Springs are among the most studied objects in physics — not because springs themselves are special, but because the spring restoring force is the simplest and most important type of restoring force in all of mechanics. Hooke's Law — that spring force is proportional to extension — underlies simple harmonic motion, elastic potential energy, and every oscillating system from guitar strings to shock absorbers to molecular bonds in a crystal.
The force exerted by a spring is proportional to its displacement from the natural length and directed toward equilibrium:
F = −kx
where k is the spring constant (N/m) and x is displacement (m). The negative sign indicates the restoring direction — the force always opposes the displacement.
The Spring Constant k
Large k = stiff spring (large force for small extension). Small k = soft spring. k is determined by the spring's material, wire thickness, coil diameter, and number of coils.
| Spring type | Typical k (N/m) |
|---|---|
| Slinky toy | ~1 |
| Mattress spring | ~10,000 |
| Car suspension | 15,000–30,000 |
| Stiff engineering spring | >100,000 |
Elastic and Plastic Deformation
A graph of force vs extension is linear in the elastic region (Hooke's Law holds). Beyond the elastic limit, the graph curves. Beyond the yield point, permanent (plastic) deformation occurs — the spring does not return to its natural length.
Elastic: material returns to original shape when force removed. Hooke's Law applies within the elastic limit.
Plastic: permanent shape change — material does not return to original dimensions. Occurs beyond the elastic limit. Stretching a spring past its elastic limit, bending metal, or squashing clay are examples.
Worked Examples
Example 1: Finding extension
k = 400 N/m, F = 60 N applied.
Example 2: Finding spring constant
2 kg mass extends spring 8 cm (0.08 m).
Example 3: Car suspension
k = 25,000 N/m. Force to compress 3 cm:
Elastic Potential Energy
A stretched/compressed spring stores energy equal to the area under the F-x graph (a triangle):
EPE scales with x² — double the extension, four times the energy. When released from extension A, all EPE converts to kinetic energy:
Springs in Series and Parallel
Series (same force, extensions add): 1/k_eff = 1/k₁ + 1/k₂ → k_eff less than smallest.
Parallel (same extension, forces add): k_eff = k₁ + k₂ → k_eff greater than largest.
Connection to Simple Harmonic Motion
Hooke's Law (F = −kx) is the condition that produces SHM. Newton's second law: ma = −kx → a = −(k/m)x. This is the SHM equation with ω = √(k/m) and period T = 2π√(m/k). Any system with a Hooke's Law-type restoring force (pendulum for small angles, molecular bonds, LC circuits) exhibits approximately SHM.
Frequently Asked Questions
What is Hooke's Law?
Hooke's Law states that the spring force is proportional to displacement from natural length: F = kx (magnitude). The restoring force opposes displacement: F = −kx. k is the spring constant (N/m). The law holds within the elastic limit — beyond that, permanent deformation occurs.
What is the spring constant?
The spring constant k (N/m) measures stiffness — force required per unit extension. A larger k means a stiffer spring. k is determined by the spring's material, wire gauge, coil diameter, and number of coils.
What is elastic potential energy?
EPE = ½kx² — energy stored in a deformed spring. Equal to the work done to deform it (area under F-x graph). When released, EPE converts to kinetic energy. Doubling extension quadruples stored energy.
What happens beyond the elastic limit?
Beyond the elastic limit, plastic deformation occurs — the spring does not return to its original length. The F-x graph curves, Hooke's Law no longer applies, and permanent elongation results. Further stretching leads to fracture.
How is Hooke's Law related to SHM?
F = −kx is the condition for SHM. Applying Newton's second law gives a = −(k/m)x — the SHM equation. Period T = 2π√(m/k). Any system with a proportional restoring force exhibits SHM.
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Written by
Dr. James CarterPhysicist and educator with 15+ years teaching classical mechanics and thermodynamics at the university level. Former MIT OpenCourseWare contributor.
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