Two vessels connected at the bottom by a thin pipe with a sliding plug contain liquid at `20^@C` and `80^@C` respectively. The coefficient of cubic expansion of liquid is `10^-3K^-1`. The ratio of height of liquid columns in the vessel `(H_(20)//H_(80))` is nearest which integer ?

Explanation

To solve the problem, we need to find the ratio of the heights of the liquid columns in two vessels containing liquid at different temperatures. Let's break down the steps:

### Step 1: Understand the problem
We have two vessels connected at the bottom by a thin pipe. One vessel contains liquid at 20°C and the other at 80°C. The coefficient of cubic expansion of the liquid is given as \( \gamma = 10^{-3} \, \text{K}^{-1} \).

### Step 2: Set up the pressure balance
Since the vessels are connected and the sliding plug is not moving, the pressures at the bottom of both vessels must be equal. The pressure due to a liquid column is given by:
\[ P = \rho g h \]
where \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid column.

### Step 3: Write the pressure equations
For the two vessels, we can write:
\[ \rho_{20} g H_{20} = \rho_{80} g H_{80} \]
We can cancel \( g \) from both sides:
\[ \rho_{20} H_{20} = \rho_{80} H_{80} \]

### Step 4: Express the ratio of heights
Rearranging gives us:
\[ \frac{H_{20}}{H_{80}} = \frac{\rho_{80}}{\rho_{20}} \]

### Step 5: Calculate the densities
The density of a liquid at a temperature \( T \) can be expressed using the coefficient of cubic expansion:
\[ \rho_T = \rho_0 (1 + \gamma \Delta T) \]
where \( \Delta T = T - 0 \).

For \( T = 20^\circ C \):
\[ \rho_{20} = \rho_0 (1 + \gamma \cdot 20) = \rho_0 (1 + 10^{-3} \cdot 20) = \rho_0 (1 + 0.02) = \rho_0 \cdot 1.02 \]

For \( T = 80^\circ C \):
\[ \rho_{80} = \rho_0 (1 + \gamma \cdot 80) = \rho_0 (1 + 10^{-3} \cdot 80) = \rho_0 (1 + 0.08) = \rho_0 \cdot 1.08 \]

### Step 6: Substitute the densities into the height ratio
Now substituting \( \rho_{20} \) and \( \rho_{80} \) into the height ratio:
\[ \frac{H_{20}}{H_{80}} = \frac{\rho_{80}}{\rho_{20}} = \frac{\rho_0 \cdot 1.08}{\rho_0 \cdot 1.02} = \frac{1.08}{1.02} \]

### Step 7: Calculate the ratio
Calculating the ratio:
\[ \frac{H_{20}}{H_{80}} = \frac{1.08}{1.02} \approx 1.0588 \]

### Step 8: Round to the nearest integer
The nearest integer to \( 1.0588 \) is \( 1 \).

### Final Answer
The ratio of the height of liquid columns in the vessels \( \frac{H_{20}}{H_{80}} \) is nearest to **1**.

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