A wire of length L and radius r is clamped at one end. On stretching the other end of the wire with a force F, the increase in its length is `l`. If another wire of same material but of length 2L and radius 2r is stretched with a force 2F, the increase in its length will be

Options

A. `(l)/(4)`
B. `(l)/(2)`
C. l
D. 2l

Explanation

To solve the problem, we will use the concept of Young's modulus, which relates stress and strain in materials.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two wires made of the same material. The first wire has a length \( L \) and radius \( r \), and when a force \( F \) is applied, it stretches by an amount \( l \). The second wire has a length \( 2L \) and radius \( 2r \), and we need to find the increase in length when a force \( 2F \) is applied.

2. **Young's Modulus Definition**:
Young's modulus \( Y \) is defined as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
where:
- Stress \( = \frac{F}{A} \) (Force per unit area)
- Strain \( = \frac{\Delta L}{L} \) (Change in length per original length)

3. **Calculate Young's Modulus for the First Wire**:
For the first wire:
- Area \( A = \pi r^2 \)
- Stress \( = \frac{F}{\pi r^2} \)
- Strain \( = \frac{l}{L} \)

Therefore, the Young's modulus \( Y \) for the first wire is:
\[ Y = \frac{F / \pi r^2}{l / L} = \frac{F L}{\pi r^2 l} \]

4. **Calculate Young's Modulus for the Second Wire**:
For the second wire:
- Length \( = 2L \)
- Radius \( = 2r \)
- Area \( A = \pi (2r)^2 = 4\pi r^2 \)
- Stress \( = \frac{2F}{4\pi r^2} = \frac{F}{2\pi r^2} \)
- Let the increase in length be \( l' \), then Strain \( = \frac{l'}{2L} \)

Therefore, the Young's modulus \( Y \) for the second wire is:
\[ Y = \frac{(F / (2\pi r^2))}{(l' / (2L))} = \frac{F L}{\pi r^2 l'} \]

5. **Equating Young's Modulus**:
Since both wires are made of the same material, their Young's moduli are equal:
\[ \frac{F L}{\pi r^2 l} = \frac{F L}{\pi r^2 l'} \]

6. **Simplifying the Equation**:
Canceling out common terms:
\[ l = l' \]

7. **Conclusion**:
The increase in length \( l' \) of the second wire is equal to the increase in length \( l \) of the first wire. Therefore, the increase in length of the second wire is:
\[ l' = l \]

### Final Answer:
The increase in length of the second wire is \( l \).

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