Question

A cylindrical piece of cork 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 **

T = 2π√{hρ/ρ1g}

where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).

Answer 1

Let l be the height of the cork which is inside the liquid. Then, at equilibrium, weight of the cork = upthrusts of the potion of the cork inside the liquid.

⇒ Ahρg = AlρLg

⇒ l = hρ/ρL

If the cork is slightly depressed below through a length of y, the net force acting in the upward direction is

F = -[A(l + y)ρLg - AlρLg] = -AρLgy

-ve sign shows that the force acts opposite to displacement.

⇒ ma = AρLgy

⇒ a = {-AρLgy}/{m} = {-AρLgy}/{Aρh}

= {ρLg}/{ρA} x y ....(ii)

Equation (ii) is the equation of *.H.M. comparing it with

a = -ω2y

we get ω2 = {ρLg}/{hρ}

⇒ ω = √{ρLg/hρ}

Time ** of oscillation is