Find the pressure requried to prevent the expansion of a copper block if it is heated from `40^(@)C` to `50^(@)C`. Coefficient of linear expansion of copper is `15xx 10^(-6)//^(@)C` and bulk modulus `= 12 xx 10^(10) N //m^(2)`

Explanation

To find the pressure required to prevent the expansion of a copper block when it is heated from 40°C to 50°C, we will use the relationship between pressure, bulk modulus, and the coefficient of linear expansion.

### Step-by-step Solution:

1. **Identify the Given Values:**
- Coefficient of linear expansion of copper, \( \alpha = 15 \times 10^{-6} \, \text{°C}^{-1} \)
- Initial temperature, \( T_1 = 40 \, \text{°C} \)
- Final temperature, \( T_2 = 50 \, \text{°C} \)
- Bulk modulus, \( K = 12 \times 10^{10} \, \text{N/m}^2 \)

2. **Calculate the Change in Temperature:**
\[ \Delta T = T_2 - T_1 = 50 \, \text{°C} - 40 \, \text{°C} = 10 \, \text{°C} \]

3. **Use the Formula Relating Pressure, Bulk Modulus, and Coefficient of Linear Expansion:**
The formula to find the pressure required to prevent expansion is given by:
\[ K = \frac{P}{3 \alpha \Delta T} \]
Rearranging this formula to solve for pressure \( P \):
\[ P = K \cdot 3 \alpha \Delta T \]

4. **Substitute the Values into the Formula:**
\[ P = 12 \times 10^{10} \, \text{N/m}^2 \cdot 3 \cdot (15 \times 10^{-6} \, \text{°C}^{-1}) \cdot (10 \, \text{°C}) \]

5. **Calculate the Pressure:**
- First, calculate \( 3 \cdot 15 \times 10^{-6} \cdot 10 \):
\[ 3 \cdot 15 \cdot 10^{-6} \cdot 10 = 4.5 \times 10^{-5} \]
- Now substitute this back into the pressure equation:
\[ P = 12 \times 10^{10} \cdot 4.5 \times 10^{-5} \]
- Calculate:
\[ P = 54 \times 10^{5} \, \text{N/m}^2 = 5.4 \times 10^{7} \, \text{N/m}^2 \]

6. **Final Result:**
The pressure required to prevent the expansion of the copper block is:
\[ P = 5.4 \times 10^{7} \, \text{N/m}^2 \]