The length of a spring is `alpha` when a force of `4N` is applied on it and the length is `beta` when `5N` is applied. Then the length of spring when `9 N` force is applied is-

Options

A. 4b – 3a
B. 5b – a
C. 5b – 4a
D. 5b – 2a

Explanation

To solve the problem step by step, we will use Hooke's Law, which states that the force exerted by a spring is proportional to its elongation (deformation) from its original length.

### Step-by-Step Solution:

1. **Understand the Problem**: We have two scenarios with known forces and corresponding lengths of a spring. We need to find the length of the spring when a force of 9 N is applied.

2. **Define Variables**:
- Let \( L_0 \) be the original (unstretched) length of the spring.
- When a force of 4 N is applied, the length of the spring is \( \alpha \).
- When a force of 5 N is applied, the length of the spring is \( \beta \).
- When a force of 9 N is applied, we need to find the length \( \gamma \).

3. **Apply Hooke's Law**:
- For the first case (4 N):
\[ 4 = k(\alpha - L_0) \quad \text{(Equation 1)} \]
- For the second case (5 N):
\[ 5 = k(\beta - L_0) \quad \text{(Equation 2)} \]
- For the third case (9 N):
\[ 9 = k(\gamma - L_0) \quad \text{(Equation 3)} \]

4. **Rearranging the Equations**:
- From Equation 1:
\[ \alpha - L_0 = \frac{4}{k} \quad \text{(Equation 4)} \]
- From Equation 2:
\[ \beta - L_0 = \frac{5}{k} \quad \text{(Equation 5)} \]
- From Equation 3:
\[ \gamma - L_0 = \frac{9}{k} \quad \text{(Equation 6)} \]

5. **Eliminate \( L_0 \)**:
- Subtract Equation 5 from Equation 6:
\[ (\gamma - L_0) - (\beta - L_0) = \frac{9}{k} - \frac{5}{k} \]
\[ \gamma - \beta = \frac{4}{k} \quad \text{(Equation 7)} \]
- Subtract Equation 4 from Equation 6:
\[ (\gamma - L_0) - (\alpha - L_0) = \frac{9}{k} - \frac{4}{k} \]
\[ \gamma - \alpha = \frac{5}{k} \quad \text{(Equation 8)} \]

6. **Set Up a Ratio**:
- From Equations 7 and 8:
\[ \frac{\gamma - \beta}{\gamma - \alpha} = \frac{4}{5} \]

7. **Cross Multiply**:
\[ 5(\gamma - \beta) = 4(\gamma - \alpha) \]
\[ 5\gamma - 5\beta = 4\gamma - 4\alpha \]

8. **Rearranging Terms**:
\[ 5\gamma - 4\gamma = 5\beta - 4\alpha \]
\[ \gamma = 5\beta - 4\alpha \]

9. **Final Result**:
The length of the spring when a force of 9 N is applied is given by:
\[ \gamma = 5\beta - 4\alpha \]