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Elastic Potential Energy Calculator
Calculate PE = 1/2kx^2 with step-by-step solutions, spring constant lookup, and series/parallel spring systems.
Formula:PE = ½kx²
Results
Elastic PE10.0000 J
Energy (J)10.0000 J
Spring Constant500.00 N/m
Displacement0.2000 m
Force at x100.00 N
PE = 1/2kx^2
Elastic potential energy
Elastic PE
10.00 J
Energy
10.0000 J
2.3901 cal
Spring Constant
500.00 N/m
Stiffness
Force (F = kx)
100.00 N
Restoring force
Elastic PE
1
Formula
PE = ½kx²
2
Calculate
PE = 10 J
Energy Conversions
Joules
10.00 J
Kilojoules
0.01000 kJ
Megajoules
0.00001000 MJ
Calories
2.390 cal
Kilocalories (food calories)
0.002390 kcal
British Thermal Units
0.009478 BTU
Hooke's Law: F = -kx
The restoring force of a spring is proportional to displacement. The negative sign indicates the force opposes displacement (pulls back when stretched, pushes back when compressed).
Current displacement (x)0.2 m
Restoring force (F = kx)100.00 N
Energy vs Force Relationship
While force increases linearly with displacement (F = kx), energy increases with the square (PE = 1/2kx^2). This means:
- Double the displacement = Double the force
- Double the displacement = Quadruple the energy
Quick Answer
Elastic potential energy stored in a spring is PE = (1/2)kx^2, where k is the spring constant (N/m) and x is displacement from equilibrium. A spring with k = 100 N/m compressed 0.1 m stores PE = 0.5 x 100 x 0.01 = 0.5 Joules. The energy doubles if you double k, but quadruples if you double x.
Key Facts
- Elastic PE formula: PE = (1/2)kx^2
- Spring constant k is measured in N/m (Newtons per meter)
- Hooke's Law: F = -kx (force proportional to displacement)
- Energy is always positive (x is squared)
- Doubling displacement quadruples stored energy
- Springs in series: 1/k_total = 1/k1 + 1/k2 (softer)
- Springs in parallel: k_total = k1 + k2 (stiffer)
Frequently Asked Questions
Elastic potential energy is energy stored in an elastic object when it is deformed (stretched or compressed). For springs, PE = (1/2)kx^2, where k is the spring constant and x is the displacement from equilibrium. When released, this stored energy converts to kinetic energy.
Hooke's Law states that the force needed to stretch or compress a spring is proportional to the displacement: F = -kx. The negative sign indicates the restoring force opposes displacement. This law holds for "ideal" springs within their elastic limit. Real springs deviate from Hooke's Law at large deformations.
Springs in series (end-to-end): The effective spring constant is 1/k_eff = 1/k1 + 1/k2, making the combination softer (easier to stretch). Springs in parallel (side-by-side): k_eff = k1 + k2, making the combination stiffer. Two identical springs in series are half as stiff; in parallel, twice as stiff.
The work done to stretch a spring is W = integral of F*dx = integral of kx*dx = (1/2)kx^2. Since force increases linearly with displacement (Hooke's Law), more work is needed for each additional unit of stretch. The quadratic relationship means small compressions store little energy, but large ones store a lot.
The spring constant (k) measures a spring's stiffness in N/m (Newtons per meter). A higher k means a stiffer spring requiring more force to stretch. Typical values: rubber band ~1-10 N/m, car suspension ~20,000-40,000 N/m, trampoline ~4,000-8,000 N/m.
Elastic PE is used in: mechanical watches (mainspring), archery (bow stores energy), vehicle suspensions (absorb bumps), trampolines (bounce), pogo sticks, mechanical keyboards, and countless springs in machinery. The stored energy can be released quickly or slowly depending on the mechanism.
Beyond the elastic limit, the spring undergoes permanent deformation and Hooke's Law no longer applies. The spring won't return to its original length when released. Eventually, the spring may break. Quality springs are designed to operate well within their elastic limits.
Temperature changes can affect spring constant slightly. Most metal springs become slightly softer (lower k) when heated due to thermal expansion and changes in material properties. For precise applications, temperature compensation may be needed.
Joseph OrdunaFounder & Software Engineer
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).
Results
Elastic PE10.0000 J
Energy (J)10.0000 J
Spring Constant500.00 N/m
Displacement0.2000 m
Force at x100.00 N