How is potential energy (PE) stored in a compressed spring calculated?

The potential energy stored in a compressed spring is calculated using the formula ( PE = \frac{1}{2} k x^2 ), where ( k ) is the spring constant, and ( x ) is the displacement from the spring's equilibrium position. This formula is derived from Hooke's law, which states that the force exerted by a spring is proportional to its displacement: ( F = kx ).

When calculating the work done to compress or stretch the spring from its equilibrium position, we consider that the force increases linearly from 0 to ( kx ) as the spring is compressed. The potential energy accumulated in the spring is equal to the work done on it, which can be mathematically represented as the area under the force vs. displacement curve. The area of this triangle-shaped graph is given by:

[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times x \times kx = \frac{1}{2} k x^2 ]

Thus, the correct formula for the potential energy stored in a compressed spring is ( PE = \frac{