Which of the following graph depicts spring constant `k` versus length `l` of the spring correctly

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which graph correctly depicts the relationship between the spring constant \( k \) and the length \( l \) of the spring, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Spring Constant**: The spring constant \( k \) is defined by Hooke's Law, which states that the force \( F \) exerted by a spring is proportional to the displacement \( x \) from its equilibrium position:
\[ F = -kx \]
Here, \( k \) is the spring constant, and \( x \) is the extension or compression of the spring.

2. **Relationship Between Force and Spring Constant**: Rearranging Hooke's Law gives us:
\[ k = \frac{F}{x} \]
This indicates that the spring constant \( k \) is dependent on the force applied and the displacement of the spring.

3. **Consider the Effect of Length on Spring Constant**: When we consider a spring of length \( l \), if we increase the length of the spring while keeping the force \( F \) constant, the displacement \( x \) will increase. This implies that:
\[ k = \frac{F}{x} \]
As \( x \) increases (due to increased length), \( k \) decreases.

4. **Establish the Inverse Relationship**: From the above reasoning, we can conclude that:
\[ k \propto \frac{1}{l} \]
This means that as the length \( l \) of the spring increases, the spring constant \( k \) decreases. This is an inverse relationship.

5. **Graphical Representation**: The relationship \( k \propto \frac{1}{l} \) can be represented graphically as a hyperbola. In a graph of \( k \) versus \( l \):
- As \( l \) increases, \( k \) decreases.
- The graph will approach the axes but never touch them, which is characteristic of a hyperbolic curve.

6. **Identify the Correct Graph**: Given the options, we need to identify which graph represents this inverse relationship. The correct graph should be a hyperbola, which shows that as the length \( l \) increases, the spring constant \( k \) decreases.

### Conclusion:
Based on the analysis, the correct option is **Option D**, which depicts the relationship between the spring constant \( k \) and the length \( l \) of the spring as a hyperbola.