Elastic Potential Energy
Elastic potential energy is the energy stored in a deformed elastic object — such as a stretched spring, a compressed rubber band, or a bent bow — that can be fully recovered when the deforming force is removed. It arises from the intermolecular forces within the elastic material that resist deformation and is equal to the work done in stretching or compressing the object. Elastic PE is pivotal in springs, archery, vehicle suspensions, and the molecular-scale understanding of elasticity.
Key Formula
U_e = (1/2) × k × x²
LaTeX: U_e = \frac{1}{2}kx^2
| Symbol | Meaning | Unit |
|---|---|---|
| U_e | Elastic potential energy | J (Joule) |
| k | Spring constant (stiffness) | N/m |
| x | Displacement from the natural (equilibrium) length | m |
Worked Example
Problem
A spring with k = 400 N/m is compressed by 0.12 m from its natural length. How much elastic potential energy is stored in the spring?
Solution
Step 1 — Identify values: k = 400 N/m, x = 0.12 m. Step 2 — Apply the formula: U_e = ½ × k × x² = 0.5 × 400 × (0.12)² = 200 × 0.0144 = 2.88 J.
Answer
The elastic potential energy stored in the spring is 2.88 J.
Elastic potential energy vs. displacement for a spring with k = 400 N/m
| Displacement x (m) | Spring Force F (N) | Elastic PE U_e (J) | Equivalent Height for 1 kg (m) |
|---|---|---|---|
| 0.00 | 0 | 0.000 | 0.000 |
| 0.04 | 16 | 0.320 | 0.033 |
| 0.08 | 32 | 1.280 | 0.130 |
| 0.12 | 48 | 2.880 | 0.294 |
| 0.16 | 64 | 5.120 | 0.522 |
| 0.20 | 80 | 8.000 | 0.816 |
Interactive Tools
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Related Terms
Hooke's Law
Hooke's Law states that, within the elastic limit of a material, the deformation (extension or compression) of a spring or elastic solid is directly proportional to the applied force. It was formulated by English scientist Robert Hooke in 1676 and is expressed as F = kx, where k is the spring constant characterising the stiffness of the material. Hooke's Law underpins the design of force gauges, seismometers, automotive suspensions, and the theory of elasticity in materials science.
Spring Force
Spring force is the restoring force exerted by a compressed or stretched elastic spring, always directed back toward the spring's natural (equilibrium) length. It is a specific application of Hooke's Law and is proportional to the displacement from equilibrium. Spring forces are fundamental to mechanical oscillators, shock absorbers, force sensors, and the microscopic bonding forces between atoms.
Conservation of Energy
The law of conservation of energy states that the total energy of an isolated system remains constant over time: energy can neither be created nor destroyed, only transformed from one form to another. In mechanical systems, this means the sum of kinetic energy and potential energy remains constant in the absence of non-conservative forces such as friction. This principle, one of the most fundamental in all of science, is derived mathematically from Noether's theorem as a consequence of the time-translation symmetry of physical laws.
"Elastic" from Greek "elastikos" (propulsive, able to propel), from "elaunein" (to drive). The derivation of U = ½kx² from Hooke's Law follows from integrating F = kx over displacement, a mathematical step formalised in the 18th century using the calculus developed by Newton and Leibniz. The term "potential energy" was coined by William Rankine in 1853.