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A vertical pole of length $l$ , density $ρ$ , area of cross section A ,floats in two immiscible liquids of densities $ρ_{1} and ρ_{2}$ . In equilibrium position the bottom end is at the interface of the liquids. When the cylinder is displaced vertically, the time period of oscillation is...............

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$π \sqrt{\frac{ρ l}{(ρ_{1} + ρ_{2}) g}}$

$π \sqrt{\frac{ρ l}{g}} (\frac{1}{\sqrt{ρ_{1}}} + \frac{1}{\sqrt{ρ_{2}}})$

$π \sqrt{\frac{(ρ_{1} + ρ_{2}) l}{ρ g}}$

$π \sqrt{\frac{l}{ρ g}} (\sqrt{ρ_{1}} + \sqrt{ρ_{2}})$

answer is B.

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Detailed Solution

Here interface acts as mean position of SHM:

$∴$ time period, $T = \frac{T_{1}}{2} + \frac{T_{2}}{2}$

$T_{1} = 2 π \sqrt{\frac{ρ l}{ρ_{2} g}} : T_{2} = 2 π \sqrt{\frac{ρ l}{ρ_{1} g}}$

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