Spring Potential Energy Calculator
The elastic potential energy stored in a spring is given by PE = (1/2) k x², where k is the spring constant in Newtons per meter (N/m) and x is the displacement from the spring's natural (equilibrium) length in meters. This energy is potential because it can do work when the spring is released. The formula is derived from Hooke's Law (F = k x) by integrating force over displacement. Springs are used in clocks, vehicles, pogo sticks, and mechanical systems of all kinds. This calculator also shows the restoring force at the given displacement.
Spring potential energy formula
PE = (1/2) × k × x²
Where k is the spring constant (N/m) and x is the displacement from equilibrium (m). The restoring force is F = k × x (Hooke's Law). Energy is in Joules (J).
Understanding spring energy
- Spring potential energy increases with the square of displacement, so doubling the stretch quadruples the energy stored.
- The spring constant k is determined by the material, wire thickness, coil diameter, and number of coils.
- In a spring-mass system, the angular frequency of oscillation is omega = sqrt(k/m) and the period is T = 2pi sqrt(m/k).
- At maximum displacement all energy is potential; at equilibrium all energy is kinetic.
- Beyond the elastic limit, Hooke's Law no longer applies and the spring may be permanently deformed.
Spring potential energy: frequently asked questions
What is spring potential energy?
Spring (elastic) potential energy is the energy stored in a spring when it is stretched or compressed from its natural length. It is calculated as PE = (1/2) k x², where k is the spring constant (N/m) and x is the displacement from equilibrium (m). This energy is released when the spring returns to its natural length.
What is the spring constant k?
The spring constant (also called stiffness coefficient) k measures how stiff a spring is. A higher k means the spring requires more force per unit of displacement. It is measured in Newtons per meter (N/m). Typical values range from about 10 N/m for a soft spring to tens of thousands of N/m for industrial springs.
How is Hooke's Law related to spring potential energy?
Hooke's Law states that the restoring force of a spring is F = -k x. The elastic potential energy PE = (1/2) k x² is derived by integrating this force over displacement from 0 to x: PE = integral from 0 to x of k x dx = (1/2) k x².
What are the units of spring constant and potential energy?
The spring constant k is in Newtons per meter (N/m). Displacement x is in meters (m). The potential energy PE = (1/2) k x² has units of N/m x m² = N m = Joules (J).
How does spring PE convert to kinetic energy?
In an ideal spring-mass system with no damping, all potential energy converts to kinetic energy at the equilibrium position and back to potential energy at maximum displacement. Total mechanical energy is conserved: (1/2) k x² + (1/2) m v² = constant.
Official sources
- OpenStax University Physics Volume 1: Potential Energy and Conservation of Energy.
- NIST Reference on Constants, Units, and Uncertainty: NIST Physics.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.