How does compression of a spring relate to the potential energy stored in it?

The relationship between the compression of a spring and the potential energy stored within it is described by Hooke's Law and the concept of elastic potential energy. When a spring is compressed (or stretched), it stores potential energy that is proportional to the square of the displacement from its equilibrium position.

The potential energy (U) stored in a spring can be formulated as U = (1/2)kx², where k is the spring constant and x is the amount of compression or extension from the spring's natural length. This equation reveals that as the amount of compression increases, the potential energy stored in the spring increases in a quadratic manner, meaning that if you double the compression, the potential energy increases by a factor of four.

This quadratic relationship is what makes the correct answer valid, as it accurately describes how potential energy behaves in relation to compression. The other options would suggest linearity, decrease, or constancy of potential energy relative to compression, which do not align with the principles of classical mechanics as applied to springs.