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

Options

A.
B.
C.
D.

Answer

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

### Step 1: Understand the relationship between force, spring constant, and displacement
The spring force can be described by Hooke's Law, which states:
\[ F = -kx \]
where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position.

### Step 2: Rearranging the equation
When we consider the magnitude of the force, we can write:
\[ F = kx \]
From this, we can express the spring constant \( k \) in terms of the force \( F \) and the displacement \( x \):
\[ k = \frac{F}{x} \]

### Step 3: Analyzing the effect of length on the spring constant
If we keep the force \( F \) constant and increase the displacement \( x \) (which corresponds to the length \( l \) of the spring), the spring constant \( k \) will change. Specifically, since \( k \) is inversely proportional to \( x \):
\[ k \propto \frac{1}{x} \]
This means that as the length \( l \) (or displacement \( x \)) increases, the value of \( k \) decreases.

### Step 4: Graphical representation
The relationship \( k \propto \frac{1}{l} \) indicates that if we were to plot \( k \) against \( l \), we would expect a hyperbolic curve that decreases as \( l \) increases. Therefore, the graph depicting \( k \) versus \( l \) should show a downward trend, indicating that as the length of the spring increases, the spring constant decreases.

### Conclusion
Based on the analysis, the correct graph that depicts the relationship between the spring constant \( k \) and the length \( l \) of the spring is a graph that shows \( k \) decreasing non-linearly as \( l \) increases.