A cylindrical piece of cork 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=2pisqrt((hrho)/(rho_(1)g))`

Explanation

Base area of the cork=A
Height of the cork =h
Density of the liuid = `p_(1)`
Density of the cork =p
In equilirium:
Weight of the cork = Weight of the liquid displaced by the floating corkLet 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 `xx` Density xx g)
Volume = Area `xx `Distance` xx` g) through which the cork is depressed
Volume=Ax
`therefore F=-Axp_(1)g` ……(i) According to the force law
F=kx
`k=(F)/(x) `
Where,k is a constant
`k=(F)/(x)=-Ap_(1)g`.....(ii)
m=Mass of the cork =Volume of the cork `xx `Density =Ahp
Hence , the expression for the time period becomes:
`T=2pisqrt((Ahp)/(Ap_(1)g))=2pisqrt((hp)/(p_(1)g))`