To solve the problem step by step, we will calculate the change in length of the wire due to both the temperature change and the weight applied to it.
### Step 1: Given Data
- Length of the wire, \( L = 3 \, \text{m} \)
- Diameter of the wire, \( d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Radius of the wire, \( r = \frac{d}{2} = \frac{1 \times 10^{-3}}{2} = 5 \times 10^{-4} \, \text{m} \)
- Initial temperature, \( T_1 = 30^\circ C \)
- Final temperature, \( T_2 = -170^\circ C \)
- Change in temperature, \( \Delta T = T_2 - T_1 = -170 - 30 = -200^\circ C \)
- Mass hung on the wire, \( m = 10 \, \text{kg} \)
- Young's modulus, \( Y = 2 \times 10^{11} \, \text{N/m}^2 \)
- Coefficient of linear expansion, \( \alpha = 1.2 \times 10^{-5} \, \text{°C}^{-1} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
### Step 2: Calculate the Force Applied
The force \( F \) applied due to the weight is given by:
\[
F = m \cdot g = 10 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 100 \, \text{N}
\]
### Step 3: Calculate the Cross-Sectional Area
The cross-sectional area \( A \) of the wire can be calculated using the formula for the area of a circle:
\[
A = \pi r^2 = \pi (5 \times 10^{-4})^2 = \pi \times 25 \times 10^{-8} \approx 7.85 \times 10^{-8} \, \text{m}^2
\]
### Step 4: Calculate the Change in Length Due to Weight
Using the formula for elongation due to applied force:
\[
\Delta L_w = \frac{F \cdot L}{A \cdot Y}
\]
Substituting the values:
\[
\Delta L_w = \frac{100 \, \text{N} \cdot 3 \, \text{m}}{7.85 \times 10^{-8} \, \text{m}^2 \cdot 2 \times 10^{11} \, \text{N/m}^2}
\]
Calculating this gives:
\[
\Delta L_w = \frac{300}{1.57 \times 10^4} \approx 1.91 \times 10^{-2} \, \text{m} = 0.0191 \, \text{m} = 19.1 \, \text{mm}
\]
### Step 5: Calculate the Change in Length Due to Temperature
Using the formula for change in length due to temperature:
\[
\Delta L_t = \alpha \cdot L \cdot \Delta T
\]
Substituting the values:
\[
\Delta L_t = 1.2 \times 10^{-5} \cdot 3 \cdot (-200)
\]
Calculating this gives:
\[
\Delta L_t = 1.2 \times 10^{-5} \cdot 3 \cdot (-200) = -7.2 \times 10^{-3} \, \text{m} = -7.2 \, \text{mm}
\]
### Step 6: Calculate the Net Change in Length
The total change in length \( \Delta L \) is the sum of the changes due to weight and temperature:
\[
\Delta L = \Delta L_w + \Delta L_t = 19.1 \, \text{mm} - 7.2 \, \text{mm} = 11.9 \, \text{mm}
\]
### Final Answer
The change in length of the wire is approximately \( 11.9 \, \text{mm} \).
