How is spring potential energy calculated?

Spring potential energy is calculated using the formula ( PE = \frac{1}{2} k x^2 ), where ( PE ) represents the potential energy stored in the spring, ( k ) is the spring constant, and ( x ) is the displacement of the spring from its equilibrium position. This formula derives from Hooke's Law, which states that the force exerted by a spring is proportional to its extension or compression, defined mathematically as ( F = -kx ).

The factor of ( \frac{1}{2} ) arises because as a spring is compressed or stretched, the force increases linearly from zero at the rest position up to ( kx ) at maximum displacement. To find the work done in stretching or compressing the spring, one must integrate this force over the distance ( x ), leading to the formula for potential energy. The ( \frac{1}{2} k x^2 ) thus represents the total work done on the spring to displace it from its equilibrium position.

In contrast, the other formulas represent different types of potential energy. The formula ( PE = mgh ) is used to calculate gravitational potential energy, where ( m ) is mass,