To determine the mass \( m \) that can be hung from the steel wire while allowing it to regain its original length after being cooled from 40°C to 30°C, we need to consider the concepts of thermal expansion and the mechanical properties of the wire. Let's break this down step by step.
Understanding Thermal Expansion
When materials are heated, they expand, and when they are cooled, they contract. The change in length \( \Delta L \) of a material due to temperature change can be calculated using the formula:
\(\Delta L = L_0 \cdot \alpha \cdot \Delta T\)
Where:
- \(L_0\) is the original length of the wire.
- \(\alpha\) is the coefficient of linear thermal expansion (for steel, \( \alpha = 10^{-5} \, \text{°C}^{-1} \)).
- \(\Delta T\) is the change in temperature (in this case, \( \Delta T = 30°C - 40°C = -10°C\)).
Calculating the Change in Length
Substituting the values into the formula, we get:
\(\Delta L = L \cdot 10^{-5} \cdot (-10)\)
This simplifies to:
\(\Delta L = -L \cdot 10^{-4}\)
This negative sign indicates that the wire contracts by \( L \cdot 10^{-4} \) when cooled from 40°C to 30°C.
Understanding Stress and Strain
When a mass \( m \) is hung from the wire, it creates a tension \( T \) in the wire, which can be expressed as:
\(T = m \cdot g\)
Where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)). The stress \( \sigma \) in the wire is given by:
\(\sigma = \frac{T}{A}\)
Where \( A \) is the cross-sectional area of the wire. The area \( A \) can be calculated using the radius \( r \) of the wire:
\(A = \pi r^2\)
Given that the radius \( r = 1 \, \text{mm} = 0.001 \, \text{m}\), we find:
\(A = \pi (0.001)^2 \approx 3.14 \times 10^{-6} \, \text{m}^2\)
Relating Stress to Strain
The relationship between stress and strain in a material is given by Young's modulus \( E \):
\(E = \frac{\sigma}{\epsilon}\)
Where \( \epsilon \) is the strain, defined as:
\(\epsilon = \frac{\Delta L}{L}\)
From the earlier calculation, we know that:
\(\Delta L = -L \cdot 10^{-4}\)
Thus, the strain can be expressed as:
\(\epsilon = \frac{-L \cdot 10^{-4}}{L} = -10^{-4}\)
Putting It All Together
Now we can relate everything back to find the mass \( m \). First, we calculate the stress:
\(\sigma = \frac{T}{A} = \frac{m \cdot g}{A}\)
Substituting this into the Young's modulus equation gives:
\(E = \frac{m \cdot g}{A \cdot (-10^{-4})}\)
Rearranging for \( m \), we have:
\(m = \frac{E \cdot A \cdot (-10^{-4})}{g}\)
Now substituting the known values:
- E = 10^{11} \, \text{N/m}^2
- A \approx 3.14 \times 10^{-6} \, \text{m}^2
- g \approx 9.81 \, \text{m/s}^2
Plugging in these values:
m = \frac{10^{11} \cdot 3.14 \times 10^{-6} \cdot (-10^{-4})}{9.81}
Calculating this gives:
m \approx \frac{-3.14 \times 10^{1}}{9.81} \approx -3.20 \, \text{kg}
Since mass cannot be negative, we take the absolute value, leading us to conclude that the mass \( m \) that can be hung from the wire is approximately 3.20 kg.
Final Thoughts
This example illustrates the interplay between thermal expansion and mechanical properties of materials. Understanding these principles is crucial in fields such as engineering and materials science, where temperature changes can significantly affect structural integrity.
