Elastic potential energy is the energy stored in elastic materials when they are stretched or compressed from their equilibrium position. This energy is recoverable - when released, the material returns to its original shape, converting stored energy back to kinetic energy or work.

Springs, rubber bands, bow strings, and trampolines all store elastic potential energy. The energy depends on the spring constant (material stiffness) and displacement (how far stretched or compressed). Understanding elastic energy is fundamental in mechanical engineering, physics, and systems involving oscillation and vibration.

The spring constant (k) measures material stiffness and resistance to deformation:

• Higher k: Stiffer spring requires more force for same displacement; stores more energy.
• Lower k: Softer spring easier to compress; stores less energy for same displacement.
• Units: Newtons per meter (N/m) or pounds per inch (lb/in).
• Material dependency: Steel springs have higher k than rubber; varies with wire diameter and coil count.
• Energy scaling: Energy increases linearly with k but quadratically with displacement (x²).

1. Select calculation mode: Choose to calculate energy, spring constant, or displacement.
2. Enter known values: Input spring constant (N/m) and displacement (m) for energy calculation.
3. Click Calculate: Compute elastic potential energy and spring force.
4. Review results: View energy (J), spring constant, displacement, and force.
5. Verify with Hooke Law: Check force calculation F = kx for consistency.

Elastic potential energy calculations apply to:

• Suspension Systems: Car and bike suspensions absorb shock energy in springs and dampers.
• Archery: Bow limbs store energy when drawn, releasing it to propel arrows.
• Trampolines: Springs store jumper energy, returning it for bounce height.
• Mechanical Watches: Mainsprings store winding energy to power timekeeping mechanisms.
• Pogo Sticks: Compression springs convert gravitational potential to kinetic energy.
• Crash Protection: Bumpers and crumple zones use elastic deformation to absorb impact energy.
• Spring-Loaded Tools: Clamps, valves, and latches use spring energy for actuation.
• Vibration Isolation: Spring mounts isolate machinery vibrations.

Elastic systems demonstrate energy conservation principles:

• Work-Energy Theorem: Work done compressing spring equals stored elastic energy.
• Energy Transfer: Kinetic energy converts to elastic potential, then back to kinetic when released.
• Simple Harmonic Motion: Springs oscillate with continuous energy exchange between kinetic and potential.
• Period of Oscillation: T = 2π × sqrt(m/k), where m is mass, k is spring constant.
• Energy Loss: Real springs lose energy to heat (damping), reducing amplitude over time.
• Maximum Compression: Limited by material yield strength; exceeding causes permanent deformation.

Scenario: A spring with constant k = 200 N/m is compressed 0.3 meters. How much energy is stored?

1. Given: k = 200 N/m, x = 0.3 m
2. Calculate energy: E = (1/2) × k × x²
3. E = 0.5 × 200 × (0.3)² = 0.5 × 200 × 0.09 = 9 joules
4. Calculate force at maximum compression: F = k × x = 200 × 0.3 = 60 N
5. The spring stores 9 J of energy and exerts 60 N restoring force when compressed 0.3 m.
6. If released, this 9 J converts to kinetic energy, launching a mass or doing work.

Different spring types have varying energy storage characteristics:

• Compression Springs: Store energy when compressed (car suspensions, pogo sticks).
• Extension Springs: Store energy when stretched (garage doors, trampolines).
• Torsion Springs: Store energy when twisted (clothespins, mousetraps).
• Leaf Springs: Flat springs used in vehicle suspensions (trucks, trailers).
• Wave Springs: Compact springs with high load capacity (valves, clutches).
• Gas Springs: Use compressed gas instead of metal (office chairs, hatchbacks).
• Constant Force Springs: Provide near-constant force over displacement range.