A cylindrical piece of cork of density of base area A and height h floats in a liquid of density `p_(l)`. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
`T=2pisqrt((hp)/(p_(1)g))`
where p is the density of cork. (Ignore damping due to viscosity of the liquid).

Explanation

Total mass of cork m = volume `xx` density
= Area `xx` height `xx` density
`therefore m = Ahp " ""……"(1)`
When the cork is depressed in liquid, the mass of displacement liquid,
m. = cross sectional area of cork `xx` length of depressed part of cork `xx` density of liquid `xx` g
`therefore m. = Alp_(l)`
From the floating law of body.
Weight of cork in air = weight of cork in liquid
`mg - m.g, Ahpg = Alp_(l)g`
`therefore h= (lp_(l))/(p)" ""........"(2)`
Suppose the cork is depressed by `PQ = y`, then the restoring force acting on it,
`F= -[" mass of cylindrical height of liquid "] xx g xx y`
`= -(Ayp_(l) g)y = -ky`
Where force constant `k= Ap_(l) g" ""......."(3)`
The time period of oscillation of cork, `T= 2pi sqrt((m)/(k))`
`= 2pi sqrt((Ahp)/(Ap_(l)g))" "[therefore " From equation (1) and (2) "]`
`therefore T= 2pi sqrt((hp)/(p_(l)g))` is proved
This formula is the time period of simple pendulum because,
`T= 2pi sqrt((p_(l)l)/(p)xx (p)/(p_(l)g)" "` (from equation (2))
`therfore T= 2pi sqrt((l)/(g))`.