A copper and a steel wire of the same diameter are connected end to end. A deforming force F is applied to this composite wire, which causes a total elongation of 1 cm. The two wires will have

Options

A. the same stress
B. different stress
C. same strain
D. different strain

Answer

To solve the problem, we need to analyze the behavior of the copper and steel wires when a force is applied to them. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
We have two wires: one made of copper and the other made of steel, both having the same diameter. They are connected end to end, and when a force \( F \) is applied, the total elongation of the composite wire is 1 cm.

### Step 2: Identify the Key Concepts
- **Stress** is defined as the force applied per unit area. For both wires, since they have the same diameter, the area \( A \) is the same.
- **Strain** is defined as the ratio of elongation to the original length of the wire.
- **Young's Modulus (Y)** relates stress and strain for a material:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
or
\[ \text{Strain} = \frac{\text{Stress}}{Y} \]

### Step 3: Analyze the Stress in Each Wire
Since both wires are subjected to the same force \( F \) and have the same cross-sectional area \( A \):
- The stress in both wires is given by:
\[ \text{Stress in copper (S1)} = \frac{F}{A} \]
\[ \text{Stress in steel (S2)} = \frac{F}{A} \]
Thus, we conclude that:
\[ S1 = S2 \]

### Step 4: Analyze the Strain in Each Wire
Using the relationship between stress, strain, and Young's modulus:
- For copper:
\[ \text{Strain in copper} = \frac{S1}{Y_{copper}} = \frac{F/A}{Y_{copper}} \]
- For steel:
\[ \text{Strain in steel} = \frac{S2}{Y_{steel}} = \frac{F/A}{Y_{steel}} \]

Since the Young's moduli for copper and steel are different (\( Y_{copper} \neq Y_{steel} \)), the strains will also be different:
\[ \text{Strain in copper} \neq \text{Strain in steel} \]

### Step 5: Conclusion
- The stresses in both wires are the same.
- The strains in both wires are different due to the different Young's moduli of copper and steel.

### Final Answer
The two wires will have the same stress but different strains.

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