What is the energy stored in a linear spring with stiffness k and displacement x?

• A. k x
• B. (1/2) x^2 / k
• C. 2 k x
• D. (1/2) k x^2

Answer: (D)

Energy stored in a linear spring is the elastic potential energy that arises from deformation. For a spring with stiffness k and displacement x from its natural length, this energy is (1/2) k x^2. The reasoning comes from the work required to stretch or compress the spring from 0 to x against the restoring force F = -k x (Hooke’s law). The work done is the integral W = ∫_0^x k ξ dξ = (1/2) k x^2, and in an ideal spring no energy is lost, so that work becomes the stored energy. The units confirm this: k in N/m and x in m give energy in N·m = J, with the 1/2 factor just a scalar. The other forms don’t represent energy—they have the dimensions of force or other quantities, not energy.