A circular hole in an aluminium plat e is `2.54xx10^(-2)`m in diamter at `0^(@)C`, its diameter when the temperature of the plate is raised to `100^(@)C` will be? `(alpha=2.55xx10^(-5)//.^(@)C)`

Options

A. `2.50xx10^(-2)m`
B. `2.546xx10^(-2)m`
C. `4.5xx10^(-2)m`
D. `2.45xx10^(-2)m`

Answer

To solve the problem of finding the diameter of a circular hole in an aluminum plate when the temperature is raised from \(0^\circ C\) to \(100^\circ C\), we can use the formula for linear expansion:

\[ L = L_0 (1 + \alpha \Delta T) \]

Where:
- \(L\) is the final length (or diameter in this case),
- \(L_0\) is the initial length (or diameter),
- \(\alpha\) is the coefficient of linear expansion,
- \(\Delta T\) is the change in temperature.

### Step-by-Step Solution:

1. **Identify the given values:**
- Initial diameter (\(L_0\)) = \(2.54 \times 10^{-2}\) m
- Coefficient of linear expansion (\(\alpha\)) = \(2.55 \times 10^{-5} \, ^\circ C^{-1}\)
- Initial temperature = \(0^\circ C\)
- Final temperature = \(100^\circ C\)

2. **Calculate the change in temperature (\(\Delta T\)):**
\[ \Delta T = 100^\circ C - 0^\circ C = 100^\circ C \]

3. **Substitute the values into the linear expansion formula:**
\[ L = L_0 (1 + \alpha \Delta T) \]
\[ L = 2.54 \times 10^{-2} \, \text{m} \left(1 + (2.55 \times 10^{-5}) \times 100\right) \]

4. **Calculate \(\alpha \Delta T\):**
\[ \alpha \Delta T = 2.55 \times 10^{-5} \times 100 = 2.55 \times 10^{-3} \]

5. **Now substitute this back into the equation:**
\[ L = 2.54 \times 10^{-2} \, \text{m} \left(1 + 2.55 \times 10^{-3}\right) \]

6. **Calculate \(1 + 2.55 \times 10^{-3}\):**
\[ 1 + 2.55 \times 10^{-3} = 1.00255 \]

7. **Now multiply to find \(L\):**
\[ L = 2.54 \times 10^{-2} \times 1.00255 \]
\[ L \approx 2.5465 \times 10^{-2} \, \text{m} \]

8. **Round the result appropriately:**
\[ L \approx 2.546 \times 10^{-2} \, \text{m} \]

### Final Answer:
The diameter of the hole when the temperature of the plate is raised to \(100^\circ C\) will be approximately \(2.546 \times 10^{-2} \, \text{m}\).

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