A wooden cylinder of length l floats vertically in a liquid of specific gravity `rho` with `((1)/(3))^(rd)` part of its length submerged. Another liquid that is immiscible with the previous liquid is poured into it, to just completely submerge the cylinder. If the density of the liquid at the bottom, density of cylinder and density of upper liquid are in G.P., find the fraction of the cylinder submerged in lower liquid.

Explanation

To solve the problem step by step, we will analyze the forces acting on the wooden cylinder and use the conditions given in the problem.

### Step 1: Understand the initial condition
A wooden cylinder of length \( L \) floats in a liquid of specific gravity \( \rho \) with \( \frac{1}{3} \) of its length submerged.

### Step 2: Calculate the buoyant force
The buoyant force \( F_b \) acting on the cylinder when it is floating is equal to the weight of the liquid displaced by the submerged part of the cylinder.

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