Question

A cylindrical piece of cork of density of base area A and height h floats in a liquid of density p₁. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period T= 2 π √( hp/p₁/g) where r is the density of cork. (Ignore damping due to viscosity of the liquid).

Answer 1

It is given that the base area of cork of a cylindrical piece is A, the height is h, the density of liquid is ρ l and the density of cork is ρ.

Initially in equilibrium, the height of cylinder inside the liquid is x.

Weight of the cylinder is equal to the up thrust due to liquid displaced.

Ahρg=Ax ρ 1 g.

When the cork cylinder is depressed slightly by Δx and then released, a restoring

Force F will act on it and it will be equal to the additional up thrust.

Then, the restoring force can be written as,

F=A( x+Δx ) ρ 1 g−Ax ρ 1 g =AΔx ρ 1 g

The formula of acceleration is,

a= F m a= AΔx ρ 1 g Ahρ a= ρ 1 g hρ ×Δx

The direction of acceleration is opposite to Δx, so acceleration will be proportional to ( −Δx ).

The time period of cork cylinder can be written as,

T=2π Δy a

Substitute the value of a in the above equation.

T=2π Δx ρ 1 g hρ ×Δx =2π hρ ρ 1 g

Hence, the time period of cork is 2π hρ ρ 1 g .