The potential energy in an elastic string of force constant k, which has an extension X is?

Analysis

The question asks for the formula for the potential energy (often called elastic potential energy or strain energy) stored in a stretched elastic string (or spring) with a force constant $k$ and an extension $x$.

Concept

According to Hooke’s Law, the force $F$ required to stretch a spring is directly proportional to the extension $x$, provided the elastic limit is not exceeded. This is expressed as:

When you stretch a spring, you do work against the restoring force. This work is stored in the spring as Elastic Potential Energy ($E_p$).

To find the energy, we look at the force-extension graph. The graph is a straight line passing through the origin. The work done (energy stored) is equal to the area under the force-extension graph.

Solution

1. Identify the shape: The graph of Force ($F$) against Extension ($x$) is a right-angled triangle.
2. Calculate the area: The area of a triangle is given by: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
3. Substitute the variables: • Base = extension ($x$) • Height = force ($F = kx$)
4. Derive the formula: $$E_p = \frac{1}{2} \times x \times (kx)$$ $$E_p = \frac{1}{2}kx^2$$

Conclusion

The potential energy stored in the elastic string is $\frac{1}{2}kx^2$. Comparing this to your options, it matches option C.

Exam Tip

Always remember that for any linear relationship (like Hooke's Law), the energy is always "half" of the product of the two variables ($F$ and $x$). If you ever forget the formula, just sketch a quick triangle graph of $F$ against $x$ and calculate the area!