Elastic Potential Energy

Elastic potential energy is the energy stored in a deformed elastic body — recoverable when the body returns to its natural shape.

For a spring or wire obeying Hooke's law: U = ½kx² (spring) or U = ½·F·ΔL (wire under tension F, extension ΔL).

Energy per unit volume (energy density): u = ½·σ·ε = ½·Y·ε² = σ²/(2Y).

All this is recovered when stress is released — provided you stay in the elastic region.

Beyond the elastic limit, some energy goes into permanent (plastic) deformation, heat, or fracture.

Springs in series and parallel: in series, k decreases (1/k_eq = Σ 1/k_i); in parallel, k increases (k_eq = Σ k_i).

Application: bow-and-arrow stores elastic energy in the bow, transferred to KE of arrow.

Earthquake faults store enormous elastic energy in stressed rocks — released suddenly during slip.