Elastic Potential Energy

The formula

For a spring obeying Hooke's law (F = -k·x):

U(x) = ½·k·x²

where x is displacement from equilibrium. Reference point: U = 0 at x = 0 (equilibrium).

Derived by integrating force over distance: U(x) = -∫₀ˣ F·dx' = -∫₀ˣ (-k·x') dx' = ½kx².

The minus sign in F means PE INCREASES as you stretch or compress (work done on spring is stored).

Parabolic potential

U is parabolic in x — symmetric about equilibrium. Same energy stored whether stretched or compressed by the same amount.

x	U (k = 100 N/m)
0	0 J
±0.05 m	0.125 J
±0.1 m	0.5 J
±0.2 m	2 J (4× more for 2× displacement)
±0.5 m	12.5 J
±1.0 m	50 J

Energy conversion in a spring system

Stretch a spring with mass m attached. Release.

At maximum displacement (x = A): all energy is PE — U = ½kA². Velocity = 0.

At equilibrium (x = 0): all energy is KE — KE = ½mv². PE = 0.

By energy conservation: ½kA² = ½m·v_max² → v_max = A·√(k/m) = A·ω.

Mechanical energy is constant throughout the oscillation:

E = ½k·x² + ½m·v² = ½kA² (constant)

Strain energy in materials

For 3D bulk materials under stress, the strain energy per unit volume:

u = ½·σ·ε

where σ is stress (force per area) and ε is strain (relative deformation). For linear elastic with Young's modulus E (σ = E·ε):

u = ½·E·ε² = ½·σ²/E

Total strain energy in a stressed object:

U = ∫u·dV  (integrate over the object's volume)

For uniformly-stressed object of volume V — U = ½·σ·ε·V. Used in structural engineering, fracture mechanics, vibration analysis.

Real-world elastic energy storage

System	Mechanism	Typical energy stored
Coil spring (typical)	Twisting wire → torsion + bending	1-100 J
Watch mainspring	Coiled flat spring under torsion	~0.001-0.01 J
Archery bow	Bent limbs (composite or wood)	50-200 J
Trampoline	Stretched fabric + perimeter springs	~500 J at full bounce
Pole vault pole	Bent fiberglass/composite	2,000-3,000 J
Climbing rope (catching a fall)	Stretched nylon	3,000-5,000 J for 60 kg climber
Bungee cord	Stretched rubber + nylon	20,000+ J for tall jump

JavaScript — elastic energy calculations

// Spring PE
function springPE(k, x) {
  return 0.5 * k * x * x;
}

// Max velocity from compressed spring
function springLaunchVelocity(k, x_compressed, mass) {
  // ½kx² = ½mv²  →  v = x·√(k/m)
  return Math.abs(x_compressed) * Math.sqrt(k / mass);
}

// 100 N/m spring compressed 10 cm with 0.05 kg ball
console.log(springLaunchVelocity(100, 0.1, 0.05).toFixed(2)); // 4.47 m/s

// Period of oscillation
function springPeriod(m, k) {
  return 2 * Math.PI * Math.sqrt(m / k);
}

// Total mechanical energy in oscillator
function totalEnergy(k, A) {
  return 0.5 * k * A * A;  // = ½k·A²
}

// Velocity at any position x in oscillation
function velocityAt(k, A, x, m) {
  // ½kA² = ½kx² + ½mv²  →  v = √((k/m)(A² - x²))
  if (Math.abs(x) > A) return null;  // outside oscillation range
  return Math.sqrt(k * (A*A - x*x) / m);
}

console.log(velocityAt(100, 0.1, 0, 0.05).toFixed(2));    // 4.47 (max at x=0)
console.log(velocityAt(100, 0.1, 0.05, 0.05).toFixed(2)); // 3.87
console.log(velocityAt(100, 0.1, 0.1, 0.05));             // 0 (turning point)

// Strain energy in a uniformly-stressed beam
function strainEnergy(stress, strain, volume) {
  return 0.5 * stress * strain * volume;
}

// Steel beam: Young's modulus E = 200 GPa, strain 0.001 (0.1%), volume 1 m³
const E_steel = 200e9;
const strain = 0.001;
const stress = E_steel * strain;  // 200 MPa
console.log(`Steel strain energy: ${strainEnergy(stress, strain, 1).toFixed(0)} J/m³`);
// 1e5 J/m³ = 100 kJ/m³

// Compare spring vs gravitational PE
function compareGravVsSpring(mass, height, k, x_spring) {
  return {
    gravitational: mass * 9.81 * height,
    elastic: 0.5 * k * x_spring * x_spring
  };
}

console.log(compareGravVsSpring(1, 0.1, 100, 0.1)); // Equal (~0.5 J each)

Where elastic PE shows up

• Mechanical springs. Watches, vehicles, machinery, valves — all storing and releasing elastic PE.
• Sports. Pole vault, archery, gymnastics, trampolines — all use elastic PE for energy boost.
• Building safety. Climbing ropes (dynamic vs static), seismic dampers in buildings, harness systems.
• Manufacturing. Stamping, forging, springs in factories — stress storage and release.
• Biology. Muscle elasticity, tendons (storing energy during running), insect leg springs (fleas can jump 200× their body length using stored elastic PE).
• Robotics. Compliant robots use elastic elements for shock absorption and energy efficiency.
• Energy storage research. Mechanical batteries (flywheels and pressurized springs) for grid-scale energy storage.

Common mistakes

• Forgetting the ½ in ½kx². Easy to forget; comes from integration. Don't use kx² as PE.
• Using ½kx² beyond elastic limit. Real springs deviate at large stretch — formula breaks down. Energy stored may be more or less than ½kx².
• Confusing PE with KE. PE = ½kx² (in spring); KE = ½mv². Both have ½ × scalar × variable², but variable is different (displacement vs velocity).
• Wrong sign convention. PE always positive in springs (or zero at equilibrium). It's energy stored, regardless of direction of stretch/compression.
• Treating elastic PE as recoverable always. Real materials have hysteresis — some PE lost to heat each cycle. Ideal elastic = no losses; real elastic has small but nonzero damping.
• Mixing 1D spring PE with 3D strain energy. For 1D mass-spring, U = ½kx². For 3D bulk material, U = ∫½σε dV. Different formulas, related concepts.