### Q.18:- Cylindrical piece of cork of density of base area *A *and height *h *floats in a liquid of density *ρ*_{1}. 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:-***A*

*h*

*ρ*

_{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 **ρ*_{1 }*g *…..**(i)**

Accroding to the force law:

*F* = *kx *

*k* = *F*/*x*

where, *k* is constant

*k* = *F*/*x = -A**ρ*_{1 }*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

= *Ah**ρ*

Hence, the expression for the time period becomes: