Introduction
Elastic potential energy is stored in deformed elastic materials like springs, rubber bands, and compressed gases. For ideal springs following Hooke's law, the potential energy is U = 1/2kx2. This simple model is fundamental to understanding oscillations, wave motion, and energy storage in mechanical systems.
Hooke's Law
Hooke's Law states that the restoring force exerted by an ideal spring is proportional to displacement from equilibrium and opposite in direction: F = -kx, where k is the spring constant (stiffness) and x is displacement from equilibrium. Valid for small deformations within the elastic limit. The spring constant k has units N/m. Stiffer springs have larger k values.
Elastic Potential Energy
The potential energy stored in a spring displaced by x from equilibrium is U = 1/2kx2. This is derived from work: W = ∫ F dx = ∫ (-kx)dx from 0 to x = -1/2kx2. The work done BY the spring is negative, so the energy stored is positive 1/2kx2. U is always positive (quadratic) and symmetric about x = 0 (same energy for compression and extension).
\nIntroduction
Elastic potential energy is stored in deformed elastic materials like springs, rubber bands, and compressed gases. For ideal springs following Hooke's law, the potential energy is U = 1/2kx2. This simple model is fundamental to understanding oscillations, wave motion, and energy storage in mechanical systems.
Hooke's Law
Hooke's Law states that the restoring force exerted by an ideal spring is proportional to displacement from equilibrium and opposite in direction: F = -kx, where k is the spring constant (stiffness) and x is displacement from equilibrium. Valid for small deformations within the elastic limit. The spring constant k has units N/m. Stiffer springs have larger k values.
Elastic Potential Energy
The potential energy stored in a spring displaced by x from equilibrium is U = 1/2kx2. This is derived from work: W = ∫ F dx = ∫ (-kx)dx from 0 to x = -1/2kx2. The work done BY the spring is negative, so the energy stored is positive 1/2kx2. U is always positive (quadratic) and symmetric about x = 0 (same energy for compression and extension).
\nWork by Spring Force
Work done by spring moving from x1 to x2: W = 1/2kx₲ - 1/2kx22. If moving toward equilibrium (|x2| < |x1|), spring does positive work. If moving away from equilibrium, spring does negative work (must apply external force). The work depends only on endpoints, confirming the spring force is conservative. This work-energy relation is essential for spring-mass oscillation analysis.
Spring Combinations
Springs in series: 1/k_eq = 1/k1 + 1/k2 + ... (more stretch per force, softer overall). Springs in parallel: k_eq = k1 + k2 + ... (less stretch per force, stiffer overall). Effective spring constant determines oscillation frequency ω = √(k_eq/m). These combinations appear in suspension systems, mattress springs, and mechanical linkages.
\nApplications
Applications: Vehicle suspensions (absorb shocks, store/return energy); clock mechanisms (mainspring stores energy); trampolines and diving boards (elastic energy conversion); molecular bonds (modeled as springs); seismometers (springs couple mass to frame); energy storage devices; vibration isolation systems. Understanding spring energy is prerequisite for simple harmonic motion analysis.
Solved Example: Spring-Loaded Launcher
A spring with k = 2000 N/m is compressed 0.15 m and used to launch a 0.5 kg block horizontally on frictionless surface. Find: (a) Speed of block when spring returns to natural length, (b) Maximum height if launched vertically upward. Solution: (a) Spring potential energy U = 1/2kx2 = 1/2 × 2000 × (0.15)2 = 22.5 J. By energy conservation, this converts to kinetic energy: 1/2mv2 = 22.5 → v = √(2×22.5/0.5) = √90 = 9.49 m/s. (b) For vertical launch, same initial KE = 22.5 J converts to gravitational PE at max height: mgh = 22.5 → h = 22.5/(0.5×9.8) = 4.59 m. If launched at angle, same speed at launch but different trajectory. Real systems have some energy loss to friction and air resistance.
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