A wooden cylinder of length L is partly submerged in a liquid of specific gravity `rho_(1)` with `n^(th)` `(nlt1)` part of it inside the liquid. Another immiscible liquid of density `rho_(2)` is poured to completely submerge the cylinder. Density of cylinder `rho` is the geometric mean of the densities of the two liquid.
When the cylinder is slightly depressed and released, it oscillates. let there be a mean position. find the time period of small oscillations below thee mean position

Options

A. `pi[((n+a)L)/(g(n-1))]^((1)/(2))`
B. `pi[(n^(2)L)/(g(1-n)^(2))]^((1)/(2))`
C. `pi[(nL)/(g(n^(2)-1))]^(1//2)`
D. `pi[(nL)/(g(1-n^(2)))]^((1)/(2))`

Answer

D

Explanation

Let cylinder bedepressed by `x`
Restoring force `F=-Ax[(rho)/(n)-nrho]g`
`(AL)rhoa=-Axrhog.((1-n^(2)))/(n)`
`a=-(xg(1-n^(2)))/(nL),(x)/(a)=-(nL)/(g(1-n^(2)))`
`(T_(1))/(2)=pisqrt(|(x)/(a)|)=pi((nL)/(g(1-n^(2))))^((1)/(2))`