The length of a rubber cord is `l_(1)` m when the tension is 4 N and `l_(2)` m when the tension is 6 N.The length when the tension is 9 N, is

Options

A. `(2.5l_(2) - 1.5l_(1))m`
B. `(6l_(2)- 1.5l_(1))m`
C. `(3l_(1) - l_(2))m`
D. `(3.5l_(2) =- 2.5l_(1))m`

Explanation

To solve the problem, we need to find the length of a rubber cord when the tension is 9 N, given the lengths at tensions of 4 N and 6 N. We can use the concept of Young's modulus and the relationship between tension and extension.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( L \) be the original length of the rubber cord.
- Let \( L_1 \) be the length when the tension is 4 N.
- Let \( L_2 \) be the length when the tension is 6 N.
- Let \( L_3 \) be the length when the tension is 9 N.

2. **Calculate the Change in Length**:
- The change in length when the tension is 4 N is given by:
\[ \Delta L_1 = L_1 - L \]
- The change in length when the tension is 6 N is given by:
\[ \Delta L_2 = L_2 - L \]
- The change in length when the tension is 9 N is given by:
\[ \Delta L_3 = L_3 - L \]

3. **Use Young's Modulus**:
- Young's modulus \( Y \) is defined as:
\[ Y = \frac{F}{A \cdot \text{strain}} = \frac{F}{A \cdot \frac{\Delta L}{L}} \]
- Since the material is the same for all cases, we can set up the equations:
\[ \frac{4}{L_1 - L} = \frac{6}{L_2 - L} = \frac{9}{L_3 - L} \]

4. **Set Up the Equations**:
- From the first two cases:
\[ 4(L_2 - L) = 6(L_1 - L) \]
Rearranging gives:
\[ 4L_2 - 4L = 6L_1 - 6L \implies 4L_2 = 6L_1 - 2L \tag{1} \]
- From the second and third cases:
\[ 6(L_3 - L) = 9(L_2 - L) \]
Rearranging gives:
\[ 6L_3 - 6L = 9L_2 - 9L \implies 6L_3 = 9L_2 - 3L \tag{2} \]

5. **Solve the Equations**:
- From equation (1):
\[ 4L_2 = 6L_1 - 2L \implies 2L = 6L_1 - 4L_2 \tag{3} \]
- From equation (2):
\[ 6L_3 = 9L_2 - 3L \implies 3L = 9L_2 - 6L_3 \tag{4} \]

6. **Equate and Solve for \( L_3 \)**:
- Substitute \( L \) from equation (3) into equation (4):
\[ 3(6L_1 - 4L_2)/2 = 9L_2 - 6L_3 \]
- Simplifying gives:
\[ 9L_1 - 6L_2 = 9L_2 - 6L_3 \]
- Rearranging gives:
\[ 6L_3 = 9L_2 - 9L_1 \implies L_3 = \frac{3}{2}L_2 - \frac{3}{2}L_1 \]

7. **Final Calculation**:
- Plug in the values of \( L_1 \) and \( L_2 \) to find \( L_3 \).