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A cylindrical piece of cork of base area...

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))`

A

`T=2pisqrt((1)/(g))`

B

`T=2pisqrt((sigma1)/(pg))`

C

`T=2pisqrt((1)/(2g))`

D

`T=2pisqrt((1)/(sigmag))`

Text Solution

Verified by Experts

The correct Answer is:
B
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))`
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