The following four wires are made of the same material which of these will have the largest extension when the same tension is applied

Options

A. L=100 cm, r= 0.2 mm
B. L=200 cm, r =0.4 mm
C. L 300 cm, r =0.6 mm
D. L 400 cm, r= 0.8 mm

Explanation

To determine which of the four wires will have the largest extension when the same tension is applied, we can follow these steps:

### Step 1: Understand the Relationship between Stress, Strain, and Extension
According to Hooke's Law, stress is directly proportional to strain. Mathematically, this can be expressed as:
\[ \text{Stress} = \frac{F}{A} \]
where \( F \) is the force applied (tension) and \( A \) is the cross-sectional area of the wire.

### Step 2: Define the Area of the Wire
The area \( A \) of a wire with radius \( r \) is given by:
\[ A = \pi r^2 \]

### Step 3: Relate Extension to Length and Radius
The extension \( \Delta L \) of the wire can be related to the original length \( L \) and radius \( r \) as follows:
\[ \Delta L \propto \frac{L}{A} = \frac{L}{\pi r^2} \]
Thus, we can express the extension as:
\[ \Delta L \propto \frac{L}{r^2} \]

### Step 4: Calculate the Ratio \( \frac{L}{r^2} \) for Each Wire
Now, we will calculate \( \frac{L}{r^2} \) for each wire based on the given lengths and radii.

1. **Wire 1**:
- Length \( L_1 = 100 \) cm = 1000 mm
- Radius \( r_1 = 0.2 \) mm
- \( \frac{L_1}{r_1^2} = \frac{1000}{(0.2)^2} = \frac{1000}{0.04} = 25000 \)

2. **Wire 2**:
- Length \( L_2 = 200 \) cm = 2000 mm
- Radius \( r_2 = 0.4 \) mm
- \( \frac{L_2}{r_2^2} = \frac{2000}{(0.4)^2} = \frac{2000}{0.16} = 12500 \)

3. **Wire 3**:
- Length \( L_3 = 300 \) cm = 3000 mm
- Radius \( r_3 = 0.6 \) mm
- \( \frac{L_3}{r_3^2} = \frac{3000}{(0.6)^2} = \frac{3000}{0.36} \approx 8333.33 \)

4. **Wire 4**:
- Length \( L_4 = 400 \) cm = 4000 mm
- Radius \( r_4 = 0.8 \) mm
- \( \frac{L_4}{r_4^2} = \frac{4000}{(0.8)^2} = \frac{4000}{0.64} = 6250 \)

### Step 5: Compare the Values
Now we compare the calculated values of \( \frac{L}{r^2} \):
- Wire 1: 25000
- Wire 2: 12500
- Wire 3: 8333.33
- Wire 4: 6250

### Conclusion
The wire with the largest value of \( \frac{L}{r^2} \) is Wire 1, which means it will have the largest extension when the same tension is applied.

**Final Answer**: Wire 1 will have the largest extension.

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