Mercury of density `rho_(m)` is poured into cylinderical communicating vessels of cross-sectional area 2A and A respectively. A solid iron cube of volume `V_(0)` and relative density 2 is dropped into the broad vessel, and as a result the level of the mercury in it rises. Then, water is poured into the broader vessel until the mercury reaches the previous level in it. Find the height of water column if it does not submerge the block.

Explanation

To solve the problem step by step, we need to analyze the situation involving the communicating vessels, the iron cube, and the mercury and water levels.

### Step 1: Understand the System
We have two communicating vessels with cross-sectional areas \(2A\) and \(A\). Mercury of density \(\rho_m\) is filled in both vessels. An iron cube with volume \(V_0\) and relative density \(2\) is dropped into the broader vessel (with area \(2A\)).

### Step 2: Calculate the Density of the Iron Cube
The density of the iron cube can be calculated using its relative density:
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