Question
Cylindrical piece of cork of density 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 period,
where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).
Solution
Given,
Base area of the cork = A
Base area of the cork = A
Height of the cork = h
Density of the liquid = ρ1
Density of the cork = ρIn equilibrium:
Weight of the cork = Weight of the liquid displaced by the floating cork
Let the cork be depressed slightly by x.
As a result, some extra water of a certain volume is displaced.
Hence, an extra up-thrust acts upward and provides the restoring force to the cork.
Up-thrust = Restoring force, F = Weight of the extra water displaced
F = –(Volume × Density × g)
Volume = Area × Distance through which the cork is depressed
Volume = Ax
Therefore,
F = – A x ρ1 g ...(i)
According to the force law:
F = kx
k = F/x
where, k is constant
k = F/x = -Aρ1 g ...(ii)
The time period of the oscillations of the cork:
T = 2π √m/k ...(iii)
where,
m = Mass of the cork
= Volume of the cork × Density
= Base area of the cork × Height of the cork × Density of the cork
= Ahρ
Hence, the expression for the time period is given by,
Up-thrust = Restoring force, F = Weight of the extra water displaced
F = –(Volume × Density × g)
Volume = Area × Distance through which the cork is depressed
Volume = Ax
Therefore,
F = – A x ρ1 g ...(i)
According to the force law:
F = kx
k = F/x
where, k is constant
k = F/x = -Aρ1 g ...(ii)
The time period of the oscillations of the cork:
T = 2π √m/k ...(iii)
where,
m = Mass of the cork
= Volume of the cork × Density
= Base area of the cork × Height of the cork × Density of the cork
= Ahρ
Hence, the expression for the time period is given by,
