Question

The cylindrical piece of the cork of density of base area A and height h floats in a liquid of density `rho_1`. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period

`T = 2pi sqrt((hrho)/(rho_1g)` 

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

Answer 1

Base area of the cork = A

Height of the cork = h

Density of the liquid = `rho_1`

Density of the cork = ρ

In 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

∴ F = – A x `rho_1` g … (i)

According to the force law:

F = kx

`k = F/x`

Where, k is a constant

`k = F/x = -Arho_1g`  ...(ii)

The time period of the oscillations of the cork:

`T = 2pi sqrt(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 becomes:

`T = 2pi sqrt((Ahrho)/(Arho_1g)) = 2 pi sqrt((hrho)/(rho_1g)`

Answer 2

Say, initially in equilibrium, y  height of cylinder is inside the liquid. Then,

Weight of the cylinder = upthrust due to liquid displaced

`:. Ahrhog = Ayrho_1g`

When the cork cylinder is depressed slightly by `triangle y` and released, a restoring force, equal to additional upthrust, act on it. The restoring force is

`F =  A(y + triangle y) rho_1 g - Ayrho_1g = Arho_1g triangle y`

:. Acceleration, `a = F/m = (Arho_1g triangle y)/(Ahrho) = (rho_1g)/(hrho). triangle y `  and the acceleration is directed in a  direction opposite to `triangle y`. Obviously, as `a prop - triangle y`, the motion of cork cylinder is SHM, whose time period is given by

`T = 2pi sqrt("displacement"/"accelertion")`

`=2pi sqrt((triangle y)/a)`

`= 2pi sqrt((hrho)/(rho_1g))`