A brass rod and a lead rod each 80 cm long at `0^@C` are clamped together at one end with their free ends coinciding. The separatioin of free ends of the rods if the system is placed in a steam bath is (`alpha_("brass")=18xx10^(-6)//^(@)C` and `alpha_("lead")=28xx10^(-6)//^(@)C`)

Options

A. 0.2 mm
B. 0.8 mm
C. 1.4 mm
D. 1.6 mm

Explanation

To solve the problem of finding the separation of the free ends of a brass rod and a lead rod when placed in a steam bath, we will follow these steps:

### Step 1: Understand the Problem
We have two rods, one made of brass and the other made of lead, both initially 80 cm long at 0°C. They are clamped at one end, and we need to find the separation of their free ends when the system is heated to 100°C (the temperature of the steam bath).

### Step 2: Identify the Coefficients of Linear Expansion
The coefficients of linear expansion for the materials are given as:
- For brass, \( \alpha_{brass} = 18 \times 10^{-6} \, /^{\circ}C \)
- For lead, \( \alpha_{lead} = 28 \times 10^{-6} \, /^{\circ}C \)

### Step 3: Calculate the Change in Length for Each Rod
We will use the formula for linear expansion:
\[ L' = L + \alpha \cdot L \cdot \Delta T \]
where:
- \( L' \) is the final length,
- \( L \) is the initial length,
- \( \alpha \) is the coefficient of linear expansion,
- \( \Delta T \) is the change in temperature.

The initial length \( L \) for both rods is 80 cm, and the change in temperature \( \Delta T \) is \( 100°C - 0°C = 100°C \).

#### For the Brass Rod:
\[ L'_{brass} = 80 \, \text{cm} + (18 \times 10^{-6}) \cdot 80 \cdot 100 \]
Calculating:
\[ L'_{brass} = 80 + 0.144 \, \text{cm} = 80.144 \, \text{cm} \]

#### For the Lead Rod:
\[ L'_{lead} = 80 \, \text{cm} + (28 \times 10^{-6}) \cdot 80 \cdot 100 \]
Calculating:
\[ L'_{lead} = 80 + 0.224 \, \text{cm} = 80.224 \, \text{cm} \]

### Step 4: Calculate the Separation of the Free Ends
The separation of the free ends of the rods is given by the difference in their final lengths:
\[ \text{Separation} = L'_{lead} - L'_{brass} \]
Substituting the values we calculated:
\[ \text{Separation} = 80.224 \, \text{cm} - 80.144 \, \text{cm} = 0.08 \, \text{cm} \]

### Step 5: Convert to Meters (if needed)
To express the separation in meters:
\[ 0.08 \, \text{cm} = 0.0008 \, \text{m} = 8 \times 10^{-4} \, \text{m} \]

### Final Answer
The separation of the free ends of the rods when placed in a steam bath is \( 0.08 \, \text{cm} \) or \( 8 \times 10^{-4} \, \text{m} \).

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