A cylindrical piece of cork of density of base area Aand height h floats in a liquid of density `rho_l`. 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_1g))` where p is the density of cork. (Ignore damping due to viscosity of the liquid)

A cylindrical log of wood of height h and area of cross-section A floasts in water. It is pressed and then released. Show that the log would execute S.H.M. with a time period T = 2pi sqrt(m//Arhog) where m is mass of the body and rho is density of the liquid.

A gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. A gas column under gravity, for example, does not have uniform density (and pressure). As you might expect, its density decreases with height. The precise dependence is given by the so-called 'law of atmospheres' n_2 = n_1 exp[-(mg)(H_2-H_1)/(kT)] where n_2,n_1 refer to number density at heights h_2 and h_1 respectively.Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column: n_2= n_1 exp[(-mgN_a)/(RT) (1-(rho)/(rho))(h_2-h_1)] where rho is the density of the suspended particle, and rho' that of surrounding medium. [N_A is Avogadro's number, and R the universal gas constant].