Elasticity and Hooke's Law
Elasticity and Hooke’s Law
Definition: Elasticity is a material’s ability to return to its original shape after deformation. Hooke’s Law states that the force needed to stretch or compress a spring is proportional to its displacement from equilibrium.
How It Works
- Hooke’s Law, F = −kx, holds only within a material’s elastic limit, where the spring constant k measures stiffness.
- The negative sign shows the force is a restoring force, always directed opposite to the displacement, pulling or pushing the object back toward equilibrium.
- Beyond the elastic limit, deformation becomes permanent (plastic), and the material no longer returns to its original shape when the load is removed.
- Stress is force per unit cross-sectional area; strain is the fractional deformation that results. Within the elastic region, stress is proportional to strain.
- Stretching or compressing an elastic object stores elastic potential energy, released as the object springs back toward its natural shape.
- Materials are characterized by elastic moduli: Young’s modulus for stretching/compression, shear modulus for shape distortion, and bulk modulus for uniform volume change.
- Some materials behave nearly perfectly elastically over a wide range (spring steel), while others, like rubber, show hysteresis, losing some energy as heat during a load-unload cycle even while staying within the elastic range.
- A stress-strain curve typically shows a straight elastic region, a yield point where plastic deformation begins, further plastic flow, and finally a fracture point.
Elastic Moduli
| Modulus | Resists | Typical use | Units |
|---|---|---|---|
| Young’s modulus (E) | Stretching or compression along one axis | Rods, wires, beams, cables | Pa |
| Shear modulus (G) | Shape distortion at constant volume | Twisting, cutting, torsion | Pa |
| Bulk modulus (K) | Uniform volume change under pressure | Fluids, submerged solids | Pa |
| Spring constant (k) | Force per unit displacement of one specific object | Springs, specific-shaped parts | N/m |
Illustration
Under the Hood
Key equations:
F = −kx (Hooke's Law)PE_spring = ½kx²σ (stress) = F / Aε (strain) = ΔL / L₀E (Young's modulus) = σ / εSprings in series: 1/k_eff = 1/k₁ + 1/k₂
Springs in parallel: k_eff = k₁ + k₂Worked Example 1: Compressed spring Given: spring constant k = 250 N/m, compression x = 0.12 m Step 1: Restoring force magnitude: F = kx = 250 × 0.12 = 30 N Step 2: Stored elastic potential energy: PE = ½kx² = 0.5 × 250 × 0.12² Answer: PE = 0.5 × 250 × 0.0144 = 1.8 J
Worked Example 2: Elongation of a steel rod Given: steel rod, Young’s modulus E = 200 × 10⁹ Pa, length L = 2 m, cross-sectional area A = 0.0005 m², applied force F = 10,000 N Step 1: Stress: σ = F/A = 10,000/0.0005 = 2.0 × 10⁷ Pa Step 2: Strain: ε = σ/E = (2.0 × 10⁷)/(200 × 10⁹) = 1.0 × 10⁻⁴ Step 3: Elongation: ΔL = ε × L = (1.0 × 10⁻⁴)(2) = 2.0 × 10⁻⁴ m Answer: The rod stretches 0.2 mm under the 10 kN load, well within steel’s elastic range.
Worked Example 3: Springs in series vs. parallel Given: two springs, k₁ = 100 N/m, k₂ = 200 N/m Step 1: Series combination: 1/k_eff = 1/100 + 1/200 = 0.01 + 0.005 = 0.015 Step 2: k_eff(series) = 1/0.015 ≈ 66.7 N/m Step 3: Parallel combination: k_eff(parallel) = 100 + 200 = 300 N/m Answer: The same two springs act four and a half times stiffer in parallel (300 N/m) than in series (66.7 N/m).
Worked Example 4: Elastic potential energy in a drawn bow Given: compound bow, peak draw force F = 300 N at full draw length x = 0.7 m, approximated as linear Step 1: Effective spring constant: k = F/x = 300/0.7 ≈ 428.6 N/m Step 2: Stored energy: PE = ½kx² = 0.5 × 428.6 × 0.7² Answer: PE ≈ 0.5 × 428.6 × 0.49 ≈ 105 J transferred to the arrow at release, ignoring losses to limb flex and vibration.
Worked Example 5: Bungee cord peak force Given: bungee cord spring constant k = 90 N/m, jumper mass m = 75 kg, maximum stretch x = 20 m Step 1: Peak restoring force: F = kx = 90 × 20 = 1800 N Step 2: Jumper’s weight: mg = 75 × 9.8 = 735 N Step 3: Load factor: F/mg = 1800/735 ≈ 2.45 Answer: At maximum stretch the cord pulls with about 2.45 times the jumper’s body weight, the peak deceleration felt at the bottom of the fall.
Worked Example 6: Simple harmonic motion period from spring constant Given: a 2 kg mass on a spring with k = 800 N/m, displaced and released Step 1: Angular frequency: ω = √(k/m) = √(800/2) = √400 Step 2: ω = 20 rad/s Step 3: Period: T = 2π/ω = 2π/20 Answer: T ≈ 0.314 s, the time for one full oscillation, showing how a stiffer spring or lighter mass produces faster oscillation.
Why It Matters
- Vehicle suspension systems, mattress springs, and shock absorbers are all designed around precise spring constants for ride comfort and control.
- Structural engineers keep working stresses on beams and cables well below the elastic limit, building in a safety margin before permanent deformation or failure.
- Medical devices like stents and orthodontic wires rely on controlled elastic recoil to apply steady, predictable force inside the body.
- Sports equipment, from archery bows to pole-vault poles to running shoe midsoles, is tuned to store and return elastic energy efficiently.
- Seismology treats the Earth’s crust as an elastic medium; earthquakes are the sudden release of elastic strain energy built up along a fault.
Common Pitfalls
- Dropping the negative sign in F = −kx and losing track of the fact that the force always opposes displacement, not aligns with it.
- Confusing the elastic limit with the breaking point; many materials deform permanently well before they actually fracture.
- Assuming Hooke’s law applies at any deformation; it’s only valid in the linear elastic region, and many real materials (rubber, biological tissue) are strongly nonlinear even at modest strain.
- Mixing up stress (a property of the material’s internal state, force per area) with simple pressure, even though both share the pascal unit.
- Getting the series and parallel spring formulas backwards, since they’re the reciprocal of the equivalent resistor rules.
- Confusing stiffness (spring constant, how much force per unit displacement) with strength (the maximum force or stress before failure); a material can be stiff and brittle, or flexible and strong.
- Treating the spring constant k as a universal material property; it depends on the specific object’s shape and dimensions, unlike Young’s modulus, which is intrinsic to the material.
Comparison
| Property | Elastic deformation | Plastic deformation |
|---|---|---|
| Reversible? | Yes, returns to original shape | No, permanent change |
| Occurs when | Stress stays below the yield point | Stress exceeds the yield point |
| Energy | Stored, fully recoverable | Partly dissipated as heat and internal rearrangement |
| Stress-strain relation | Linear (Hooke’s law region) | Nonlinear |
| Example | Stretched rubber band, compressed spring | Bent paperclip, dented car panel |
FAQ
Does Hooke’s law apply to every elastic material? No. It’s a good approximation for small deformations in many solids, but materials like rubber and biological tissue show pronounced nonlinearity even within their elastic range.
What’s the difference between the elastic limit and yield strength? The elastic limit is the exact point where permanent deformation begins. Yield strength is a practical, measurable approximation of that point, often defined as the stress producing a small (commonly 0.2%) permanent strain.
Can a material be elastic and brittle at the same time? Yes. Glass deforms elastically almost perfectly up until it suddenly fractures, with essentially no plastic deformation in between.
Is stretching a rubber band governed by simple Hooke’s law? Not really. Rubber’s elasticity is dominated by entropy changes in its polymer chains, producing a strongly nonlinear stress-strain curve and hysteresis, unlike a metal spring.
Example
A diving board flexes downward as a diver jumps, storing elastic potential energy in its bent fiberglass structure. As the diver’s weight lifts off at the bottom of the bounce, the board springs back, releasing that stored energy to launch the diver upward, converting elastic potential energy into kinetic energy in a fraction of a second.
Related Terms
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