A spring is held between two supports and it is in its...

A spring is held between two supports and it is in its normal length. If the center is displaced by $\delta \mathrm{x}$ normal to its length, the energy stored is the spring in proportional to $(\delta \mathrm{x})^{\mathrm{n}}$ where $\mathrm{n}$ is
(a) 1
(b) 2
(c) 3
(d) 4

Key Concepts

Elastic Potential Energy

This concept refers to the energy stored in a system when it is deformed, such as stretching or bending. In the context of elastic materials and springs, the energy is a function of the displacement from equilibrium and is determined by how the force applied relates to that displacement.

Hooke's Law

Hooke's Law states that, for small deformations, the force exerted by an elastic object is directly proportional to the displacement from its equilibrium position. This linear relationship implies that the work done (or energy stored) is proportional to the square of the displacement, reinforcing a quadratic dependence.

Small Displacement Approximation

When displacements are small, the changes in the geometry of the system (including any bending or stretching) can be approximated using a Taylor expansion. The leading term in such an expansion is quadratic, meaning that the energy stored due to the displacement scales as the square of the displacement magnitude.

Geometric Considerations in Elastic Deformation

Even when the displacement is applied in a direction normal to an element's length, the geometric changes—such as those involving bending or slight stretching—produce an energy change that, for small deflections, is dominated by quadratic terms. This results from the inherent symmetry and the interplay between the geometry and elastic response of the material.

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