At `50^(@)C` , a brass rod has a length 50 cm and a diameter 2 mm . It is joined to a steel rod of the same length and diameter at the same temperature . The change in the length of the composite rod when it is heated to `250^(@)C` is (Coefficient of linear expansion of brass = `2.0 xx 10^(-5)"^(@) C^(-1)` , coefficient of linear expansion of steel = `1.2 xx 10^(-5) "^(@) C^(-1)`)

Options

A. `0.28` cm
B. `0.30` cm
C. `0.32` cm
D. `0.34` cm

Explanation

To find the change in length of the composite rod made of brass and steel when heated from \(50^\circ C\) to \(250^\circ C\), we will use the formula for linear expansion:

\[ \Delta L = \alpha \cdot L_0 \cdot \Delta T \]

where:
- \(\Delta L\) is the change in length,
- \(\alpha\) is the coefficient of linear expansion,
- \(L_0\) is the original length,
- \(\Delta T\) is the change in temperature.

### Step 1: Calculate the change in temperature
The change in temperature (\(\Delta T\)) is calculated as follows:

\[ \Delta T = 250^\circ C - 50^\circ C = 200^\circ C \]

### Step 2: Calculate the change in length for the brass rod
Using the formula for linear expansion, we can calculate the change in length of the brass rod:

Given:
- Coefficient of linear expansion of brass (\(\alpha_b\)) = \(2.0 \times 10^{-5} \, ^\circ C^{-1}\)
- Original length of brass rod (\(L_{0b}\)) = 50 cm
- Change in temperature (\(\Delta T\)) = 200°C

Now, substituting the values:

\[ \Delta L_b = \alpha_b \cdot L_{0b} \cdot \Delta T \]

\[ \Delta L_b = (2.0 \times 10^{-5}) \cdot (50) \cdot (200) \]

Calculating this gives:

\[ \Delta L_b = 2.0 \times 10^{-5} \cdot 50 \cdot 200 = 0.2 \, \text{cm} \]

### Step 3: Calculate the change in length for the steel rod
Now, we calculate the change in length of the steel rod:

Given:
- Coefficient of linear expansion of steel (\(\alpha_s\)) = \(1.2 \times 10^{-5} \, ^\circ C^{-1}\)
- Original length of steel rod (\(L_{0s}\)) = 50 cm

Using the same formula:

\[ \Delta L_s = \alpha_s \cdot L_{0s} \cdot \Delta T \]

Substituting the values:

\[ \Delta L_s = (1.2 \times 10^{-5}) \cdot (50) \cdot (200) \]

Calculating this gives:

\[ \Delta L_s = 1.2 \times 10^{-5} \cdot 50 \cdot 200 = 0.12 \, \text{cm} \]

### Step 4: Calculate the total change in length of the composite rod
Now, we can find the total change in length of the composite rod by adding the changes in length of both rods:

\[ \Delta L_{total} = \Delta L_b + \Delta L_s \]

Substituting the values:

\[ \Delta L_{total} = 0.2 \, \text{cm} + 0.12 \, \text{cm} = 0.32 \, \text{cm} \]

### Final Answer
The change in length of the composite rod when heated from \(50^\circ C\) to \(250^\circ C\) is:

\[ \Delta L_{total} = 0.32 \, \text{cm} \]