Calculate energy stored in springs using PE = ½kx². Includes spring arrangements, oscillation frequency, velocity analysis, and energy-displacement tables.
Elastic potential energy is the energy stored in an object when it is deformed — stretched, compressed, bent, or twisted — within its elastic limit. The most common example is a spring obeying Hooke's Law, where the energy stored is PE = ½kx², with k being the spring constant and x the displacement from equilibrium.
This type of energy storage is fundamental across physics and engineering: from car suspension systems absorbing road bumps, to mechanical watches storing energy in mainsprings, to archery bows converting elastic PE into kinetic energy of the arrow. The quadratic relationship between energy and displacement means doubling the deformation quadruples the stored energy.
This calculator computes elastic potential energy along with the restoring force, release velocity, equivalent gravitational height, and oscillation frequency for spring-mass systems. It handles parallel and series spring arrangements and provides comprehensive comparison tables. Check the example with realistic values before reporting. It keeps the spring constant, displacement, and derived motion values together so the result is easier to interpret in the same setup.
While PE = ½kx² is simple, real problems often involve unit conversions, multiple springs in parallel or series, or follow-up calculations like oscillation frequency and release velocity. This calculator automates everything and provides context through comparison tables and visual energy-displacement charts that aid intuition about quadratic energy scaling. Keep these notes focused on your operational context.
Elastic Potential Energy: PE = ½kx² Hooke's Law Force: F = kx Release Velocity: v = √(2PE/m) Oscillation: f = (1/2π)√(k/m), T = 1/f Parallel Springs: k_eff = n × k Series Springs: k_eff = k / n Where: k = spring constant (N/m) x = displacement from equilibrium (m) m = attached mass (kg)
Result: 2.5 J
A spring with k = 500 N/m compressed by 0.1 m stores PE = 0.5 × 500 × 0.01 = 2.5 J. If released with a 1 kg mass, maximum velocity is v = √(2 × 2.5 / 1) = 2.24 m/s.
When a compressed or stretched spring is released, its elastic potential energy converts to kinetic energy. For an ideal spring-mass system on a frictionless surface, the total mechanical energy remains constant: E_total = ½kx² + ½mv² = constant. At maximum displacement, all energy is potential; at the equilibrium position, all energy is kinetic and the mass reaches its maximum speed.
This energy oscillates back and forth between potential and kinetic forms indefinitely in an ideal system. In reality, damping from friction and air resistance gradually dissipates the energy as heat, causing the oscillations to decay.
Elastic energy storage is used in countless engineering applications. Car suspensions use springs to absorb road irregularities, converting kinetic energy of vertical motion into elastic PE and back. Mechanical clocks and watches store energy in coiled mainsprings. Trampolines convert the kinetic energy of a jumping person into elastic PE in the springs, then launch them back up. Archery bows store elastic PE when drawn and release it as kinetic energy of the arrow.
Real materials often exhibit nonlinear elastic behavior. Rubber, biological tissues, and many polymers show stress-strain curves that deviate significantly from the linear Hooke's Law prediction. For these materials, the energy integral ∫F(x)dx must be evaluated with the actual force-displacement relationship, which is often determined experimentally.
Hooke's Law states F = kx: the restoring force of a spring is proportional to its displacement. It holds only within the elastic limit — beyond that, permanent deformation occurs.
Because the force increases linearly with displacement (F = kx). The energy is the integral of force over distance: ∫kx dx = ½kx². So doubling displacement doubles the force but quadruples the energy.
Beyond the elastic limit, the material yields permanently and does not return to its original shape. The stored energy is partially dissipated as heat and plastic deformation.
Parallel springs share the load, so the effective spring constant increases (k_eff = nk). Series springs each experience the full load, so the system is more compliant (k_eff = k/n).
The spring constant k (in N/m) measures stiffness: how much force per unit of displacement. Stiff springs have large k; soft springs have small k.
No. Since PE = ½kx² and both k and x² are always non-negative, elastic PE is always zero or positive regardless of whether the spring is stretched or compressed.