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 S.H.M. comparing it with
a = -ω2y
we get ω2 = {ρLg}/{hρ}
⇒ ω = √{ρLg/hρ}
Time period of oscillation is