibrium: Weight of the cork = Weight of the liquid displaced by the floating cork Let the cork be depressed slightly by x. As a result, some extra water of a certain volume is displaced. Hence, an extra up-thrust acts upward and provides the restoring force to the cork. Up-thrust = Restoring force, F= Weight of the extra water displaced F=−(Volume×Density×g) Volume = Area × Distance through which the cork is depressed Volume =Ax ∴F=−Ax×ρ1g ...(i) Accroding to the force law: F=kx k=F/x where, k is constant k=F/x=−Aρ1g ...(ii) The time period of the oscillations of the cork: T=2πkm ....(iii) where, m= Mass of the cork = Volume of the cork × Density = Base area of the cork × Height of the cork × Density of the cork =Ahρ Hence, the expression for the time period becomes: T=2πAρ1gAhρ=2πρ1ghρ
