Two wires are made of the same material and have the same volume. However wire 1 has cross-sectional area A and wire 2 has cross-sectional area 3A. If the length of wire 1 increases by `Deltax` on applying force F, how much force is needed to stretch wire 2 by the same amount?

Options

A. 4F
B. 6F
C. 9F
D. F

Answer

To solve the problem, we need to analyze the relationship between stress, strain, and Young's modulus for the two wires.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two wires made of the same material and having the same volume.
- Wire 1 has a cross-sectional area \( A \) and wire 2 has a cross-sectional area \( 3A \).
- Wire 1 stretches by \( \Delta x \) when a force \( F \) is applied. We need to find the force \( F' \) required to stretch wire 2 by the same amount \( \Delta x \).

2. **Volume of the Wires**:
- Since both wires have the same volume, we can express the volume \( V \) in terms of their lengths and cross-sectional areas:
\[ V_1 = A \cdot L_1 \quad \text{and} \quad V_2 = 3A \cdot L_2 \]
- Setting the volumes equal gives:
\[ A \cdot L_1 = 3A \cdot L_2 \implies L_1 = 3L_2 \]

3. **Using Young's Modulus**:
- Young's modulus \( Y \) relates stress and strain:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta x/L} \]
- Rearranging gives:
\[ \Delta x = \frac{F \cdot L}{Y \cdot A} \]

4. **Applying to Wire 1**:
- For wire 1, we have:
\[ \Delta x = \frac{F \cdot L_1}{Y \cdot A} \]

5. **Applying to Wire 2**:
- For wire 2, we want to stretch it by the same amount \( \Delta x \):
\[ \Delta x = \frac{F' \cdot L_2}{Y \cdot (3A)} \]

6. **Setting the Two Equations Equal**:
- Since both wires stretch by the same amount:
\[ \frac{F \cdot L_1}{Y \cdot A} = \frac{F' \cdot L_2}{Y \cdot (3A)} \]
- Canceling \( Y \) and \( A \) from both sides gives:
\[ F \cdot L_1 = \frac{F'}{3} \cdot L_2 \]

7. **Substituting \( L_1 \) in Terms of \( L_2 \)**:
- From step 2, we know \( L_1 = 3L_2 \):
\[ F \cdot (3L_2) = \frac{F'}{3} \cdot L_2 \]
- Dividing both sides by \( L_2 \) (assuming \( L_2 \neq 0 \)):
\[ 3F = \frac{F'}{3} \]

8. **Solving for \( F' \)**:
- Multiplying both sides by 3 gives:
\[ F' = 9F \]

### Final Answer:
The force needed to stretch wire 2 by the same amount \( \Delta x \) is \( F' = 9F \).