Question

Cylindrical piece of cork of density of base area A and height h floats in a liquid of density. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period


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

Answer 1

Mass of cylindrical piece of cork (m) = volume × density
= area × height × density
= Ah.d

In floating condition the weight of the cylinder balanced by upthrust of the liquid .
see the figure .
Let cylinderical cork is displaced down by a depth x further then,
Restoring force acting on it = weigh of extra liquid displaced
F = - ( volume × density )× g
= - ( Area × depth × density) × g
= - ( AxDl)g

A/c to Newton's 2nd law ,
F = ma
So , ma = - AxDl× g
a = - (A.Dl.g/m)x
Compare this equation to a= -w²x
w² = (A.Dl.g/m)
w = √(ADl.g/m)
2π/T = √(ADl.g/m)
T = 2π√(m/A.Dl.g)
now, m = AhD [ mass = volume × density]
T = 2π√(Dh/Dl.g)
Hence, proved //