Cylindrical piece of cork of density of base area A and height h floats in a liquid. 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 is the density of cork [`P_(1)`is the density of the liquid.] (Ignore damping due to viscosity of the liquid).

Options

A. `T=2pisqrt((1)/(g))`
B. `T=2pisqrt((sigma1)/(pg))`
C. `T=2pisqrt((1)/(2g))`
D. `T=2pisqrt((1)/(sigmag))`

Answer

B

Explanation

Let `d_(0)` be the height of cylinder below water surface at equilibrium.
Then , if A is the area of cross section of the cylinder `(Ad_(0)pg)Rightarrow d_(0)=(m)/(p_(A))`
Let x be the small displacement from this equilibrium depth.
Then the unbalance force =`-p^(Agx)`, being downward when x (upward) is positive and upward when x (downward)is negative.
Hence the equation of motion is
`m(d^(2)x)/(dt^(2))=-p^(Agx)`
Now, m= mass of wood =`AIsigma`
Hence , `AIsigma(d^(2)x)/(dt^(2))=p^(Agx)` or `(d^(2)x)/(dt^(2))=-((pg)/(sigmaI))X`
The time period `T=2pisqrt((sigmaI)/(pg))`