Advanced · 13 min
Springs and oscillations
Use Hooke's law and elastic energy to explain an ideal spring's motion.
The restoring force
Within its linear range, an ideal spring follows Hooke's law. The restoring force is F = −k × x, where x is displacement from equilibrium and k is stiffness in N/m. The minus sign means the force opposes displacement. For a slowly stretched spring, the required applied force has magnitude k × x. Use extension, not total spring length, and convert centimetres to metres.
Worked example
A spring with k = 100 N/m is stretched by 0.03 m. Restoring force magnitude = 100 × 0.03 = 3 N
Stored energy and repeating motion
The elastic potential energy stored relative to an unstretched ideal spring is E = ½ × k × x². For a mass on a horizontal ideal spring with no friction, energy alternates between elastic potential energy and kinetic energy. The motion is simple harmonic: acceleration is proportional to displacement and points towards equilibrium. At the endpoints, speed is zero; at equilibrium, speed is greatest.
Worked example
A spring with k = 80 N/m is stretched by 0.1 m. Elastic energy = ½ × 80 × 0.1² = 0.4 J
Track an ideal oscillator
- For a spring with k = 50 N/m, compare the stored energy at extensions of 0.1 m and 0.2 m.
- Sketch a horizontal mass–spring system at both endpoints and at equilibrium. Mark where speed is zero and where it is greatest.